Open Math Problems Claimed to Be Solved with AI

Original 2000 problem paper

The paper also corrects one false statement and two incomplete proofs in the 2000 source. The Lean artifact is substantial but should not be read as an end-to-end formalization of every analytic result.

Twenty-six-year-old conjecture proved; partly formalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude (version undisclosed)
Verification: Author-checked proof, independent computations, partial Lean coverage
Claim audit: Issue documented

1 Sep 2026Arithmetic incidence matrices

Depth-three Smith profile for affine Hjelmslev incidence

For the line-by-point incidence matrix of the affine plane over $\mathbb Z/p^3\mathbb Z$ , the paper gives all-prime rank formulas and the complete $p$ -primary Smith profile, including exact exceptional rows for $p=2$ and $p=3$ .

Details and sources

AI contribution

Oleksiy Babanskyy reports directing GPT-5.6 Sol through the proof, manuscript, and exact companion construction, then checking and packaging the result.

Problem origin

Łaba and Trainor asked for rank and Smith-type information for affine incidence matrices over prime-power rings; this result treats the specialization $k=3,n=2$ .

Verification

Author-checked manuscript plus exact symbolic companion

Claim audit

The exact companion validates formulas and bounded exceptional cases but is not a proof of the uniform all-prime theorem. Independent specialist review is not yet public.

Publication

Pinned public manuscript, sources, and reproducible exact companion

Preprint / manuscript

Pinned manuscript and exact companion

This resolves the depth-three affine-plane specialization only. General depth, higher-dimensional cases, and the broader incidence-rank program remain open.

Original affine-incidence problem source Community scope and reproducibility audit

Depth-three plane case computed

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked manuscript plus exact symbolic companion
Claim audit: Issue documented

3 Sep 2026Algebra

Köthe conjecture

A unital algebra over $\overline{\mathbb F_2}$ contains a nil two-sided ideal $I$ for which $M_2(I)$ contains a nonnilpotent matrix, refuting Krempa's matrix formulation and hence Köthe's conjecture.

Details and sources

AI contribution

The autonomous model constructed the counterexample and wrote its Lean proof without human steering; Tom Adamczewski published the run and Claude later generated documentation.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Lean/Comparator checked against an independently written benchmark statement

Lean proof repository

Claim audit

No ring theorist has reviewed the construction. The elementary implication from Köthe's original formulation to the matrix form is explained but not itself formalized.

Publication

Public Lean disproof repository

Preprint / manuscript

No public preprint or manuscript located.

The repository audit reports 3,331 Lean lines, no added axioms or unsafe escape hatches, a byte-identical benchmark statement, and only standard foundational axioms. The result remains provisional outside the kernel-checked formulation.

Formal Conjectures source statement Independent formal-artifact audit

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Lean/Comparator checked against an independently written benchmark statement
Claim audit: Issue documented

3 Sep 2026 revisionAlgebra

Topological invariance of unibranch motivic superpolynomials

The substantially revised preprint claims a general proof that motivic superpolynomials of unibranch plane-curve singularities are topological invariants, extending earlier torus-knot, selected two-cable, and $a=0,t=1$ cases.

Details and sources

AI contribution

The author credits ChatGPT as an important lead and joint contributor to the appendix proving topological invariance, with substantial mathematical and computational assistance. The author checked the arguments and assumes responsibility.

Problem origin

The revised manuscript calls topological invariance a fundamental prior problem and one of the main conjectures on compactified Jacobians and motivic superpolynomials.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The claim is for unibranch plane-curve singularities; broader instanton-slice, arbitrary-Young-diagram, mixed-characteristic, and Riemann-hypothesis-style statements elsewhere in the 106-page paper retain additional restrictions or remain conjectural. No public proof code or independent specialist review was located.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI ChatGPT/Codex; version unspecified
Verification: Author-checked proof; independent review pending

25 Aug 2026Algebra

Fröberg's conjecture for equal quintics and septics in four variables

The predicted Hilbert series holds for every number of general equal-degree generators in four variables over characteristic-zero fields when the degree is five or seven.

Details and sources

AI contribution

The revised disclosure credits an extensive workflow for ideas, proof strategies, exact certificates, intermediate checks, and manuscript development.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Exact rank certificates described in the preprint; independent replay pending

Publication

Public research preprint

Preprint / manuscript

Certificate source package

The August 27 revision reduces the new cases to explicitly recorded nonzero modular minors and Zariski openness. The unrestricted Fröberg conjecture remains open; characteristic zero is essential to the stated scope.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol, Claude Fable 5, and Grok 4.6
Verification: Exact rank certificates described in the preprint; independent replay pending

25 Aug 2026Algebra

Kalck's global-dimension bound for weighted projective lines

A weighted projective line of type $(2,3,3)$ is derived equivalent to a 13-dimensional radical-square-zero algebra of global dimension four, exceeding the conjectured maximum weight three.

Details and sources

AI contribution

The model assisted discovery of the counterexample; the authors independently verified the arguments and references.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The manuscript identifies the target as Conjecture 13.43 in Schröer's Atlas. The example combines established derived-equivalence constructions; no claim that those underlying ingredients are new is made.

Claimed resolution in a public preprint

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-prepared proof; independent review pending

14 Aug 2026Commutative algebra and ideal theory

Purely-prime ideals need not be purely-maximal

Problem statement

Must every purely-prime ideal of a commutative ring be purely-maximal?

An explicit commutative ring contains a purely-prime ideal that is not purely-maximal, disproving the conjecture that the two notions always coincide.

Details and sources

AI contribution

The author states that Brian Conrad posed the question to ChatGPT Pro and relayed the resulting construction. The public manuscript presents and checks the counterexample; the exact underlying model version is not disclosed.

Problem origin

The counterexample targets Conjecture 5.8 from the author's 2021 paper on purely-prime ideals, after the question had remained unresolved for several years.

Verification

Author-presented construction; independent review pending

Publication

Public counterexample preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the explicit disproof of a published conjecture after several years without a known example.

The record attributes only the explicit counterexample reported in the manuscript. The model identity is retained at the level disclosed by the author and is not inferred more precisely.

Conjecture 5.8 disproved by an explicit commutative-ring example

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT Pro (OpenAI; exact model undisclosed)
Verification: Author-presented construction; independent review pending
Open for: Conjecture 5.8 from the 2021 purely-prime-ideal paper
Research activity*: 3/5

8 Aug 2026Amenable algebras and modules

Base-field independence of algebraic amenability

Problem statement

Can amenability of an associative algebra or module depend on the choice of its ground field?

If $K\subseteq L$ are fields, $A$ is an $L$ -algebra, and $M$ is an $A$ -module, then amenability of $M$ is the same whether $A$ and $M$ are viewed over $K$ or over $L$ . In particular, amenability and full amenability of $A$ do not depend on the ground field.

Details and sources

AI contribution

Greenfeld states that the work was heavily based on back-and-forth discussions with ChatGPT 5.6 Sol and that the model supplied the core ideas for the going-up argument. The human author wrote the paper, checked the argument, and accepts responsibility.

Problem origin

Yves Cornulier raised the base-field-dependence question in a 2018 MathOverflow discussion, eight years before the AI-assisted proof.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint with a detailed methodology statement

Preprint / manuscript

Original MathOverflow discussion

Activity evidence

A documented editorial estimate based on an eight-year-old specialist question and the theorem's extension from algebras to arbitrary modules.

The theorem applies to arbitrary modules as well as algebras. The paper records that the model initially predicted the opposite answer before the collaborative argument converged.

MathOverflow question answered affirmatively

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Author-checked proof preprint; external review pending
Open for: Raised on MathOverflow in May 2018
Research activity*: 3/5

7 Aug 2026Polynomial differential operators

Generalized Vanishing Conjecture in five variables

Problem statement

Does pure-power vanishing under a constant-coefficient differential operator force vanishing after multiplication by every polynomial

Q

An explicit five-variable triple $(\Lambda,P,Q)$ has $\Lambda^m(P^m)=0$ for all $m\ge1$ but $\Lambda^m(QP^m)\ne0$ for every $m\ge1$ , disproving the conjecture.

Details and sources

AI contribution

The model served as an interactive research assistant: it tested the homogenized construction, generated checking code, and located literature. The author attributes the final construction and proof responsibility to the human work.

Problem origin

The Generalized Vanishing Conjecture is a human-posed algebraic statement related to the Jacobian Conjecture.

Verification

Symbolic identities and author proof; external review pending

Publication

Public short counterexample preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture's relationship to the Jacobian Conjecture and prior vanishing-conjecture literature.

The AI contribution was supporting rather than autonomous. The explicit identities are independently reproducible, but no proof-assistant development or peer review was located.

Conjecture disproved by explicit example

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Symbolic identities and author proof; external review pending
Research activity*: 4/5

6 Aug 2026Cayley graphs and chromatic number

Babai's minimal Cayley graph problem

Problem statement

Are chromatic numbers of finite minimal Cayley graphs bounded?

Finite minimal Cayley graphs can have arbitrarily large chromatic number, refuting the proposed boundedness phenomenon.

Details and sources

AI contribution

The authors state that the initial proof was found by GPT-5.6 Sol after it was given the relevant recent construction. They checked, simplified, and integrated the argument.

Problem origin

László Babai posed the minimal Cayley graph question in the human algebraic-combinatorics literature.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Lean project README Terminal Lean theorem Aristotle replay receipt

Activity evidence

A documented editorial estimate based on Babai's problem and the active interface of group theory, Ramsey theory, and graph colouring.

The theorem settles the unbounded-chromatic-number question for finite minimal Cayley graphs; it does not classify which chromatic numbers occur for each group.

Problem answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof preprint; external review pending
Research activity*: 4/5

29 Jul 2026Algebraic geometry and commutative algebra

Separable Jacobian conjecture in dimension two and characteristic two

Problem statement

In characteristic

p

, must an étale polynomial endomorphism whose finite generic degree is prime to

p

be an automorphism, at least in dimension two?

An explicit étale polynomial map $F:\mathbb A_k^2\to\mathbb A_k^2$ over $k=\overline{\mathbb F}_2$ has $\det JF=1$ , separable generic degree $3$ , and three distinct points with the same image, so it is not an automorphism.

Details and sources

AI contribution

ChatGPT and Codex assisted with proof organization, code navigation, and exposition. Lean and Mathlib kernel-checked the stated theorem, and Harmonic Aristotle independently replayed the digest-locked sources. The author retains responsibility for the mathematics.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean kernel checked from recorded sources + independent Aristotle replay

Publication

Public proof preprint with complete Lean ancillary files and reproducibility receipts

Preprint / manuscript

Lean certificate for the non-sofic group Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Activity evidence

A documented editorial estimate based on the long-running Jacobian-conjecture program, the delicate characteristic- $p$ variants, and the complete public formal artifact.

This does not address the classical characteristic-zero Jacobian conjecture. The plane map is derived from a coordinate-permuted three-variable construction of Irit Huq-Kuruvilla; the paper makes no broader priority claim. The recorded local build is historical rather than a newly repeated cache-free build, and neither computational receipt is human peer review.

Characteristic-two conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI ChatGPT + Codex; Harmonic Aristotle replay
Verification: Lean kernel checked from recorded sources + independent Aristotle replay
Open for: Positive-characteristic separable Jacobian conjecture
Research activity*: 5/5

1 Aug 2026Geometric and measured group theory

Existence of a non-sofic group

Problem statement

Does there exist a countable group that is not sofic?

The unit group $L_{\mathbb F_2}(1,2)^\times$ of the binary Leavitt algebra is not sofic, giving an explicit negative answer to the question whether every countable group is sofic.

Details and sources

AI contribution

OpenAI reports that Astra developed the mathematical result, helped prepare the human-readable manuscript, and translated the final claim into Lean. OpenAI researchers curated the release and its formal verification package.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Claim audit

The theorem has a public Lean certificate and a human follow-up using its key criterion, but no peer-reviewed version has yet been located. The announcement's original literature-history wording was corrected on 3 August 2026.

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

Composite research manuscript Human torsion-free follow-up Human residually finite wreath-product follow-up

Activity evidence

A documented editorial estimate based on the question’s prominence since Gromov and Weiss introduced sofic groups and its links to approximation, dynamics, and operator algebras.

The result addresses soficity, not the still-separate question of whether this group is hyperlinear. Two subsequent human-authored notes apply the proof mechanism to torsion-free examples and to explicit non-sofic generalized wreath products built from residually finite Kazhdan groups, providing external mathematical follow-up. The first note also records that OpenAI redacted its original historical claim of no progress for at least a decade after earlier work was identified.

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Explicit non-sofic group constructed

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Claim audit: Issue documented
Open for: 27 years
Research activity*: 5/5

21 Jul 2026Lie algebras and Galois descent

Explicit minimal-dimensional counterexample to Deré’s real-form conjecture

Problem statement

Can one construct explicitly a complex Lie algebra isomorphic to its conjugate but not definable over

\mathbb R

, and determine the smallest two-step nilpotent dimension where this occurs?

An explicit $10$ -dimensional two-step nilpotent complex Lie algebra is isomorphic to its complex conjugate but has no real form; dimension $10$ is minimal within the two-step nilpotent class.

Details and sources

AI contribution

Skip Garibaldi posed the key question to Claude Fable. The model autonomously proved the stabilizer theorem underlying the construction; the authors rewrote a complete proof using its ideas and their own calculations.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; independent review pending

Publication

Three-author public preprint with explicit construction

Preprint / manuscript

Explicit construction preprint

Activity evidence

A documented editorial estimate based on a 2019 conjecture and a 2026 nonconstructive disproof followed by the explicit-construction question.

This is not the first disproof of Deré's conjecture: Demarche had already established nonexistence of a real form nonconstructively. The AI-assisted contribution is the explicit example, its two-step nilpotent structure, and minimality in that class.

Prior nonconstructive disproof

Explicit minimal witness constructed

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Claude Fable
Verification: Author checked; independent review pending
Open for: 7 years
Research activity*: 2/5

1 Aug 2026Polynomial moments and transcendence theory

Two-variable Factorial Conjecture

Problem statement

\mathcal L(x^a y^b)=a!b!

and

\mathcal L(f^m)=0

for every positive integer

m

, must a polynomial

f\in\mathbb C[x,y]

be zero?

The draft claims that for the factorial functional $\mathcal L(x^a y^b)=a!b!$ on $\mathbb C[x,y]$ , the vanishing $\mathcal L(f^m)=0$ for every $m\geq1$ forces $f=0$ , establishing the two-variable Factorial Conjecture.

Details and sources

AI contribution

The author reports that ChatGPT assisted proof discovery, organization, symbolic checks, reference verification, adversarial auditing, and drafting. Claude contributed a semisimple-projector strategy, a reduction to phase-polynomial moments, and additional adversarial audits.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author draft; not peer reviewed or formally verified

Publication

Public 37-page research draft and LaTeX source

Preprint / manuscript

Research draft, source, and explicit status disclaimer

Activity evidence

A documented editorial estimate based on the Factorial Conjecture's connection to the Image and Jacobian conjecture programs, tempered by the preliminary status of this draft.

The repository explicitly labels the manuscript a research draft that is neither peer reviewed nor formally verified. The index records the claim without endorsing it; several deep transcendence and $E$ -function steps require specialist scrutiny.

Initial public commit

Full proof claimed in an unreviewed draft

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Sol + Claude Opus 5
Verification: Author draft; not peer reviewed or formally verified
Open for: Two-variable case of the Factorial Conjecture
Research activity*: 3/5

27 Jun 2026Commutative algebra and matrix combinatorics

Principal-minor-one matrices under taking powers

Problem statement

For which commutative rings is the property that all principal minors equal

1

inherited by every integer power of a matrix?

If every principal minor of a square matrix is $1$ , then every power has the same property over reduced rings, over $\mathbb Z/d\mathbb Z$ , and over several further ring classes. A stronger nullcyclic property is shown to be power-stable over arbitrary commutative rings.

Details and sources

AI contribution

The author reports that GPT-5.5 supplied most ideas and most of the writing. The author fully proofread, edited, and took responsibility for the resulting argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author proofread; independent review pending

Publication

Public revised preprint

Preprint / manuscript

Matrix-powers theorem preprint

Activity evidence

A documented editorial estimate based on a natural algebraic generalization of a contest theorem and its links to cyclic products and integral dependence.

This generalizes a Putnam 2021 problem from fields to broad classes of rings and identifies the stronger property that survives over arbitrary commutative rings. It is indexed as a theorem-level advance, not as a named open-problem closure.

Ring-theoretic inheritance theorem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5
Verification: Author proofread; independent review pending
Open for: Ring-generalization question following Putnam 2021 B5
Research activity*: 2/5

21 May 2026Commutative algebra

Log-concavity of codimension-three pure O-sequences

Problem statement

For a pure O-sequence

h=(h_0,\ldots,h_e)

of codimension three and type two, is

h_i^2\geq h_{i-1}h_{i+1}

for every interior index

i

Every pure O-sequence of codimension three and type two is proved log-concave. The broader nonmonomial level-Hilbert-function case remains open.

Details and sources

AI contribution

The system reformulated the combinatorial structure and supplied a substantial case analysis, then translated the argument into Lean.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Formal development

Publication

AP Nexus preprint and public formal development

Preprint / manuscript

AP Nexus paper Original conjecture paper

Activity evidence

A recent explicit conjecture inside a sustained specialist program on Hilbert functions and pure O-sequences.

The monomial pure-O-sequence statement is complete, but it must not be conflated with the wider level-algebra conjecture.

Precise monomial case proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AlphaProof Nexus
Verification: Lean checked
Open for: 4 years
Research activity*: 3/5

10 Jun 2026Algebraic combinatorics

Hook coefficients of universal Hilbert series

Problem statement

For the universal Schur coefficients

c_\lambda(n)

in the multigraded Hilbert series of the coinvariant algebra

R_n^{(m)}

, give a combinatorial interpretation when

\lambda

is a hook partition.

ProofCouncil, UCLA Moonshot, and ChatGPT 5.5 Pro each produced a correct combinatorial interpretation. The editors requested only minor revisions, mostly for exposition and citation placement.

Details and sources

AI contribution

Three systems independently derived hook-shape formulas using ordered set partitions or related combinatorial encodings; the reviewers noted that several differed from the human solution.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Double-blind expert review; minor revisions

Publication

First Proof Second Batch report, complete submissions, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

The problem combines active work on diagonal coinvariants, superspace, Schur expansions, and sign-reversing involutions.

The result belongs to a forthcoming project on universal Hilbert-series coefficients of superspace coinvariant rings. The public report supplies both the intended result and detailed referee judgments.

Research problem solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro / ProofCouncil / UCLA Moonshot
Verification: Double-blind expert review; minor revisions
Open for: Unpublished algebraic-combinatorics problem
Research activity*: 3/5

12 Jun 2026Finite group theory

Solubilizer Conjecture A.1

Problem statement

G

is nonsolvable and

\operatorname{Sol}_G(x)\cap\operatorname{Sol}_G(y)

is nonempty, must that intersection contain a nontrivial normal subgroup of

G

The alternating group $A_5$ and a five-cycle give a counterexample: the relevant solubilizer is $D_{10}$ and contains no nontrivial normal subgroup of $A_5$ .

Details and sources

AI contribution

The system generated an explicit group-theoretic certificate, independent recomputations, a clean-room checker, and mutation tests.

Problem origin

Appendix Conjecture A.1 was generated in the ChatGPT, Gemini, and Claude conjecture-mining study arXiv:2412.16177.

Verification

Dual computational routes; not externally refereed

Certificate and checker Verification catalog and runnable checks Verification method and limitations

Publication

Public certificate, checker, source snapshot, and write-up

Preprint / manuscript

Original conjecture collection Result write-up

Activity evidence

A recent appendix conjecture produced in an AI-math workshop project; the mathematical weight is modest and the wording matters.

This refutes the printed phrase “normal subgroup of $G$ .” A charitable alternative meaning “normal in the intersection” is not refuted.

Conjecture refuted as printed

Claimed outcome: Disproved
Problem origin: AI-generated problem
System: Demonstrandum multi-agent pipeline
Verification: Dual computational routes; not externally refereed
Open for: LLM-generated workshop-paper conjecture
Research activity*: 1/5

12 Jun 2026Finite group theory

Solubilizer Conjecture A.13

Problem statement

\operatorname{Sol}_G(x)

is a proper subgroup of a nonsolvable finite group

G

, must the intersection of all its conjugates lie in the hypercenter of

G

For $G=A_5\times S_3$ and a five-cycle in the first factor, the solubilizer’s normal core contains $1\times S_3$ while the hypercenter of $G$ is trivial.

Details and sources

AI contribution

The pipeline produced a finite certificate, a mutation-tested checker, and an independent clean-room recomputation.

Problem origin

Appendix Conjecture A.13 was generated in the ChatGPT, Gemini, and Claude conjecture-mining study arXiv:2412.16177.

Verification

Audit-panel grade; one banked checker; not externally refereed

Certificate, checker, and audit Verification catalog and runnable checks Verification method and limitations

Publication

Public certificate, checker, source snapshot, and audit log

Preprint / manuscript

Original conjecture collection Result write-up

Activity evidence

A recent AI-generated conjecture that the original authors reported they could not computationally validate.

The two plausible quantifier readings of “its conjugates” are checked to be equivalent here. Novelty evidence is literature-search based.

Conjecture refuted as printed

Claimed outcome: Disproved
Problem origin: AI-generated problem
System: Demonstrandum multi-agent pipeline
Verification: Audit-panel grade; one banked checker; not externally refereed
Open for: Previously unvalidated workshop-paper conjecture
Research activity*: 1/5

12 Jun 2026Finite group theory

Solubilizer Conjecture A.16

Problem statement

\operatorname{Sol}_G(x)

is a proper subgroup of a nonsolvable finite group

G

, must its intersection with its normalizer be metabelian?

For $G=A_5\times S_4$ , the relevant proper self-normalizing solubilizer is $D_{10}\times S_4$ , whose derived length is three rather than metabelian.

Details and sources

AI contribution

The system supplied a brute-force finite-group certificate, a mutation-tested checker, and an independent recomputation.

Problem origin

Appendix Conjecture A.16 was generated in the ChatGPT, Gemini, and Claude conjecture-mining study arXiv:2412.16177.

Verification

Audit-panel grade; one banked checker; not externally refereed

Certificate, checker, and audit Verification catalog and runnable checks Verification method and limitations

Publication

Public certificate, checker, source snapshot, and audit log

Preprint / manuscript

Original conjecture collection Result write-up

Activity evidence

A recent AI-generated appendix conjecture with a direct finite counterexample.

The concrete group is completely checked. The accompanying product template is explanatory and does not turn every related solubilizer conjecture into a theorem.

Conjecture refuted as printed

Claimed outcome: Disproved
Problem origin: AI-generated problem
System: Demonstrandum multi-agent pipeline
Verification: Audit-panel grade; one banked checker; not externally refereed
Open for: Previously unvalidated workshop-paper conjecture
Research activity*: 1/5

20 Jul 2026Infinite group theory

Kourovka Problem 3.46 — maximal locally soluble normal subgroups

Problem statement

Does there exist a group having more than one but only finitely many maximal locally soluble normal subgroups?

An explicit group is constructed with exactly two maximal locally soluble normal subgroups, showing that the number of such subgroups need not be one or infinite.

Details and sources

AI contribution

Aristotle autonomously developed the mathematical argument in Lean. The authors monitored the process, clarified definitions when needed, built supporting infrastructure, and translated and polished the resulting formal proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Lean development for Problem 3.46

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Lean development for Problem 18.50

Activity evidence

A long-standing group-theory question rooted in classical work on products of locally soluble normal subgroups.

The construction uses a semidirect product built from finitely supported infinite unipotent matrices. This is one of eight independent headline results in the paper.

Problem solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Asked in the 1969 Kourovka Notebook
Research activity*: 3/5

20 Jul 2026Combinatorial group theory

Kourovka Problem 18.50 — prescribed permuted-product cardinality

Problem statement

Given positive integers

n

and

1\leq k\leq n!

, can

n

distinct group elements have exactly

k

distinct values among all their permuted products?

For every $1\leq k\leq n!$ , the authors construct a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Lean development for Problem 19.25

Activity evidence

A focused combinatorial group-theory problem with a public notebook record and a now-complete formal construction.

The construction records permutation inversion codes inside a central extension and then takes a finite central quotient to obtain exactly the requested number of products.

Problem solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 18.50
Research activity*: 2/5

20 Jul 2026Finite group theory

Kourovka Problem 19.25 — totient sums and simplicity

Problem statement

Do a finite group’s order and the sum

\sum_{g\in G}\varphi(|g|)

determine whether the group is simple?

A simple group and a non-simple group of order $6048$ are exhibited with the same statistic $\sum_{g\in G}\varphi(|g|)=23984$ .

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Lean development for Problem 20.125

Activity evidence

A precise finite-group characterization question supported by explicit machine-checkable examples.

The explicit finite groups show that group order together with the stated totient sum does not determine whether the group is simple.

Question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 19.25
Research activity*: 2/5

20 Jul 2026Rota–Baxter operators on groups

Kourovka Problem 20.125 — noninjective Rota–Baxter operator

Problem statement

Can a nonabelian group admit a Rota–Baxter operator that is surjective but not injective?

A surjective but non-injective Rota–Baxter operator is constructed on a nonabelian group.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Lean development for Problem 21.8

Activity evidence

A recent specialist existence question in the developing theory of Rota–Baxter operators on groups.

The paper supplies both a conventional construction and a public Lean development of the group and operator identities.

Problem solved by construction

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 20.125
Research activity*: 2/5

20 Jul 2026Permutation groups

Kourovka Problem 21.8 — horizontal class transpositions

Problem statement

\operatorname{CT}_{(k)}

is generated by all horizontal class transpositions with modulus at most

k

, is

\operatorname{CT}_{(k)}\cong S_{\operatorname{lcm}(2,\ldots,k)}

for every

k\geq4

For every $k\geq4$ , the horizontal class transpositions of moduli at most $k$ generate a group isomorphic to $S_{\operatorname{lcm}(2,\ldots,k)}$ .

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Publication

Public arXiv paper, accepted Kourovka solution, Lean development, and an independent proof

Preprint / manuscript

Research paper Pan’s independent preprint

Activity evidence

The conjecture grew from computational results in the theory of class transpositions and now has two independent proofs.

J. Pan independently proved the same result using multiple transitivity. The index credits both routes rather than treating the AI-assisted proof as exclusive priority.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 21.8
Research activity*: 3/5

20 Jul 2026Finite groups and power graphs

Kourovka Problem 21.24 — cograph power graphs are chordal

Problem statement

If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four?

For every finite group, a cograph power graph is chordal; equivalently, an induced $4$ -cycle in the power graph forces an induced path on four vertices.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Lean development for Problem 21.24

Publication

Public arXiv paper, accepted Kourovka solution, Lean development, and an independent solution note

Preprint / manuscript

Rundström’s independent solution note

Activity evidence

The problem connects recent classification work on cograph and chordal power graphs and was solved independently by two routes.

M. Rundström obtained the result independently, and both solutions are credited by the Kourovka Notebook.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 21.24
Research activity*: 3/5

20 Jul 2026Ordered groups and lattice theory

Kourovka Problem 21.147 — relatively convex subgroups

Problem statement

Must the right-relatively convex subgroups of a right-orderable nonabelian group form a sublattice of its subgroup lattice?

A nonabelian right-orderable group is constructed whose right-relatively convex subgroups do not form a sublattice of its subgroup lattice.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Lean development for Problem 21.147

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Lean development for Problem 21.150

Activity evidence

A recent specialist problem whose review exposed an unstated nonabelian requirement and prompted a corrected construction.

Aristotle first found an abelian example that the problem authors said was already known. It then embedded that example by a semidirect-product construction to resolve the intended nonabelian question.

Intended nonabelian version solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 21.147
Research activity*: 2/5

20 Jul 2026Finite p-groups

Kourovka Problem 21.150 — rank inequality for $p$ -group extensions

Problem statement

For an extension

G=A\rtimes B

of elementary abelian

p

-groups with an element

a\in A

satisfying

C_B(a)=1

, must

H=\langle a,B\rangle

satisfy

\operatorname{rank}(Z(H)\cap H')\leq\operatorname{rank}(B)

An explicit $p$ -group extension is constructed for which $\operatorname{rank}(Z(H)\cap H')>\operatorname{rank}(B)$ , contradicting the proposed bound.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; author-curated preprint

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Lean development for Problem 21.149

Activity evidence

A newly recorded finite-group inequality with a complete explicit counterexample and formal certificate.

The counterexample satisfies the Notebook’s accompanying elementary-abelian and trivial-centralizer hypotheses; the formal development checks the finite construction.

Proposed inequality disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Lean checked; author-curated preprint
Open for: Kourovka Notebook Problem 21.150
Research activity*: 2/5

20 Jul 2026Ordered groups and automorphisms

Kourovka Problem 21.149 — Dlab-group formulation pitfall

Problem statement

Original wording: do Dlab groups have order automorphisms that are not inner? Strengthened wording: can such an automorphism fail to be induced by conjugation in any possibly larger Dlab group?

Aristotle proved that some order automorphisms of Dlab groups are not inner. The problem author confirmed that result but clarified a stronger intended question, which remains open.

Details and sources

AI contribution

Aristotle formalized and solved the literal version-43 statement. Review by the problem author then revealed that the intended quantification allowed conjugation inside a possibly larger Dlab group.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked for original wording

Claim audit

The checked theorem answers only the original formulation. It must not be presented as a resolution of the strengthened version now recorded in the Kourovka Notebook.

Publication

Public Lean development and paper-level post-mortem; strengthened version remains open

Preprint / manuscript

Exact certificate checker Reproduction instructions

Activity evidence

A recent specialist problem whose main significance here is the documented statement mismatch uncovered during review.

This entry is intentionally retained as a formulation audit. It illustrates that kernel checking establishes the encoded theorem, not that the encoded theorem matches every unstated intention behind a research question.

Original wording solved; intended problem open

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Aristotle (Harmonic)
Verification: Lean checked for original wording
Claim audit: Issue documented
Open for: Strengthened formulation remains open
Research activity*: 2/5

26 Jul 2026Finite-group cohomology and commutative algebra

Carlson’s associated-prime depth conjecture

Problem statement

Is the depth of every finite-group cohomology ring realized by the dimension of one of its associated primes?

For $G=\operatorname{SmallGroup}(128,859)$ over $\overline{\mathbb F}_2$ , the cohomology ring has depth $2$ , while every associated-prime quotient has dimension at least $3$ .

Details and sources

AI contribution

TARS autonomously identified the group. Xinan Dai reconstructed and completed the mathematical chain and independently checked the finite computations and presentation certificates.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Exact GAP/Singular certificate + human proof audit

Publication

Public arXiv proof with hash-pinned group data and an exact ancillary verifier

Preprint / manuscript

Structural follow-up and infinite-family extension

Activity evidence

Carlson posed the question in 1995 within the specialist but mature theory of depth, transfer, and finite-group cohomology.

The certificate uses exact $\mathbb F_2$ arithmetic and enumerates all 75 rank-two elementary abelian subgroups. A subsequent paper explains the failure through centralizer excess and extends it to $G\times(C_2)^n$ . The certificate is reproducible computational verification, not a proof-assistant formalization.

Conjecture disproved by an exact finite example

Claimed outcome: Disproved
Problem origin: Human-source problem
System: TARS agent system (foundation model not disclosed)
Verification: Exact GAP/Singular certificate + human proof audit
Open for: 31 years
Research activity*: 4/5

22 Jul 2026Representation zeta functions and root systems

Au’s gamma-product conjecture for Witten zeta functions

Problem statement

Do the leading residues of normalized Witten zeta functions for all irreducible crystallographic root systems admit the gamma-product form predicted by Au?

For every irreducible crystallographic root system $\Phi$ , the normalized Witten zeta function has a simple pole at $2/h$ with an explicit universal gamma-product residue. This proves Au’s predicted gamma-product shape, including the type- $A_4$ value.

Details and sources

AI contribution

The manuscript states that Codex generated all derivations, proofs, exposition, code, and publication materials, with GPT-5.6 Pro used separately as a reviewer. Jonas Matuzas reports independently checking the mathematics and citations and takes responsibility.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human-checked AI proof plus exact Mathematica checks

Publication

Public revised arXiv manuscript; no independent specialist review or formal proof located

Preprint / manuscript

Research paper, revised 22 July

Activity evidence

Au formulated the general prediction in 2024 after computations in ranks two and three; it lies in an active representation-growth literature.

The record is unusually explicit about near-total AI generation. Exact symbolic and numerical checks support consistency, but they are not a formal proof or independent peer review.

Au’s original conjecture paper

Universal residue formula claimed

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Codex (GPT-5.6 Sol Ultra) / GPT-5.6 Pro
Verification: Human-checked AI proof plus exact Mathematica checks
Open for: 2 years
Research activity*: 4/5

30 Jul 2026Lie groups, invariant integration, and Mathieu–Zhao spaces

Mathieu property for compact connected Lie groups

Problem statement

For which compact connected Lie groups is the kernel of Haar integration a Mathieu–Zhao space?

For a compact connected Lie group $G$ , the kernel of Haar integration is claimed to be a Mathieu–Zhao space exactly when $G$ is a torus. Every nonabelian $G$ receives explicit witnesses with all pure moments zero but infinitely many nonzero marked moments.

Details and sources

AI contribution

Christopher D. Long reports using the models for exploration, calculations, proof organization, literature work, and editing. He says he checked the arguments and citations and takes responsibility for the manuscript.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked manuscript and exact identity verifier

Exact identity verifier

Claim audit

The global Lie-theoretic transfer has not yet received independent specialist or formal verification.

Publication

Public manuscript, TeX source, and Python verifier; no independent specialist review located

Preprint / manuscript

Author manuscript

Activity evidence

A documented editorial estimate based on the Mathieu conjecture program, compact-group harmonic analysis, and the scope of the claimed classification.

The Python artifact checks the universal algebraic identities, but the global highest-weight and quotient-transfer argument is not formalized. This record therefore reports a claimed classification, not an independently confirmed theorem.

LaTeX source Author announcement July 30 source summary

Classification claimed; specialist review pending

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Sol Pro / Claude Opus 5
Verification: Author-checked manuscript and exact identity verifier
Claim audit: Issue documented
Open for: General compact-group classification problem
Research activity*: 4/5

23 Jul 2026Poisson algebras and Weyl algebras

Rank-two Poisson Conjecture

Problem statement

Must every polynomial endomorphism of a rank-

n

symplectic Poisson algebra that preserves the canonical bracket be an automorphism?

Explicit polynomials $R,T,D,S\in\mathbb Q[x,q,p,z]$ define a Poisson endomorphism for two canonical pairs with Jacobian determinant one and an exact three-point fiber. Stabilization would disprove $PC(n)$ for every $n\geq2$ ; an appendix also gives a nonautomorphic endomorphism of the fourth Weyl algebra.

Details and sources

AI contribution

Long reports that ChatGPT produced the four-variable construction, Hamiltonian correction, and proof organization. Claude Fable 5 supplied an adversarial audit; Long checked the final formulas.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked manuscript and four exact computational audits

Exact verification artifacts

Claim audit

The finite identities are reproducibly checked, but the manuscript has not yet received independent specialist or formal review.

Publication

Public manuscript and verification-artifact directory; no independent specialist review located

Preprint / manuscript

Original author claim Artifact announcement

Activity evidence

A documented editorial estimate based on the conjecture’s relationship with Weyl-algebra automorphisms, the Dixmier conjecture, and the Jacobian conjecture.

The repository’s exact Python, SymPy, and sparse audits check the Poisson brackets, determinant, factorization, and three-point collision. Contrary to an earlier social description, no Lean proof is currently public.

Explicit counterexample claimed for every rank

n\geq2

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Sol / Claude Fable 5
Verification: Author-checked manuscript and four exact computational audits
Claim audit: Issue documented
Open for: Poisson analogue of the Dixmier/Jacobian program
Research activity*: 4/5

30 Jul 2026Finite group theory and invariable generation

Kourovka Problem 21.142 — invariable generation embedding

Problem statement

For fixed distinct primes

p,q

, does every finite group embed into a finite group invariably generated by elements of orders

p

and

q

Albilich reports that for fixed distinct primes $p,q$ , some alternating group $A_m$ cannot embed into any finite group invariably generated by elements of orders $p$ and $q$ .

Details and sources

AI contribution

A human-steerable Albilich run used multiple proof-search, advisor, verifier, retrieval, and computer-algebra roles to develop the proposed negative answer.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Internal verification only; blocking debts remain

Proof report and debt ledger

Claim audit

The experiment report labels the run solved but also records active blocking debts; the system paper says 10 of 11 claims were verified and acknowledges residual debts. No independent or formal verification was located.

Publication

Public system paper, experiment directory, and proof ledger

Preprint / manuscript

Albilich paper

Activity evidence

A documented editorial estimate based on the Kourovka Notebook record and the newly public proof-state archive.

This is included as a claimed disproof under review, not as an independently established resolution. The outcome flag describes the target of the claim; the verification and audit labels describe its current evidentiary state.

Public experiment archive

Counterexample claimed; blocking proof debts remain

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Albilich (GPT-5.6 Sol xhigh)
Verification: Internal verification only; blocking debts remain
Claim audit: Issue documented
Research activity*: 3/5

30 Jul 2026Finite simple groups and closure properties

Kourovka Problem 20.2 — totally $3$ -closed simple groups

Problem statement

Does there exist a nonabelian simple group of Lie type that is totally

3

-closed?

Albilich proposes $\mathrm{PSL}_2(7)$ as an explicit nonabelian simple group of Lie type that is totally $3$ -closed, together with a stronger family-level analysis.

Details and sources

AI contribution

Two Albilich runs, with and without the advisor, developed the proof using GPT-5.6 Sol roles and GAP-assisted finite checks.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Internal proof ledger; independent review pending

Proof report and debt ledger

Claim audit

The report’s certification section claims no unresolved proof debt, but its live branch snapshot still lists one blocking complete pair-coset enumeration. No independent expert or proof-assistant check was located.

Publication

Public system paper, experiment directory, and GAP-assisted report

Preprint / manuscript

Albilich paper

Activity evidence

A documented editorial estimate based on the Kourovka problem record and the reproducible experiment archive.

This is included as a claimed affirmative resolution under review. It remains visible under Claim issues until the recorded blocking obligation is discharged or independently checked.

Public experiment archive

Positive witness claimed; one blocking debt remains

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Albilich (GPT-5.6 Sol xhigh)
Verification: Internal proof ledger; independent review pending
Claim audit: Issue documented
Research activity*: 3/5

4 Apr 2026Commutative algebra

Anderson’s quasi-completeness question

A weakly quasi-complete Noetherian local ring that is not quasi-complete gives a counterexample to D. D. Anderson’s 2014 question.

Details and sources

AI contribution

Rethlas found the construction using mathematical retrieval; Archon translated it into Lean and filled nontrivial gaps.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked + statement comparator

Publication

Public preprint, complete Lean project, and open-source agents

Preprint / manuscript

Rethlas source Archon source

This is an unusually complete audit trail: the informal proof, formal statement, kernel-checked development, references, and raw model output are all available.

Question answered negatively

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Rethlas + Archon
Verification: Lean checked + statement comparator

3 Feb 2026Numerical semigroups

Fel’s conjecture on syzygies

The conjectured universal formula for normalized alternating syzygy power sums of numerical semigroup rings was proved for every index.

Details and sources

AI contribution

Starting from a natural-language specification, AxiomProver generated the mathematical proof and a Lean/Mathlib development.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked + multi-author audit

Publication

Public arXiv preprint with a complete formal proof

Preprint / manuscript

The theorem was open as a mathematical conjecture, rather than merely an already-known result newly encoded in Lean.

Conjecture proved and formalized

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: AxiomProver
Verification: Lean checked + multi-author audit

Analysis

8 Sep 2026Operator algebras

The hyperfinite $\mathrm{II}_1$ factor is not quasidiagonal

There is a unital separable MF $\mathrm C^*$ -algebra $A$ for which $A\otimes\mathcal R$ contains a proper isometry. Consequently stable finiteness is not preserved by tensor products in general, and the hyperfinite $\mathrm{II}_1$ factor $\mathcal R$ is not quasidiagonal.

Details and sources

AI contribution

Narutaka Ozawa says he had conceived the outline long ago but could not complete it; most of the proofs in the finished argument were supplied through interactive work with ChatGPT Pro 6.0.

Problem origin

The paper identifies Brown's Section 6.6 problem, Brown--Ozawa Problem 10.4.9, and an earlier operator-algebra problem list as the human sources.

Verification

Author-checked preprint; external review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

Formal Conjectures source statement Independent formal-artifact audit

The constructed algebra is non-simple and has only non-faithful traces. The result does not resolve the simple-algebra tensor-product problem or Kaplansky's problem.

Brown's quasidiagonality problem answered negatively

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT Pro 6.0
Verification: Author-checked preprint; external review pending

3 Sep 2026Analysis

Smale's mean-value conjecture at $K=1$

A complex polynomial with $p(0)=0$ and $p'(0)=1$ has $|p(c)/c|>1$ at every critical point $c$ , refuting the proposed universal constant $K=1$ .

Details and sources

AI contribution

An autonomous benchmark run found the counterexample construction and wrote the Lean proof; Tom Adamczewski released the artifacts and Claude later drafted documentation.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Lean/Comparator checked against an independently written benchmark statement

Lean proof repository

Claim audit

No independent complex analyst has reviewed the nonconstructive, large-unspecified-degree witness; the machine-written proof account is unaudited.

Publication

Public Lean disproof repository

Preprint / manuscript

No public preprint or manuscript located.

This disproves only the $K=1$ conjecture. It does not affect Smale's proved $K=4$ theorem or determine the optimal universal constant.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Lean/Comparator checked against an independently written benchmark statement
Claim audit: Issue documented

28 Aug 2026Analysis

Non-MF groups and non-finite full group $C^*$ -algebras

Explicit constructions show that a full group $C^*$ -algebra need not be finite and that a stably finite reduced group $C^*$ -algebra need not be MF.

Details and sources

AI contribution

Caleb Eckhardt states that the mathematical ideas and proofs were generated by the model; he internalized, refined, and rewrote them.

Problem origin

The preprint cites the prior operator-algebra questions asking whether all full group $C^*$ -algebras are finite and whether stably finite algebras are MF.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The seven-page preprint is author-checked but unrefereed. Input acknowledged from several experts is not described as verification of the complete proof.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

13 Aug 2026Analysis

Radchenko--Viazovska Fourier-interpolation question

A nonzero continuous integrable and square-integrable Fourier-invariant function vanishes at every $\sqrt n$ ; more generally, such functions exist for the denser points $\sqrt n/[\log(e+n)]^\beta$ when $0\leq\beta\leq1/2$ .

Details and sources

AI contribution

Bondarenko and Seip say ChatGPT's exploratory input and calculations were essential, without assigning a particular proof step or naming the model version.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The case $\beta=0$ answers the 2019 question negatively; the stronger logarithmically denser zero sets are additional. This is a conventional author proof with no formalization or independent review located.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT (OpenAI; version unspecified)
Verification: Author-checked proof; independent review pending

2 Sep 2026Analysis

Basso's stability question for maximal projection constants

For every fixed positive integer $r$ , the maximal relative projection constants satisfy $\lambda(r,n)=\lambda(r)$ for all $n\geq2^r\binom{r+1}{2}$ , proving that the sequence always stabilizes.

Details and sources

AI contribution

The authors disclose using ChatGPT solely for preliminary brainstorming and exploration of proof strategies. They wrote and checked every formal statement and proof themselves.

Problem origin

Basso formally posed the stability question after correcting an erroneous earlier proof; the new paper states and answers Question A.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The yes-or-no stability question is resolved. The smallest stabilization threshold remains unknown, and the paper newly asks whether it is $O(r^2)$ . The AI contribution is limited and no public transcript is available.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5
Verification: Author-checked proof; independent review pending

1 Sep 2026Analysis

Asymptotic safety regions for Hermite Gabor frames

For every $\eta>0$ and all sufficiently large Hermite orders $n$ , the lattice Gabor system is a frame whenever $ab\leq n^{-2/3-\eta}$ ; a separate near-axis theorem is asymptotically sharp.

Details and sources

AI contribution

The authors' original argument gave an $n^{-5/6}$ region. ChatGPT helped refine it to the stronger $n^{-2/3-\eta}$ range, which the authors independently verified.

Problem origin

The paper develops the established human research problem of determining frame sets for Hermite-generated Gabor systems and cites the published frame-set conjecture literature.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This enlarges certified frame regions but does not characterize the full frame set or resolve the general frame-set conjecture. No code, formalization, or public transcript was located.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT (OpenAI; version unspecified)
Verification: Author-checked proof; independent review pending

1 Sep 2026Analysis

Unique preduals of Lipschitz spaces after Weaver's proof gap

An explicit counterexample disproves the codimension-one lemma used in Weaver's 2018 proof. A new argument still proves that $\operatorname{Lip}_0(M)$ has a strongly unique predual when $M$ is a convex subset of a finite-dimensional normed space.

Details and sources

AI contribution

The authors state that ChatGPT was used to obtain the results and double-check the final text; they wrote the manuscript and independently verified its contents.

Problem origin

The target is Weaver's published unique-predual program and a recently identified gap in its crucial lemma; the paper links both the 2018 article and its 2026 corrigendum.

Verification

Author-checked correction and replacement proof; independent review pending

Claim audit

A published lemma is false. The broad bounded/geodesic claims cannot be treated as settled by the original proof; only the newly proved finite-dimensional convex case is recorded positively here.

Publication

Public research preprint

Preprint / manuscript

Weaver's original preprint

The counterexample invalidates the lemma that strong uniqueness passes to weak-star-closed codimension-one subspaces. The paper says the rest of Weaver's arguments remain valid to the authors' knowledge, but it does not supply a replacement proof for every original scope.

Proved partial advance; general problem remains open

Claimed outcome: ProvedDisproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro
Verification: Author-checked correction and replacement proof; independent review pending
Claim audit: Issue documented

3 Sep 2026Analysis

Endpoint fractional Riesz estimate on the Hamming cube

For $1<p<2$ , the discrete gradient satisfies $\|\nabla f\|_{L_p(\ell_2)}\lesssim_p\|\Delta^{1/p}f\|_p$ with a constant independent of dimension, answering the endpoint conjecture of Naor, Eskenazis, and Ivanisvili.

Details and sources

AI contribution

Some proof ideas arose in model interactions; the authors examined, reformulated, and independently verified the incorporated arguments.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The endpoint removes the previous exponent loss or dimension-dependent logarithm. The cited unpublished proof of Naor concerns the earlier non-endpoint estimate, not this endpoint theorem.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-prepared proof; independent review pending

24 Aug 2026Analysis

One-dimensional sign-uncertainty upper bound of 0.3089

An exact-rational polynomial times a Gaussian gives $C_{SU}\leq0.3089$ , beyond the restricted prescribed-double-root family used by the original evaluator.

Details and sources

AI contribution

Agents explored constructions and proofs in the Station environment. The paper distinguishes novel outputs, concurrent discoveries, and human contributions; the linked dataset records the individual runs.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public proofs and exact verification artifacts; independent review pending

Publication

Public research preprint

Preprint / manuscript

This is a one-dimensional bound; the exact constant remains unknown. It is distinct from the high-dimensional sign-uncertainty asymptotics in OpenAI's earlier work.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Station agents using OpenAI, Anthropic, and Google models; see per-run artifacts
Verification: Public proofs and exact verification artifacts; independent review pending

24 Aug 2026Analysis

Erdős minimum-overlap constant exceeds 0.380552

A phase-sensitive Fourier argument improves the lower bound to $\mu>0.380552$ , against the cited upper bound 0.380868, closing about 82% of the previously published interval.

Details and sources

AI contribution

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public proofs and exact verification artifacts; independent review pending

Publication

Public research preprint

Preprint / manuscript

This is a lower-bound advance, not an exact evaluation. The percentage compares the specific published bounds reported in the paper.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Station agents using OpenAI, Anthropic, and Google models; see per-run artifacts
Verification: Public proofs and exact verification artifacts; independent review pending

6 Aug 2026Operator semigroups

Inverse generator problem on Hilbert space

Problem statement

If a densely ranged operator generates a bounded

C_0

-semigroup on a Hilbert space, must its inverse generate a

C_0

-semigroup?

A bounded, strongly stable $C_0$ -semigroup generator with dense range is constructed whose inverse does not generate a $C_0$ -semigroup, giving a negative answer even on Hilbert space.

Details and sources

AI contribution

The authors used the models to explore constructions, check calculations, and detect mistakes. They selected the final route, repaired the proof, and take responsibility for the theorem.

Problem origin

The inverse generator problem is a longstanding human question in semigroup theory.

Verification

Author-verified counterexample preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Lean 4 counterexample development

Activity evidence

A documented editorial estimate based on the long-running inverse-generator literature and its consequences for semigroup stability.

The paper also derives counterexamples concerning inverse-semigroup growth, Kreiss bounds, and Crank–Nicolson stability; those closely related consequences are not counted as separate records.

Inverse generator problem answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro / Claude Fable 5
Verification: Author-verified counterexample preprint; external review pending
Research activity*: 4/5

5 Aug 2026Spectral geometry and overdetermined eigenvalue problems

Schiffer's conjecture and the planar Pompeiu problem

Problem statement

Must every smooth planar domain admitting the Schiffer overdetermined eigenfunction property be a disk, equivalently have the Pompeiu property?

Infinitely many smooth planar domains that are not disks satisfy the Schiffer overdetermined-eigenfunction condition and the corresponding Pompeiu failure, disproving both conjectures in $\mathbb R^2$ .

Details and sources

AI contribution

The models assisted numerical exploration, supplied early Bessel-function proof drafts, and generated a Lean proof from an early manuscript. The authors developed the bifurcation construction, checked the analysis, and wrote the paper.

Problem origin

Schiffer's conjecture and Pompeiu's problem are classical human problems in spectral geometry and harmonic analysis.

Verification

Lean-checked non-disk counterexample plus author proof preprint

Publication

Public proof preprint and Lean 4 development

Preprint / manuscript

Reference formal statement for Pompeiu's problem

Activity evidence

A documented editorial estimate based on the conjectures' age and central links among spectral geometry, harmonic analysis, and free-boundary methods.

The Lean development formalizes a planar non-disk counterexample without relying on numerics; the analytic paper proves an infinite family. The site keeps those scopes explicit.

Both planar conjectures disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 / GPT-5.6 / Claude Fable 5
Verification: Lean-checked non-disk counterexample plus author proof preprint
Open for: 70 years
Research activity*: 5/5

5 Aug 2026Fractional Laplacians and quadratic forms

Nazarov's truncation conjecture for fractional Laplacians

Problem statement

Does the spectral fractional Dirichlet quadratic form strictly increase after replacing a sign-changing function by its absolute value when

1<s<3/2

For $1<s<3/2$ , the spectral fractional Dirichlet quadratic form strictly increases under $u\mapsto|u|$ for sign-changing $u$ , and the paper proves a broader family of truncation inequalities.

Details and sources

AI contribution

Under joint direction by the authors, Claude produced the original proof of the key spectral corollary. The authors supplied the broader framework, verified the argument, and proved the remaining results.

Problem origin

A. I. Nazarov posed the conjecture in the human fractional-Laplacian literature.

Verification

Author-verified proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Certified 12-point Python audit

Activity evidence

A documented editorial estimate based on the conjecture's role in the analysis of nonlocal operators and sign-changing functions.

The AI-generated proof concerns a key corollary of the main theorem; the broader generalization is attributed to the authors.

Conjecture proved and generalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude
Verification: Author-verified proof preprint; external review pending
Research activity*: 3/5

5 Aug 2026Time-frequency analysis

Heil–Ramanathan–Topiwala conjecture

Problem statement

Are all finite sets of distinct time-frequency shifts of every nonzero

L^2(\mathbb R)

function linearly independent?

Twelve distinct time-frequency shifts of an explicit Schwartz function are linearly dependent, disproving the HRT conjecture.

Details and sources

AI contribution

The counterexample configuration and proof strategy were developed by the model in dialogue with the authors. The authors replaced complex arguments, filled substantial gaps, repaired an error, and independently rewrote the paper.

Problem origin

The HRT conjecture is a longstanding human conjecture in time-frequency analysis.

Verification

Author-verified analytic proof plus certified numerical 12-point construction; external review pending

Publication

Public revised proof preprint with analytic and certified-numerical routes

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture's age and centrality in Gabor and time-frequency analysis.

Version 2 adds a separate purely analytic qualitative counterexample while retaining the certified 12-point construction and public Python audit. No proof-assistant formalization was located; the authors explicitly report that some model gaps were substantial.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Author-verified analytic proof plus certified numerical 12-point construction; external review pending
Open for: 30 years
Research activity*: 5/5

5 Aug 2026Gabor analysis and shift-invariant spaces

Frame set of totally positive functions

Problem statement

For which lattice parameters does a continuous integrable totally positive window generate a Gabor frame for

L^2(\mathbb R)

For every continuous integrable totally positive function, the Gabor system is a frame exactly when $\alpha\beta<1$ ; a sharp Kadets-type theorem is also proved.

Details and sources

AI contribution

GPT-5.4 suggested the productive limit-operator literature connection and technical simplifications. Codex and Claude assisted the Lean formalization; the authors checked every argument and formal file.

Problem origin

The frame-set classification is an established human problem in Gabor analysis.

Verification

Sorry-free Lean proof of the positive direction plus author proof of the iff theorem

Lean 4 formalization

Publication

Public proof preprint and Lean 4 development

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the long-standing frame-set program and the complete characterization for a broad window class.

The Lean repository proves the nontrivial $\alpha\beta<1$ direction with an axiom-clean bridge. The converse and full paper-level scope are supplied by the author-checked manuscript.

Frame set completely characterized

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 / Codex 5.5 / Claude Opus 4.7
Verification: Sorry-free Lean proof of the positive direction plus author proof of the iff theorem
Research activity*: 4/5

4 Aug 2026Numerical range and operator inequalities

Crouzeix's conjecture

Problem statement

Is the numerical range a

2

-spectral set for every matrix?

For every square matrix $A$ and polynomial $p$ , $\|p(A)\|\le2\max_{z\in W(A)}|p(z)|$ .

Details and sources

AI contribution

The authors used the model to explore proof strategies. They selected the successful route, wrote the note, and take responsibility for the proof.

Problem origin

Crouzeix's conjecture is a longstanding human conjecture in matrix and operator analysis.

Verification

Author-written proof preprint; independent review pending

Publication

Public proof note

Preprint / manuscript

Support-growth and rigidity preprint

Activity evidence

A documented editorial estimate based on the conjecture's prominence in numerical linear algebra and operator theory.

The paper explicitly says its argument does not directly prove the completely bounded matrix-amplified variant, which remains outside this record.

Scalar conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro
Verification: Author-written proof preprint; independent review pending
Open for: 25 years
Research activity*: 5/5

3 Aug 2026Discrete harmonic analysis and quantitative unique continuation

Sparse-support growth for lattice eigenfunctions

Problem statement

How sparse can the support of a nonzero lattice eigenfunction be inside growing boxes, and what algebraic dimension must its full support have?

The revised proof establishes $m_3(n)\asymp n^2$ . For every $d\ge4$ , it gives $c_d n^{d^2/(2d-1)}\le m_d(n)\le(2n+1)^{\lfloor d/2\rfloor+1}$ , leaving a gap of less than $3/4$ in even dimensions and less than $1/4$ in odd dimensions.

Details and sources

AI contribution

The author states that all proofs were obtained through a human-guided GPT-5.6 Sol Ultra workflow and then checked by the author. The paper supplies the full arguments and a dedicated AI-disclosure section.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; independent review pending

Publication

Public 33-page revised preprint with complete proofs

Preprint / manuscript

Activity evidence

A documented editorial estimate based on recent work on discrete unique continuation, lattice Schrödinger operators, and support-dimension bounds.

Version 2 replaces the original multiscale and endpoint-collision arguments with a nearly optimal Hilbert-function method. The dimension-three order and Zariski bounds are sharp, but the exact support-growth exponent for every $d\ge4$ remains open.

Dimension three sharp; nearly matching higher-dimensional exponents

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.6 Sol Ultra (OpenAI)
Verification: Author checked; independent review pending
Open for: Quantitative support-growth problem following earlier unique-continuation bounds
Research activity*: 3/5

1 Aug 2026Operator algebras and rigidity

Connes’s group-factor rigidity conjecture

Problem statement

If countable ICC property-

(T)

groups

G

and

H

have isomorphic group factors

L(G)\cong L(H)

, must

G\cong H

There are infinitely many pairwise nonisomorphic, mutually commensurable, finitely generated ICC property- $(T)$ groups with isomorphic group von Neumann algebras. This disproves Connes’s conjecture and the finite-to-one weakening asked by Popa.

Details and sources

AI contribution

OpenAI reports that Astra developed its counterexample and Lean certificate. Shuoxing Zhou independently and concurrently obtained a different explicit counterexample with GPT-5.6 Sol assistance and supplied a conventional author-written proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for the rigidity counterexample Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript and Lean certificate plus an independent concurrent author-written preprint

Preprint / manuscript

Composite research manuscript Independent concurrent counterexample, revised v2

Activity evidence

A documented editorial estimate based on the conjecture’s roots in Connes’s 1982 rigidity program, its explicit 1994 formulation, and the extensive literature on von Neumann superrigidity.

The theorem concerns the bare group-factor functor within the ICC property- $(T)$ class; it does not contradict the stronger hypotheses in known $W^*$ -superrigidity theorems. Zhou's independently and concurrently developed construction provides a second public proof path. Its substantially revised v2 says the author reorganized and checked the AI-assisted proof and incorporated specialist comments; neither path is yet peer reviewed.

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Rigidity conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI Astra + GPT-5.6 Sol
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: 44 years
Research activity*: 5/5

19 Jul 2026Banach space geometry

Toroidal Elton–Odell theorem

Problem statement

Does every infinite-dimensional complex normed space admit a uniformly separated sequence after quotienting unit vectors by unimodular scalars?

Every infinite-dimensional complex normed space contains unit vectors $(x_n)$ and some $\varepsilon>0$ such that $\lVert x_n-\theta x_m\rVert\geq1+\varepsilon$ whenever $n\ne m$ and $|\theta|=1$ .

Details and sources

AI contribution

The models generated the proof ideas and essentially complete arguments before the authors checked, edited, and integrated them into self-contained proofs.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; no formal verification

Publication

Public author-checked preprint with five selected problem papers

Preprint / manuscript

Original AI outputs and project record

Activity evidence

A documented editorial estimate based on the classical Elton--Odell theorem and sustained work on separated sequences in Banach spaces.

The authors say the included proof was produced essentially by the model, then manually verified and edited. A second generated proof is public but was not checked to the same standard and is not part of this verification label.

Complete affirmative solution

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.5 Pro + Codex
Verification: Author checked; no formal verification
Open for: Raised explicitly in 2018 literature
Research activity*: 3/5

19 Jul 2026Banach algebras and operator theory

A unital Banach algebra that is not a Calkin algebra

Problem statement

Is there a unital Banach algebra that cannot occur as the Calkin algebra of any Banach space?

There exists a unital Banach algebra $A$ that is not isomorphic to $\mathcal B(X)/\mathcal K(X)$ for any Banach space $X$ , including nonseparable spaces.

Details and sources

AI contribution

The models generated the proof ideas and essentially complete arguments before the authors checked, edited, and integrated them into self-contained proofs.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; no formal verification

Publication

Public author-checked preprint with five selected problem papers

Preprint / manuscript

Original AI outputs and project record

Activity evidence

A documented editorial estimate based on a recognized realization problem with substantial prior construction results.

Earlier work ruled out realization only over separable Banach spaces. The new proof supplies the missing nonseparable obstruction and was author-checked after model generation.

Existence question answered affirmatively

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.5 Pro + Codex
Verification: Author checked; no formal verification
Open for: Longstanding Calkin-algebra realization question
Research activity*: 3/5

19 Jul 2026Banach space operator ideals

Strict cosingularity and adjoints for separable range

Problem statement

Y

is separable, is

T:X\to Y

strictly cosingular exactly when

T^*

is strictly singular?

When $Y$ is separable, an operator $T:X\to Y$ is strictly cosingular if and only if its adjoint $T^*:Y^*\to X^*$ is strictly singular.

Details and sources

AI contribution

The models generated the proof ideas and essentially complete arguments before the authors checked, edited, and integrated them into self-contained proofs.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; no formal verification

Publication

Public author-checked preprint with five selected problem papers

Preprint / manuscript

Original AI outputs and project record Related MathOverflow discussion

Activity evidence

A documented editorial estimate based on a specialist operator-ideal question arising from descriptive-set-theoretic work.

The proof resolves the separable-range form needed for the operator-ideal hierarchy. The authors describe the model output as essentially correct after small fixes and human checking.

Duality question answered affirmatively

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.5 Pro + Codex
Verification: Author checked; no formal verification
Open for: Related MathOverflow question posted in 2012
Research activity*: 2/5

19 Jul 2026Banach space factorization

Weakly compact factorization through a space with a basis

Problem statement

Can the DFJP factorization space always be chosen to have a Schauder basis whenever the range space has one?

If $T:X\to Y$ is weakly compact and $Y$ has a Schauder basis, then $T$ factors through a reflexive Banach space that also has a Schauder basis.

Details and sources

AI contribution

The models generated the proof ideas and essentially complete arguments before the authors checked, edited, and integrated them into self-contained proofs.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; no formal verification

Publication

Public author-checked preprint with five selected problem papers

Preprint / manuscript

Original AI outputs and project record Earlier special-case discussion

Activity evidence

A documented editorial estimate based on the central DFJP factorization method and multiple earlier special cases.

This extends earlier shrinking-basis, unconditional-basis, $C[0,1]$ , and $L_1$ cases. The authors report only small corrections and editing after the model-generated proof.

General basis case proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.5 Pro + Codex
Verification: Author checked; no formal verification
Open for: Generalization of a question discussed since at least 2016
Research activity*: 3/5

19 Jul 2026Banach space decomposition theory

Primariness of $L_p(L_1)$

Problem statement

L_p(L_1)

primary for every

1<p<\infty

For every $1<p<\infty$ , the mixed-norm space $L_p(L_1)$ has the uniform primary factorization property and is therefore primary.

Details and sources

AI contribution

The model supplied the proof architecture and most technical content. The authors substantially reorganized the argument, completed indicated steps, and checked the final proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; no formal verification

Publication

Public author-checked preprint with five selected problem papers

Preprint / manuscript

Original AI outputs and project record

Activity evidence

A documented editorial estimate based on the established primariness program for classical and mixed-norm Banach spaces.

Unlike the other four selected Banach problems, the raw model output was not a complete proof as written. The record therefore credits a stronger human repair and completion role while retaining the model's substantive proof architecture.

Primariness question proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.5 Pro + Codex
Verification: Author checked; no formal verification
Open for: Prominent remaining mixed-norm primariness case
Research activity*: 4/5

6 May 2026Functional analysis and inequalities

Carbery’s almost-orthogonality inequality in $L^p$

Problem statement

For p ≥ 2, does Carbery’s proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power 2? If not, what is the largest possible exponent?

The proposed exponent 2 fails for every p > 2. The paper identifies the necessary critical exponent p′ and proves the corresponding inequality for every integer p ≥ 2, together with an optimal three-function bound.

Details and sources

AI contribution

Grok supplied the structural counterexample and a central kernel inequality. The later theorems were developed through a documented human–AI collaboration, with the authors correcting numerical inaccuracies in the first model output.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked public proof

Publication

Complete arXiv paper with conventional proofs and disclosed Grok conversations

Preprint / manuscript

Erdős case-study paper Aletheia system paper

Activity evidence

A 2009 question connected to sustained work on sharpened triangle inequalities, Schatten classes, and optimal moment comparisons.

The authors state that they already expected a counterexample, but Grok found the conceptual construction that exposed the sharp exponent. The published argument is the authors’ checked and edited version.

Question disproved; sharp form proved

Claimed outcome: ProvedDisproved
Problem origin: Human-source problem
System: Grok Heavy / Grok 4.20 Heavy
Verification: Author-checked public proof
Open for: 17 years
Research activity*: 4/5

Feb 2026Potential theory and polynomials

Erdős Problem #1040 — polynomial sublevel-set measure

Problem statement

For a closed infinite set

F\subseteq\mathbb C

, let

\mu(F)

be the infimum of

|\{z:|f(z)|<1\}|

over monic polynomials whose zeros lie in

F

. Is

\mu(F)

determined only by the transfinite diameter of

F

Aletheia produced closed sets of equal transfinite diameter but sharply different values of the polynomial sublevel-set invariant, disproving the claim that capacity alone determines it. The zero-measure clause remains separate.

Details and sources

AI contribution

The agent autonomously searched for contrasting sets, tested the construction, and wrote the released argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert-reviewed project result

Publication

DeepMind paper, released output, and official problem discussion

Preprint / manuscript

Activity evidence

A long-standing specialist problem linked to classical logarithmic potential theory and extremal polynomial questions.

The compound Erdős entry contains more than one question. This result settles only the capacity-determination clause.

Official problem record Released Aletheia outputs

First question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Aletheia / Gemini Deep Think
Verification: Expert-reviewed project result
Open for: 68 years
Research activity*: 3/5

9 Jun 2026Global inversion and neural networks

Neural Jacobian Conjecture at width $N=n+1$

Problem statement

If a one-hidden-layer affine-ridge sigmoid network

F:\mathbb R^n\to\mathbb R^n

has strictly positive Jacobian determinant everywhere, must it be globally injective? The proved case has width

N=n+1

Two models independently proved the newly proposed conjecture for one-hidden-layer affine-ridge sigmoid networks at width $N=n+1$ . The cases $N\geq n+2$ remain open.

Details and sources

AI contribution

Moonshine generated the conjecture and invoked the two models independently; an additional ChatGPT web-session collaboration produced a geometric-topological proof.

Problem origin

The Moonshine paper explicitly presents the Neural Jacobian Conjecture as a conjecture autonomously formulated by the Moonshine research agent.

Verification

Multiple proofs in public preprint

Publication

Public Moonshine arXiv paper

Preprint / manuscript

Moonshine paper

Activity evidence

A new conjecture introduced in the same paper as its first special-case proofs, with little independent follow-up yet.

Because the conjecture and proof appeared together in 2026, the activity and age fields deliberately remain low.

First nontrivial width proved

Claimed outcome: Proved
Problem origin: AI-generated problem
System: GPT-5.5 Pro / DeepSeek-V4 Pro
Verification: Multiple proofs in public preprint
Open for: Newly posed in 2026
Research activity*: 1/5

17 May 2026Complex analysis

Erdős Problem #1039 — inradius of polynomial lemniscates

Problem statement

For

f(z)=\prod_{i=1}^n(z-z_i)

with

|z_i|\leq1

, let

\rho(f)

be the radius of the largest disc contained in

\{z:|f(z)|<1\}

. Is

\rho(f)\gg1/n

, and what is its exact extremal behavior?

The largest-disc radius is proved to have worst-case order $\Theta(1/n)$ , including the explicit lower bound $\rho(f)\geq(\log2)/n$ . The exact asymptotic constant remains unknown.

Details and sources

AI contribution

GPT-5.5 Pro supplied the proof; Codex assisted the subsequent Lean formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert-vouched and Lean checked

Official problem record Research discussion

Publication

Public output, official discussion, and Lean repository

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A 1958 complex-analysis problem with Pommerenke’s classical bound, a 2025 paper, and extensive current expert discussion.

This fully answers the order-of-magnitude subquestion but not the complete request to determine the extremal radius.

Order of magnitude determined

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro / Codex 5.5
Verification: Expert-vouched and Lean checked
Open for: 68 years
Research activity*: 4/5

3 Mar 2026Complex analysis

Derivative bounds for polynomial lemniscates

Problem statement

For a degree-

n

polynomial with connected unit lemniscate, is the maximum of

|p'|

at most

(1/2+o(1))n^2

The Eremenko–Lempert theorem was formalized in Lean with an open-source proof scaffold and proprietary model backends.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / Claude Opus 4.6 / Claude Sonnet 4.6 / Gemini 3 Flash / Gemini 3.1 Pro / ulam.ai scaffold with Gemini 3 Flash and Gemini 3.1 Pro
Verification: Lean checked

30 Dec 2025Complex analysis

Projection lengths of polynomial lemniscates

Problem statement

Must every monic non-constant polynomial have a straight line onto which its unit lemniscate projects with length at most two?

Aristotle produced and checked the explicit counterexample $p(z)=z^{16}-1$ ; GPT located related prior literature.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / GPT
Verification: Lean checked

21 Jan 2026Complex analysis

Convexity of small lemniscate components

Problem statement

Let

f\in\mathbb C[x]

be monic with

m

distinct roots, and let

c>0

be small enough that

\{z:|f(z)|\leq c\}

has

m

connected components. Must all those components be convex?

Aristotle generated a formal counterexample and verified it in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

28 Jan 2026Complex analysis

Diameter of a lemniscate component

Problem statement

If every root of a monic polynomial lies in

|z|\leq r<2

, must some component of its unit lemniscate have diameter greater than

2-r

The bound is false for $r>1$ but true for $0<r\leq1$ ; the counterexample and positive range were formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

29 Dec 2025Complex analysis

Entire functions preserving rationality

Problem statement

Does a nonlinear entire function exist such that

x

is rational exactly when

f(x)

is rational?

The classical affirmative construction was reconstructed and checked in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

28 Dec 2025Entire functions

Prescribed derivative-zero sets

Problem statement

Given discrete sets

S_n\subset\mathbb C

, can one transcendental entire function have some derivative vanish on every point of each

S_n

The Barth–Schneider affirmative theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

20 Feb 2026Automorphic forms and representation theory

First Proof Problem 2 — uniform Whittaker test vector

Problem statement

Does there exist one Whittaker-model vector

W

for

\mathrm{GL}_{n+1}(F)

that yields a finite, nonzero local Rankin–Selberg integral for every generic representation

\pi

\mathrm{GL}_n(F)

and every

s\in\mathbb C

OpenAI initially described its submission as likely correct, then withdrew that assessment after the official commentary and community analysis exposed a false support condition for a Whittaker vector.

Details and sources

AI contribution

The model identified a promising test vector and the key nonvanishing reduction, but its proposed support condition contradicts the representation’s central character.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Error documented by problem authors and acknowledged by OpenAI

Claim audit

The attempted “standard Howe-vector existence result” imposes support properties that cannot hold because they conflict with the vector’s central character.

Publication

OpenAI submission, official First Proof solutions, and author commentary

Preprint / manuscript

Original First Proof submission

Activity evidence

A specialized research question whose public failure analysis illustrates how plausible local representation-theoretic arguments can break.

First Proof Batch 1 had ten questions but no formal grading process. OpenAI currently identifies five other attempts—Problems 4, 5, 6, 9, and 10—as having a high chance of correctness, not as formally accepted solutions.

OpenAI correction and submissions Official First Proof Batch 1 record Author solutions and error commentary

Initially favored proof attempt withdrawn

Problem origin: Human-source problem
System: OpenAI internal reasoning model
Verification: Error documented by problem authors and acknowledged by OpenAI
Claim audit: Issue documented
Open for: Solved-but-unpublished benchmark question
Research activity*: 3/5

24 Jul 2026Operator theory and convex functional calculus

Planar strictly convex hyperrigidity

Problem statement

Let

X,Y

be commuting positive contractions and

P

a projection. If

A=PXP|_{\operatorname{ran}P}

and

B=PYP|_{\operatorname{ran}P}

commute and equality holds in the compressed functional calculus for a strictly convex function

f

, must

P

reduce both

X

and

Y

A new manuscript claims an affirmative answer to the two-variable equality problem and strengthens it to a uniqueness theorem for POVMs and PVMs on compact convex sets of affine dimension at most two.

Details and sources

AI contribution

GPT-5.6 Pro generated and repaired the ordinary proof under human direction and performed hostile audits. Codex packaged the release; Aristotle formalized only a foundational layer.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

AI-audited manuscript; no specialist review

Proof and evidence repository Final corrective audit

Claim audit

The accompanying Lean work checks only the foundational layer. Three declarations representing the headline theorem and corollaries still contain `sorry`, so the claimed resolution is not end-to-end formally verified.

Publication

Public proof manuscript, audit packet, disclosure, and partial Lean development

Preprint / manuscript

Original open problem AI-use and formalization disclosure

Activity evidence

A focused recent problem in multivariable operator theory with a public problem statement and proof packet, but limited documented prior discussion.

The frozen proof packet reports an overall audit pass, but this is not independent human review. The authors explicitly do not certify historical priority.

Proof released; specialist review pending

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.6 Pro / Codex / Aristotle
Verification: AI-audited manuscript; no specialist review
Claim audit: Issue documented
Open for: Published as Open Problem 2
Research activity*: 2/5

17 Jul 2026Polynomial analysis

Erdős Problem #119 — cumulative maxima on the unit circle

Problem statement

For unit-circle zeros

z_i

, put

M_n=\max_{|z|=1}|\prod_{i\leq n}(z-z_i)|

. Must

\sum_{k\leq n}M_k>n^{1+c}

eventually for some

c>0

Samuel Korsky and GPT-5.6 Sol proved that for every sequence of zeros on the unit circle there is a constant $c>0$ such that $\sum_{k\leq n}M_k>n^{1+c}$ for all sufficiently large $n$ .

Details and sources

AI contribution

Korsky prompted GPT-5.6 Sol and developed the resulting one-page harmonic-analysis argument; Thomas Bloom checked the proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Checked by Thomas Bloom; official record updated

Publication

Official problem record and public discussion; no stable standalone manuscript located

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A multi-part problem with older specialist literature and recent public scrutiny; the score is an editorial estimate.

This resolves the third cumulative-growth question. It does not replace Beck’s 1991 proof of a different, second question on the same problem page.

Official problem and proof record Discussion and review thread Canonical status database

Third question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol + Samuel Korsky
Verification: Checked by Thomas Bloom; official record updated
Research activity*: 3/5

27 Jul 2026Asymptotic convex geometry

KLS conjecture for quadratic forms

Problem statement

Does the Kannan–Lovász–Simonovits variance inequality hold with a universal constant, and in particular for every quadratic form under an isotropic log-concave measure?

For isotropic log-concave $X$ and symmetric $M$ , the paper proves $\operatorname{Var}\langle MX,X\rangle\leq2\,\mathbb E|\nabla\langle MX,X\rangle|^2$ and derives the global estimate $\psi_n\leq C\log^{1/4}n$ .

Details and sources

AI contribution

The model identified a useful theorem variant and developed the Monge–Ampère differentiation argument; Letwin checked, generalized, and wrote the final manuscript.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked arXiv proof

Publication

Public arXiv manuscript and TeX source; peer review pending

Preprint / manuscript

Activity evidence

KLS is a central long-running problem in convex geometry, probability, sampling, and high-dimensional algorithms.

The quadratic-form case is proved and the best general logarithmic exponent is improved. The full dimension-free KLS conjecture remains open.

Quadratic-form case proved; global bound improved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.6 Pro
Verification: Author-checked arXiv proof
Open for: 31 years
Research activity*: 5/5

22 Jul 2026Complex hypercontractivity

Weissler’s missing two-point inequality

Problem statement

Do the remaining strict two-point inequalities in Weissler’s complex hypercontractivity parameter region hold?

The paper proves the missing strict off-diagonal two-point inequalities needed in Weissler’s complex hypercontractivity program.

Details and sources

AI contribution

The authors report AI assistance in developing the proof but do not identify the system or publish a reconstructable attribution of individual steps.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked arXiv proof

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Activity evidence

Weissler’s inequalities date to the late 1970s and underpin a broad hypercontractivity literature.

The paper addresses the missing strict off-diagonal ranges; diagonal and boundary cases have separate prior treatments. The public AI provenance is limited.

Remaining strict parameter ranges proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AI tools (not specified)
Verification: Author-checked arXiv proof
Open for: 47 years
Research activity*: 4/5

26 Jul 2026Fourier analysis and topological degree

Brezis’s Fourier-degree problem below $\alpha=1/3$

Problem statement

Is there a universal Fourier summation process that recovers the topological degree for every

C^{0,\alpha}

circle map when

\alpha\leq1/3

For every $0<\alpha<1/3$ and every Brezis summation process, there is $f\in C^{0,\alpha}(S^1;S^1)$ for which the weighted Fourier sum fails to converge to $\deg f$ . This answers Open Problem 5.6 negatively below the critical exponent; $C^{0,1/3}$ remains unresolved.

Details and sources

AI contribution

ChatGPT generated preliminary proof drafts. Michał Cieszyński reports checking every argument, revising proofs where needed, checking sources, and determining the final statements.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-verified analytical proof; peer review pending

Publication

Public arXiv manuscript with an AI-use declaration

Preprint / manuscript

Versioned Arb verification archive

Activity evidence

A documented editorial estimate based on the threshold’s role in Fourier definitions of degree and the surrounding Sobolev and VMO literature.

The paper also gives literal answers to Brezis Open Problems 5.7 and 5.8. The principal unresolved boundary is the endpoint $\alpha=1/3$ .

Negative below the threshold; endpoint remains open

Problem origin: Human-source problem
System: OpenAI ChatGPT (model not disclosed)
Verification: Author-verified analytical proof; peer review pending
Open for: Problem from Brezis’s favourite open-problems list
Research activity*: 4/5

20 Jul 2026Bergman spaces and complex analysis

Lower bound for Korenblum’s constant

Problem statement

How large can the universal annular threshold in Korenblum’s maximum principle for the Bergman space

A^2(\mathbb D)

be?

A moment-duality criterion and an explicit rational eight-atom measure prove $c_2\geq0.4263$ , improving the previous lower bound $0.3554$ . A stand-alone Arb ball-arithmetic program certifies the infinite family of moment inequalities.

Details and sources

AI contribution

Codex assisted mathematical brainstorming, numerical exploration of the moment problem, and development of the interval-arithmetic verifier. Frank Wikström independently reviewed the arguments and computations.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author proof and rigorous public Arb certificate

Publication

Public arXiv manuscript and versioned verification archive; peer review pending

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the long-running program to determine Korenblum’s constant and the sequence of published numerical bounds.

This is a substantial certified bound, not a determination of Korenblum’s constant. Floating-point optimization found candidate data but plays no role in the proof certificate.

Certified record lower bound; exact constant remains open

Problem origin: Origin not yet traced
System: OpenAI Codex (GPT-5.6 Sol)
Verification: Author proof and rigorous public Arb certificate
Open for: Long-standing quantitative Bergman-space problem
Research activity*: 4/5

29 Jul 2026Orthogonal polynomials on the unit circle and spectral theory

Lukic’s weighted-entropy decomposition conjecture

Problem statement

Is Lukic’s weighted entropy condition for finitely many critical points equivalent to decomposing the Verblunsky coefficients into components localized at those points?

A two-phase sequence of Verblunsky coefficients with common decay exponent $3/20$ satisfies Lukic’s proposed decomposition conditions, while the corresponding two-point weighted entropy is $-\infty$ . This gives a mixed-resonance counterexample with two critical points of multiplicity three.

Details and sources

AI contribution

Jun Yan states in the abstract that GPT-5.6 found the counterexample. The public manuscript supplies the construction and analytical proof; no more granular transcript or model-role statement was located.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-presented analytical proof; peer review pending

Publication

Public arXiv counterexample manuscript

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture’s role in higher-order Szegő theorems and orthogonal-polynomial spectral theory.

The finite-multiplicity counterexample refutes the equivalence in general. It does not classify which weights or separated phase configurations still satisfy a corrected theorem.

Conjecture disproved for two triple critical points

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: GPT-5.6
Verification: Author-presented analytical proof; peer review pending
Open for: Conjecture in higher-order Szegő theory
Research activity*: 4/5

30 Jul 2026Continuous inverse algebras

Glöckner–Neeb multiplication-growth question

Problem statement

Is the Glöckner--Neeb multiplication-growth condition automatic for every Mackey-complete continuous inverse algebra?

The same complete complex continuous inverse algebra supplies a negative answer: the Glöckner--Neeb multiplication-growth condition is not automatic for Mackey-complete continuous inverse algebras, even under the stronger assumption of completeness.

Details and sources

AI contribution

The authors credit AI agents with completing the construction and generating the manuscript from their initial ideas, followed by manual author checking.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked preprint; independent review pending

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the focused literature on regularity and multiplication growth in continuous inverse algebras.

This is separated from the Lie-group regularity record because it answers a distinct question about topological algebras, although both use the same construction.

Automatic-growth question answered negatively

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: DeepMath agents / GPT models
Verification: Author-checked preprint; independent review pending
Research activity*: 3/5

Apr 2026Entire functions and maximum modulus

Erdős Problem #514 — escape paths for entire functions

Problem statement

For a transcendental entire function, how fast can

|f(z)|

be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus

M(r,f)

The note obtains an escape path on which $|f(z)|$ dominates every power of $|z|$ , bounds its initial length by $O(M(R,f)^\varepsilon)$ , and rules out any universal positive-power lower bound in terms of $M(r,f)$ .

Details and sources

AI contribution

Przemek Chojecki’s public research index credits GPT-5.5 Pro with the claimed solution. This record follows the scope and caveats stated in the manuscript and community discussion.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public manuscript; community discussion

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Primary problem record Community discussion Ulam research index

Activity evidence

The official record has 17 discussion comments and connects several classical path-growth theorems. The 2026 note settles specific formulations while preserving the broader open variant.

The first two conclusions are assembled from published theorems of Wu and earlier work. The broader third question, allowing an arbitrary slower comparison function of $M(r,f)$ , is explicitly outside the manuscript’s scope.

Two questions settled; broader variant open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Public manuscript; community discussion
Open for: Asked by 1961; broader variant open
Research activity*: 3/5

29 Apr 2026Approximation theory and interpolation

Erdős Problem #1133 — robust polynomial interpolation obstruction

Problem statement

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below

(1+\varepsilon)n

to have arbitrarily large uniform norm?

Using Beurling density for Bernstein spaces, the draft proves the requested robust obstruction for arbitrary node multisets and bounded low-degree interpolants.

Details and sources

AI contribution

Przemek Chojecki’s public research index credits GPT-5.5 Pro with the claimed solution. This record follows the scope and caveats stated in the manuscript and community discussion.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public manuscript + community standard check

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Primary problem record Community discussion Ulam research index

Activity evidence

A focused specialist problem with a short public discussion and one documented screening of the proof note.

The proof depends on Beurling’s interpolation theorem as its only external theorem. A community screening reported no issue, but it is not a substitute for peer review.

Claimed full resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Public manuscript + community standard check
Open for: Asked by 1967
Research activity*: 2/5

29 Apr 2026Lagrange interpolation

Erdős Problem #1151 — Chebyshev–Lagrange limit sets

Problem statement

For Chebyshev interpolation nodes and a fixed evaluation point

x_0

, which closed subsets of

[-1,1]

can occur as the finite cluster set of

L_nf(x_0)

For each fixed $x_0$ , every nonempty closed $A\subset[-1,1]$ occurs as the cluster set of the Chebyshev-root interpolants; the empty set occurs exactly at the odd-rational exceptional points.

Details and sources

AI contribution

Przemek Chojecki’s public research index credits GPT-5.5 Pro with the claimed solution. This record follows the scope and caveats stated in the manuscript and community discussion.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public manuscript; interpretation-sensitive

Claim audit

The target statement has multiple natural readings; this record is explicitly scoped to the fixed-point interpretation.

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Primary problem record Community discussion Ulam research index

Activity evidence

The official record has seven comments and active interest in formalization, but the central issue is disambiguating the historical target.

The database wording is ambiguous about whether $x$ is fixed. The manuscript resolves and classifies the scalar fixed-point reading, while showing that several stronger domain-wide readings are false. It should not be presented as an unqualified resolution of every interpretation.

Fixed-point interpretation classified

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Public manuscript; interpretation-sensitive
Claim audit: Issue documented
Open for: Historical statement is ambiguous
Research activity*: 3/5

Jul 2026Extremal polynomials

Erdős Problem #1038 — polynomial lemniscates

Problem statement

Among all nonconstant monic polynomials

f

whose roots lie in

[-1,1]

, determine

\inf_f\left|\{x\in\mathbb{R}:|f(x)|<1\}\right|.

A July 2026 manuscript by Darvas, Peng, and Tao claims an exact extremal value and measure for the real length of a monic polynomial’s unit lemniscate.

Details and sources

AI contribution

The initial main argument was generated in a GPT-5.5 Pro solver–verifier framework, then substantially revised, checked, and packaged by the human authors.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked manuscript; official record open

Verification artifacts

Publication

Public manuscript, exact symbolic checks, and interval-arithmetic certificates

Preprint / manuscript

Activity evidence

A problem originating with Erdős, Herzog, and Piranian in 1958, with historic work and especially intense expert activity in 2025–26.

The manuscript includes exact symbolic checks and interval-arithmetic certificates, but the official Erdős Problems record still listed #1038 as open when checked on 25 July 2026.

Preprint and model record

Candidate exact determination

Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author-checked manuscript; official record open
Open for: 68 years
Research activity*: 4/5

24 Feb 2026Matrix analysis

Ran–Teng Conjecture 20

The exact nonreal spectral region is determined for a four-cycle family of row-stochastic nonnegative matrices, resolving Conjecture 20 of Ran and Teng.

Details and sources

AI contribution

Seven model threads generated candidate reductions and proof components; the human authors selected, corrected, and closed the argument.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human-checked mathematical proof

Publication

Detailed arXiv preprint; no formal proof assistant artifact located

Preprint / manuscript

This is a human–AI collaboration rather than a one-shot autonomous solution. The paper explicitly separates the model’s useful structural ideas from gaps repaired during correctness review.

Resolved in preprint

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.2 Thinking
Verification: Human-checked mathematical proof

24 Mar 2026Polynomial analysis

Erdős Problem #1153

Problem statement

For arbitrary interpolation nodes in

[-1,1]

, must the Lebesgue function on every fixed subinterval attain at least

(2/\pi-o(1))\log n

A sixty-five-year-old problem of Erdős and Turán concerning polynomials was resolved through sustained human–AI collaboration.

Details and sources

AI contribution

Program search, language models, and human mathematicians contributed complementary experimental and proof components.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Community-accepted proof

Publication

Public problem record and collaboration history

Preprint / manuscript

Activity evidence

1 cited source record and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This is a full resolution, but the provenance is distributed across tools and participants rather than a single model output.

Fully resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AlphaEvolve + Claude + Gemini Pro + GPT-5.2/5.4
Verification: Community-accepted proof
Open for: 65 years
Research activity*: 1/5

20 Jan 2026Analysis

Effective Brascamp–Lieb inequalities

The Numina-Lean-Agent paper reports a successful Lean formalization of a Brascamp–Lieb theorem through interactive human–agent proof engineering.

Details and sources

AI contribution

A general coding agent interacted with Lean, retrieval tools, and human experts to build the formal development.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Paper report; complete source not located

Publication

Public system paper and agent repository; no complete Brascamp–Lieb source located in the linked repository

Preprint / manuscript

This is an author-reported autoformalization milestone for an established theorem, not a newly discovered mathematical resolution. The public paper’s displayed Lean excerpt still ends in `sorry`, so this index does not label it independently verified.

Agent and proofs

Author-reported formalization

Problem origin: Origin not yet traced
System: Numina-Lean-Agent / Claude Opus 4.5
Verification: Paper report; complete source not located

9 Apr 2026Analysis, Discrepancy

Erdős Problem #987

Problem statement

For an infinite sequence

x_j\in(0,1)

, must the limsup exponential-sum amplitudes

A_k

be unbounded as

k\to\infty

? Can one at least have

A_k=o(k)

The problem was proved after remaining open for 62 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

3 cited source records and 7 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Erdős Problems site confirmed
Open for: 62 years
Research activity*: 2/5

9 Apr 2026Analysis

Erdős Problem #990

Problem statement

Can the angular discrepancy of the roots of a sparse complex polynomial be bounded by a constant times

\sqrt{n\log M}

, where

n

is its number of nonzero coefficients and

M

its normalized coefficient mass?

The problem was disproved after remaining open for 62 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Lean checked
Open for: 62 years
Research activity*: 1/5

31 Mar 2026Analysis, Discrepancy, Primes

Erdős Problem #997

Problem statement

For every real

\alpha

, is the sequence of fractional parts

\{\alpha p_n\}

along the primes necessarily not well-distributed in the strong sliding-window sense?

The problem was proved after remaining open for 62 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Lean checked
Open for: 62 years
Research activity*: 1/5

19 Apr 2026Analysis, Number Theory

Erdős Problem #1195

Problem statement

Let

S\subset\mathbb R

have infinite measure and suppose

x/y

is never an integer for distinct

x,y\in S

. How fast can

|S\cap(0,x)|

tend to infinity?

The problem was resolved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Independent constructions by Haight and Szemerédi, followed by a quantitative question.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Erdős Problems site confirmed
Open for: 46 years
Research activity*: 3/5

21 Jun 2026Analysis

Erdős Problem #1197

Problem statement

For a positive-measure set

E\subset(0,\infty)

, let

E'=\bigcup_{r\geq1}rE

. Is it true that for almost every

x

there is

M(x)

such that

nx\in E'

for every integer

n>M(x)

The problem was disproved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, Claude Opus 4.7, GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Specialist Haight question with limited documented follow-up.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Aristotle, Claude Opus 4.7, GPT-5.4 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

Combinatorics

8 Sep 2026Induced subgraphs and chromatic number

Improved exponential coloring bound for $P_t$ -free graphs

For every $t\ge5$ , each $P_t$ -free graph satisfies $\chi(G)<c_t\lambda_t^{\omega(G)-1}$ , where $\lambda_t=(t-2+\sqrt{t(t-4)})/2<t-2$ and $c_t=\sqrt{1+4/(t(t-4))}$ .

Details and sources

AI contribution

Sang-il Oum states that Claude Fable 5.1 found the refinement of the classical Gyárfás path argument.

Problem origin

The bound belongs to Gyárfás's longstanding human-origin $\chi$ -boundedness program for induced-path-free graphs.

Verification

Author-checked preprint; external review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

This lowers the best general exponential base. It does not settle whether $P_t$ -free graphs are polynomially $\chi$ -bounded for $t\ge6$ .

Exponential base improved; polynomial bound remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5.1
Verification: Author-checked preprint; external review pending

7 Sep 2026Extremal hypergraph theory

Kalai's conjecture for tight trees

If $T$ is an $r$ -uniform tight tree with $t$ edges and $H$ is $T$ -free, then $|E(H)|\le (t-1)|\partial H|/r$ . The bound is tight infinitely often.

Details and sources

AI contribution

Mubayi and Verstraëte prompted GPT-6 Astra first on tight paths and then on the full conjecture. They checked the proof, rewrote much of the model's text, and added explanations.

Problem origin

Kalai's extremal conjecture for uniform tight trees is a longstanding human-origin hypergraph problem whose graph case is the Erdős--Sós conjecture.

Verification

Two-author checked preprint; external review pending

Publication

Public arXiv proof manuscript with detailed AI declaration

Preprint / manuscript

Ancillary Lean and verification files

The paper proves Kalai's stated edge--shadow inequality. A stronger copy-count conjecture of the authors remains open.

Claimed full proof; stronger copy-count conjecture open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra
Verification: Two-author checked preprint; external review pending

1 Sep 2026Graph domination

Teschner's bondage-number conjecture

A connected cubic bipartite graph on 18 vertices has domination number $6$ and bondage number $5$ , contradicting the conjectured upper bound $3\Delta/2=4.5$ .

Details and sources

AI contribution

Yousof Yavari says GPT-5.6 Sol Max generated the counterexample after being prompted to solve the question. He revised and expanded the proof and verifier; Eric Hou independently checked the proof.

Problem origin

The bound $b(G)\le 3\Delta(G)/2$ was conjectured by Teschner in 1995 and studied subsequently as Teschner's conjecture.

Verification

Exact exhaustive verifier and named independent check

Publication

Public arXiv proof with complete Python verifier

Preprint / manuscript

Preprint and verifier

The appendix deterministically enumerates 297 minimum dominating sets and all 17,550 four-edge deletions. This is exact finite verification, not Lean formalization or peer review.

Conjecture disproved by exact counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol Max
Verification: Exact exhaustive verifier and named independent check

4 Sep 2026Combinatorial game theory

Pure periodicity in three-move subtraction games

For primitive three-move sets $\{a,b,c\}$ , the paper gives an explicit sufficient test for pure periodicity and, when it holds, determines the least period, all P-positions, and all nim-values; it proves the proposed necessity only in the $a+b$ -period range with $c\ge2(a+b)$ .

Details and sources

AI contribution

Hikaru Manabe credits Claude Fable 5 with numerical exploration, verification scripts, ancillary Lean formalization, and drafting, and GPT-5.6 Sol with adversarial review. The author says the principal mathematical ideas and analysis are human and checked every output.

Problem origin

The work belongs to the longstanding classification program for periods and pre-periods of three-move subtraction games, including questions of Guy and conjectures of Flammenkamp and Ward.

Verification

Author-checked proof with documented ancillary Lean coverage

Publication

Public arXiv preprint with frozen ancillary Lean and scripts

Preprint / manuscript

Independent reconstruction and exact checks

The sufficient direction and explicit classifications are proved. The paper itself labels necessity as a conjecture outside the stated $a+b$ -period range, so the general three-move classification remains open.

Explicit sufficient criterion; necessity remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5 + GPT-5.6 Sol
Verification: Author-checked proof with documented ancillary Lean coverage

18 Aug 2026Combinatorics

Albertson–Berman induced-forest conjecture

An explicit 31-vertex plane triangulation has maximum induced-forest order 15. Gluing copies gives infinitely many 31 $k$ -vertex maximal planar graphs with largest induced forest of order 15 $k$ , disproving the one-half conjecture.

Details and sources

AI contribution

Heejae Jung reports that GPT-5.6 Sol found the decisive 14-vertex two-terminal gadget and supplied substantial proof strategy. The author selected the problem, directed the search, checked the mathematics and certificates, and completed the paper and verifier.

Problem origin

The paper traces the conjecture that every planar graph has an induced forest on at least half its vertices to Albertson and Berman's 1979 work.

Verification

31-vertex certificate independently reconstructed; two exact optimizers agree

Publication

Public research preprint with ancillary Python verifier

Preprint / manuscript

AI-assisted preprint and ancillary verifier

VibeMathed independently rebuilt the finite graph and obtained maximum 15 using both integer programming and branch-and-bound, then exhaustively checked the gadget profile. A concurrent human-only construction was also posted independently. A later human-only preprint improves the smallest known counterexample to 29 vertices and the asymptotic upper ratio to 25/52, so the AI-assisted paper's 31 vertices and 15/31 ratio are not current optima. These later improvements do not undo the disproof.

Original problem record Later human-only 29-vertex improvement Concurrent independent human counterexample

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: 31-vertex certificate independently reconstructed; two exact optimizers agree

6 Sep 2026Combinatorics

Optimality of Ho's added-vector code in the 19-dimensional kissing construction

Every minimum-distance-five subset of the fixed 4,096-word ambient code in Ho's construction has at most 1,280 words, matching Ho's construction and settling Conjecture 16 for that code.

Details and sources

AI contribution

Kolosov credits ChatGPT with literature and optimization search, finding the certificate, developing and checking the proof and verifier, and drafting and translation.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Exact certificate independently replayed by VibeMathed

Preprint and exact certificate

Publication

Public preprint and exact verification repository

Preprint / manuscript

No public preprint or manuscript located.

This is an exact restricted-code result. It neither determines the unrestricted kissing number $k(19)$ nor improves the known 11,948-point construction. Independent expert review remains pending.

Ho's source construction Independent certificate replay

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT (OpenAI; version unspecified)
Verification: Exact certificate independently replayed by VibeMathed

5 Sep 2026Combinatorics

$\gamma$ -- $\theta$ conjecture in eternal domination

The complement of the 243-vertex ternary Golay graph satisfies $\gamma(G)=\gamma^{\infty}(G)=3<\theta(G)$ , giving an explicit counterexample in the one-guard-moves model.

Details and sources

AI contribution

An autonomous model run identified the classical Golay graph complement, developed the eternal-defense argument, and wrote the Lean proof; Tom Adamczewski published the artifacts.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Lean-checked construction; statement fidelity not independently audited

Lean proof and recorded audit

Claim audit

The statement was AI-autoformalized; the Python enumeration was not independently replayed and no domination-theory specialist has reviewed the result.

Publication

Public Lean disproof repository

Preprint / manuscript

No public preprint or manuscript located.

The repository reports a clean Lean proof with only standard foundational axioms. The classical graph is prior work; the asserted novelty is recognizing and proving that its complement refutes the conjecture.

Original Question 7.1 Independent repository audit

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Lean-checked construction; statement fidelity not independently audited
Claim audit: Issue documented

4 Sep 2026Combinatorics

Cycle-residue stability at minimum degree five

The 82-page manuscript claims an exact block classification of finite graphs of minimum degree at least five whose cycle lengths omit a residue modulo five, closing the $k=5$ case left by Luo, Ma, and Zhao.

Details and sources

AI contribution

Elias Botsford reports substantial model assistance in proof development and auditing, repairing intermediate arguments, replacement lemmas, and finite computations; the author reviewed the result.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Public proof and computational supplement; independent review pending

Computational supplement

Claim audit

An earlier version was withdrawn for errors. The replacement depends on the same author's unreviewed Dean--5 theorem, and its 23-operation computational supplement was not independently replayed.

Publication

Zenodo preprint and code archive

Preprint / manuscript

Replacement proof manuscript

Retained as a provisional claimed resolution, not as verified fact. The classification is stronger than merely showing which residue can be absent.

Dependency and withdrawal audit

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol and GPT-6 Astra
Verification: Public proof and computational supplement; independent review pending
Claim audit: Issue documented

2 Sep 2026Combinatorics

Daykin--Frankl conjecture for convex subsets of the Boolean lattice

Every convex $P\subseteq Q_n$ has width at least $|P|\binom{n}{\lfloor n/2\rfloor}2^{-n}$ ; the note proves a stronger product inequality for $P\times Q_k$ .

Details and sources

AI contribution

Kada Williams explicitly describes the proof as model-generated, then verified and communicated it in a four-page note.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This is an author-checked conventional proof of the 1983 conjecture. No formalization, full transcript, or independent review was located.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro
Verification: Author-checked proof; independent review pending

2 Sep 2026Combinatorics

Logarithmic basis number of graphs

Every finite $n$ -vertex multigraph has basis number $O(\log n)$ , with the sharper $O(\log\beta(G))$ cycle-rank bound and optimal logarithmic dependence on Euler genus.

Details and sources

AI contribution

Kolja Knauer says the proof was found with the model's help; it also assisted strategy, literature search, drafting, and revision. The author checked the result and references.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

Original question Conjecture 12

This resolves the 2024 simple-graph question and the subsequent conjecture, improving the previous $O(\log^2 n)$ general bound. No independent review or formalization was found.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

12 Aug 2026 revisionCombinatorics

Tu--Deng conjecture

For $N=2^k-1$ and every $1\leq t<N$ , at most $2^{k-1}$ pairs $(a,b)$ with $a+b\equiv t\pmod N$ have total binary Hamming weight below $k$ .

Details and sources

AI contribution

Liu, Luo, and Xie thank the model for assistance with some mathematical work and state that they subsequently verified the argument.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Public end-to-end Lean development; independent statement-fidelity audit pending

Lean formalization Independent concurrent proof

Publication

Public research preprint

Preprint / manuscript

Earlier almost-all result

The authors describe the repository as formalizing the main intermediate results and the reduction to the original modular-counting statement. A concurrent independent human proof by Thomas Cusick appeared in August. The public Lean project was not rebuilt in this audit.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro
Verification: Public end-to-end Lean development; independent statement-fidelity audit pending

2 Sep 2026Combinatorics

Canonical dyadic candidate for Erdős Problem #197

The canonical alternating-dyadic-block set admits no order-type- $\omega$ permutation avoiding monotone three-term arithmetic progressions, so the dyadic partition cannot solve the two-set permutation problem.

Details and sources

AI contribution

Claude assisted proof auditing, code, and editorial revision, while SAT solvers drove experiments and discovery. The final universal theorem is a human-readable, solver-independent proof checked by the author.

Problem origin

Davis, Entringer, Graham, and Simmons asked whether two admissibly permutable sets suffice; it is catalogued as Erdős Problem #197 and has a separate Lean statement formalization.

Verification

Author proof with pinned reproducible SAT/DRAT corroboration

Pinned proof-checking and certificate repository

Publication

Public research preprint

Preprint / manuscript

Erdős Problem #197 Lean statement formalization

Problem #197 itself remains open. The repository has DRAT certificates for two finite C3-core instances and reproducible checks for others; these corroborate but do not prove the universal theorem. Jesse Geneson corrected two preliminary statements but did not verify the main theorem.

Proved partial advance; general problem remains open

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Claude (Anthropic; version unspecified), CaDiCaL, and OR-Tools
Verification: Author proof with pinned reproducible SAT/DRAT corroboration

1 Sep 2026Combinatorics

Erdős Problem #74 — locally nearly bipartite graphs

Epoch reports that some diverging function $f(n)$ forces a graph to be 3-colorable whenever every finite $n$ -vertex subgraph becomes bipartite after deleting at most $f(n)$ edges, refuting the conjectured arbitrarily slow local obstruction.

Details and sources

AI contribution

Epoch reports autonomous model work in its fixed mathematical agent harness. Thomas Bloom reviewed the informal targets and translations; public human explanations of the resulting proofs are still being developed.

Problem origin

Adamczewski and Bloom identify this numbered, previously open Erdős problem in Appendix B of FrontierMath: Erdős. It was not generated by the model.

Verification

Public Lean/Comparator proof independently replayed; canonical tracker accepted

Publication

Public formal proof and FrontierMath: Erdős report

Preprint / manuscript

Benchmark announcement and experimental scope Public statements and benchmark harness Canonical Erdős Problems record Independent build and statement audit

One of the two successes in the 68-target benchmark. Existence of a suitable function is the formal claim; the suggested explicit asymptotic rate in the exposition is not part of the certified theorem. The canonical tracker now accepts the Lean disproof. The public audit reports 11,481 Lean lines and six successful resolutions; the certified conclusion is the stronger 3-colorability statement, not the heuristic rate discussed in the exposition. Independent builds found zero sorry outside statement stubs, no added axioms, unsafe features, or native_decide, and only Lean's standard foundational axioms. Formal statements for 68 targets do not mean 68 solved problems.

Claimed full resolution; disproof reported

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Public Lean/Comparator proof independently replayed; canonical tracker accepted

1 Sep 2026Combinatorics

Erdős Problem #548 — Erdős–Sós tree embedding conjecture

Epoch reports that every $n$ -vertex graph with more than $(k-2)n/2$ edges contains every $k$ -vertex tree, for $n\geq k$ .

Details and sources

AI contribution

Problem origin

Adamczewski and Bloom identify this numbered, previously open Erdős problem in Appendix B of FrontierMath: Erdős. It was not generated by the model.

Verification

Public Lean/Comparator proof independently replayed; canonical tracker accepted

Publication

Public formal proof and FrontierMath: Erdős report

Preprint / manuscript

Benchmark announcement and experimental scope Public statements and benchmark harness Canonical Erdős Problems record Independent build and statement audit

An additional, enlarged-budget experiment, outside the benchmark's two successful runs. The claimed scope is the full tree-embedding conjecture. The canonical tracker now accepts the Lean proof; specialist exposition and review remain pending. The public audit reports 1,311 Lean lines and documents resolution of the parity-margin statement comparison before canonical acceptance. Independent builds found zero sorry outside statement stubs, no added axioms, unsafe features, or native_decide, and only Lean's standard foundational axioms. Formal statements for 68 targets do not mean 68 solved problems.

Claimed full resolution; proof reported

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Public Lean/Comparator proof independently replayed; canonical tracker accepted

1 Sep 2026Combinatorics

Erdős Problem #571 — rational Turán exponents

Epoch reports that every rational $\alpha\in[1,2)$ occurs as a Turán exponent of a bipartite graph: $\operatorname{ex}(n,G)\asymp n^\alpha$ for some $G$ .

Details and sources

AI contribution

Problem origin

Adamczewski and Bloom identify this numbered, previously open Erdős problem in Appendix B of FrontierMath: Erdős. It was not generated by the model.

Verification

Public Lean/Comparator proof independently replayed; canonical tracker accepted

Publication

Public formal proof and FrontierMath: Erdős report

Preprint / manuscript

Benchmark announcement and experimental scope Public statements and benchmark harness Canonical Erdős Problems record Independent build and statement audit

An additional, enlarged-budget experiment. A public Lean repository is now independently replayable, Thomas Bloom reviewed the statement and supplied a proof exposition, and the canonical Erdős Problems tracker marks the problem PROVED (LEAN). The public audit reports 10,460 Lean lines; Thomas Bloom reviewed the statement and supplied a proof exposition. Independent builds found zero sorry outside statement stubs, no added axioms, unsafe features, or native_decide, and only Lean's standard foundational axioms. Formal statements for 68 targets do not mean 68 solved problems.

Claimed full resolution; proof reported

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Public Lean/Comparator proof independently replayed; canonical tracker accepted

27 Aug 2026Combinatorics

Non-Cancelling Intersections Conjecture

A finite lattice provides a counterexample even with unrestricted dot-algebra expression trees, removing the earlier left-linear restriction.

Details and sources

AI contribution

Hermann Wilhelm conceived the concepts and strategy; Claude expanded the author's notes and proof sketch into detailed proofs that the author checked.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The construction works for every prime $p\geq10^5$ (100,000), not 105. The August 31 revision adds a comparison with an independent equivalent result by A. Walz. The disclosure's word 'formalize' refers to developing written proofs; no proof-assistant certificate is claimed here.

Original conjecture

Claimed resolution in a public preprint

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Claude (version unspecified)
Verification: Author-prepared proof; independent review pending

27 Aug 2026Combinatorics

Spielman–Teng eigenvalue conjecture for bounded-genus graphs

For an $n$ -vertex graph of orientable genus $g\geq1$ and maximum degree $\Delta$ , $\lambda_2(L_G)\lesssim\Delta g/n$ . The paper also improves minor-free bounds to $O(\Delta h^2(\log h)^2/n)$ .

Details and sources

AI contribution

GPT-5.5 independently produced two proofs of a limited second-eigenvalue result. Kolbe and Spalding-Jamieson generalized and simplified it; GPT-5.6 contributed a technical formulation.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The genus conjecture is resolved in the claimed theorem. The minor-free estimate retains logarithmic factors and is not a full sharp resolution. The authors acknowledge independent work by Zhan and Zhou.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.5 extra high and GPT 5.6 medium
Verification: Author-prepared proof; independent review pending

22 Aug 2026Combinatorics

Chouinard's finiteness conjecture for graphical uniform designs

For fixed index, only finitely many nontrivial graphical $t$ -designs with $t>1$ exist; the theorem permits repeated blocks and gives explicit polynomial parameter bounds.

Details and sources

AI contribution

Yeow Meng Chee attributes the results and proofs to himself. Claude suggested proof simplifications and checked constants in addition to exposition and routine calculations.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This covers equal-size blocks. Chouinard's full conjecture for arbitrary graphical t-wise balanced designs is not claimed solved. AI's documented role is supporting proof refinement and checking.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude (version unspecified)
Verification: Author-prepared proof; independent review pending

21 Aug 2026Combinatorics

Henning–Yeo identifying vertex cover conjecture

Explicit connected graphs refute the proposed degree-size-order upper bound. At every fixed maximum degree at least four, the additive violation is unbounded; the normalized extremal gap is asymptotic to the reciprocal degree.

Details and sources

AI contribution

The author discloses extensive AI-assisted construction searches, proof suggestions, code, and drafting, followed by independent reconstruction and checking.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

Backfilled from the August 21 revision, which the previous audit missed. This refutes the cited 2012 bound rather than every possible estimate for identifying vertex covers.

Claimed resolution in a public preprint

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Generative AI; model names not disclosed
Verification: Author-prepared proof; independent review pending

24 Aug 2026Combinatorics

Smaller finite-field Kakeya sets for primes congruent to three modulo four

For primes $p\equiv3\pmod4$ , a new family in $\mathbb F_p^3$ has $(2p^3+7p^2+3)/8$ points, improving the prior family by $(p-3)/4$ points.

Details and sources

AI contribution

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public proofs and exact verification artifacts; independent review pending

Publication

Public research preprint

Preprint / manuscript

The leading asymptotic constant was already known. The analogous family for primes congruent to one modulo four is explicitly identified as a rediscovery.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Station agents using OpenAI, Anthropic, and Google models; see per-run artifacts
Verification: Public proofs and exact verification artifacts; independent review pending

24 Aug 2026Combinatorics

New infinite families for adjacent book Ramsey numbers

Three construction families prove $R(B_{n-1},B_n)=4n-1$ at infinitely many parameters, covering 43 values through 200, including 28 previously open cases in the paper's comparison.

Details and sources

AI contribution

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public proofs and exact verification artifacts; independent review pending

Publication

Public research preprint

Preprint / manuscript

Two families are attributed to autonomous agents; an external human expert derived the third from their constructions and an earlier identity. The all-parameter conjecture remains open; the 28 finite cases are not inflated into 28 independent discoveries.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Station agents using OpenAI, Anthropic, and Google models; see per-run artifacts
Verification: Public proofs and exact verification artifacts; independent review pending

19 Aug 2026Structural graph theory and coloring

Linear versus centered coloring

Problem statement

Is every graph's centered chromatic number at most twice its linear chromatic number?

There is a family $G_k$ with $\chi_{\mathrm{lin}}(G_k)\leq k$ but $\chi_{\mathrm{cen}}(G_k)=\Omega(k^2/\log k)$ , giving a superlinear separation.

Details and sources

AI contribution

The authors state that the model found the construction. They checked the proof, developed the presentation, and accept responsibility for the result.

Problem origin

Kun, O'Brien, Pilipczuk, and Sullivan conjectured that centered chromatic number is at most twice linear chromatic number.

Verification

Author-checked complete counterexample proof; independent review pending

Publication

Public counterexample preprint

Preprint / manuscript

Original coloring-conjecture paper

Activity evidence

A documented editorial estimate based on disproving a named structural graph-theory conjecture with an asymptotically superlinear family.

The construction disproves the proposed universal linear bound and is nearly tight for the chordal-graph setting discussed in the paper. Determining the optimal general separation remains open.

Kun--O'Brien--Pilipczuk--Sullivan linear-bound conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro (OpenAI)
Verification: Author-checked complete counterexample proof; independent review pending
Open for: Kun--O'Brien--Pilipczuk--Sullivan conjecture (2020)
Research activity*: 4/5

17 Aug 2026Extremal graph theory and the hard-core model

Maximum versus average independent-set size in triangle-free graphs

Problem statement

Must every triangle-free graph of minimum degree

d

satisfy

\alpha(G)/\overline{\alpha}_G(1)\geq2-o_d(1)

The triangle-free $(m+2)$ -regular graphs $C_5\,\square\,K_{m,m}$ satisfy $\alpha(G_m)/\overline{\alpha}_{G_m}(1)\to24/13<2$ , contradicting the proposed $2-o_d(1)$ lower bound.

Details and sources

AI contribution

FAR recovered and status-checked the conjecture from the literature. A single GPT-5.5 xhigh opencode run found the product-graph counterexample and exact asymptotic count; automated judging preceded expert review.

Problem origin

The asymptotic ratio conjecture was stated in the 2018 Davies--Jenssen--Perkins--Roberts paper and remained open in later surveys.

Verification

Checked by a paper author or domain expert; public complete proof; independent peer review pending

Publication

Public counterexample proof in the reviewed-solutions appendix

Preprint / manuscript

FAR pipeline and released data Original conjecture paper

Activity evidence

A documented editorial estimate based on the connection to hard-core-model estimates and consequences proposed for Ramsey bounds.

This refutes Conjecture 2 and its fixed- $r$ minimum-degree analogue. It does not refute the separate $4/3$ conjecture, the general-fugacity conjecture, or the sparse-independent-set regime relevant to the sharp Ramsey constant.

Davies--Jenssen--Perkins--Roberts Conjecture 2 disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 xhigh in the FAR pipeline (OpenAI)
Verification: Checked by a paper author or domain expert; public complete proof; independent peer review pending
Open for: Davies--Jenssen--Perkins--Roberts Conjecture 2, restated in 2025--26 surveys
Research activity*: 4/5

17 Aug 2026Finite-field incidence geometry

Small unions of non-flat line families over finite fields

Problem statement

Must a sufficiently large, non-flat family of affine lines in

\mathbb F_q^3

cover all but

o(q^3)

points?

For every odd prime power $q$ , a family of $q^2(q+1)/2$ affine lines in $\mathbb F_q^3$ , with at most $q+1$ in any plane, has a union of exactly $q^2(q+1)/2$ points, disproving both proposed near-full-union bounds.

Details and sources

AI contribution

A GPT-5.5 xhigh FAR run connected the conjectures to a half-tangent family of an elliptic quadric and returned the counterexample. Automated judging preceded author review and exhaustive checks for $q\leq13$ .

Problem origin

Lund, Saraf, and Wolf posed the two line-union conjectures in their work on three-dimensional Kakeya and Nikodym sets.

Verification

Author-checked proof plus exhaustive small-field counts; independent peer review pending

Publication

Public counterexample proof in the reviewed-solutions appendix

Preprint / manuscript

FAR pipeline and released data Original conjecture paper

Activity evidence

A documented editorial estimate based on closing a proposed route to the optimal three-dimensional Nikodym bound while leaving the target conjecture open.

The half-tangent construction was already known in finite geometry; the new contribution claimed by the authors is linking it to these conjectures. The three-dimensional Nikodym conjecture itself remains open.

Lund--Saraf--Wolf Conjectures 1.4 and 1.5 disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 xhigh in the FAR pipeline (OpenAI)
Verification: Author-checked proof plus exhaustive small-field counts; independent peer review pending
Open for: Lund--Saraf--Wolf Conjectures 1.4 and 1.5; Nikodym consequence remains open
Research activity*: 4/5

18 Aug 2026Extremal set systems and hypergraph Turán theory

Anstee--Sali conjecture for forbidden configurations

Problem statement

\operatorname{forb}(m,\mathcal F)=\Theta(m^{X(\mathcal F)-1})

for every fixed forbidden configuration

\mathcal F

For $\mathcal F_2=\{xyab,xybc,xycd,xyda\}$ one has $X(\mathcal F_2)=4$ , so the conjecture predicts $\operatorname{forb}(m,\mathcal F_2)=\Theta(m^3)$ , while Mubayi's construction gives $\Omega(m^{7/2})$ .

Details and sources

AI contribution

The author states that GPT-5.6 Sol helped identify the six-vertex example and assisted with exposition. The author independently reviewed and revised the output and verified every mathematical claim.

Problem origin

Anstee and Sali formulated the forbidden-configuration growth conjecture in the human extremal-set-systems literature.

Verification

Author-verified explicit certificate and construction; independent review pending

Publication

Public revised counterexample preprint

Preprint / manuscript

Anstee--Sali survey and conjecture source

Activity evidence

A documented editorial estimate based on a compact counterexample to a broad asymptotic conjecture in forbidden-configuration theory.

The contradiction combines a direct computation of $X(\mathcal F_2)$ with an explicit specialization of Mubayi's established complete-multipartite hypergraph construction.

Conjecture disproved by a six-vertex family

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol (OpenAI)
Verification: Author-verified explicit certificate and construction; independent review pending
Open for: Anstee--Sali conjecture
Research activity*: 4/5

11 Aug 2026Finite Ramsey numbers for cycles versus stars

Exact cycle-versus-star Ramsey number $R(C_4,K_{1,39})$

Problem statement

R(C_4,K_{1,39})

equal to

46

47

The repository claims $R(C_4,K_{1,39})=46$ . It supplies a 45-vertex lower-bound witness and an exact spectral certificate excluding the remaining 46-vertex case, with independent exact-arithmetic checkers.

Details and sources

AI contribution

The repository discloses that Claude developed the structure theory, proofs, witness constructions, and verification code. Parallel GPT-5.6 sessions identified the decisive fifth-moment constraint, independently derived a key identity, and produced independent verifiers. The human repository owner directed the work and accepts responsibility.

Problem origin

The unresolved value $R(C_4,K_{1,39})\in\{46,47\}$ was recorded in the human-authored dynamic survey of small Ramsey numbers before this AI-assisted work.

Verification

Exact public checkers and independent project re-audit; no human peer review

Exact proof and verification repository Independent reconstruction and audit note

Publication

Public proof repository with exact-arithmetic certificates and reproducible verifiers

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A documented editorial estimate based on closing a published finite Ramsey-table gap, the exact certificate bundle, and the absence of independent human review.

This resolves one finite value within the $C_4$ -versus-star Ramsey problem, not the asymptotic form of Erdős Problem #552. The repository labels the result a preprint seeking independent review and explicitly says it is not proof-assistant formalized or human peer reviewed.

Classical Ramsey-number survey and open-value table Erdős Problem #552 community record

Previously two-valued finite Ramsey case resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude (Anthropic) and GPT-5.6 (OpenAI)
Verification: Exact public checkers and independent project re-audit; no human peer review
Open for: Listed as one of two possible values in the classical survey
Research activity*: 3/5

11 Aug 2026Finite Ramsey numbers for cycles versus stars

Exact cycle-versus-star Ramsey number $R(C_4,K_{1,51})$

Problem statement

R(C_4,K_{1,51})

equal to

59

60

The repository claims $R(C_4,K_{1,51})=59$ . It supplies a 58-vertex lower-bound witness and an exact argument excluding the remaining 59-vertex case, including deficiency-graph and cyclotomic computations checked by public exact-arithmetic scripts.

Details and sources

AI contribution

The repository discloses that Claude developed the structure theory, proofs, witness constructions, and verification code. Parallel GPT-5.6 sessions contributed independent derivations and verifier implementations. The human repository owner directed the work and accepts responsibility.

Problem origin

The unresolved value $R(C_4,K_{1,51})\in\{59,60\}$ was recorded in the human-authored dynamic survey of small Ramsey numbers before this AI-assisted work.

Verification

Exact public checker bundle; independent human review pending

Exact proof and verification repository

Publication

Public proof repository with exact-arithmetic certificates and reproducible verifiers

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A documented editorial estimate based on closing a published finite Ramsey-table gap, the exact certificate bundle, and the absence of independent human review.

Classical Ramsey-number survey and open-value table Erdős Problem #552 community record

Previously two-valued finite Ramsey case resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude (Anthropic) and GPT-5.6 (OpenAI)
Verification: Exact public checker bundle; independent human review pending
Open for: Listed as one of two possible values in the classical survey
Research activity*: 3/5

10 Aug 2026Graph product structure and pathwidth

Row pathwidth of complete binary trees

Problem statement

Does the row pathwidth of the complete binary tree of height

h

grow linearly with

h

For the complete binary tree $T_h$ of height $h$ , every representation $T_h\subseteq H\boxtimes P$ satisfies $\operatorname{pw}(H)=\Omega(h)$ . The proof gives the explicit bound $\operatorname{rpw}(T_h)\geq\lfloor(h+1)/16\rfloor$ .

Details and sources

AI contribution

Hodor and Micek state that OpenAI's GPT-5.6 Sol Pro found the proof. The authors present the argument and take responsibility for its mathematical correctness.

Problem origin

Bose, Dujmović, Javarsineh, Morin, and Wood posed the question in their 2022 paper on layered and row width parameters.

Verification

Two-author checked proof preprint; external review pending

Publication

Public four-page proof preprint

Preprint / manuscript

Original 2022 problem source

Activity evidence

A documented editorial estimate based on the role of product-structure parameters in graph layout and the explicit four-year-old problem statement.

The result supplies the missing linear lower bound for complete binary trees. It concerns row pathwidth, not the already-settled separation between layered pathwidth and row pathwidth for general trees.

2022 row-pathwidth problem solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro
Verification: Two-author checked proof preprint; external review pending
Open for: Problem posed in the 2022 product-structure literature
Research activity*: 3/5

11 Aug 2026Euclidean Ramsey theory

Nearcircumsphere-Ramsey property for solvable transitive configurations

Problem statement

Which finite spherical configurations are nearcircumsphere-Ramsey, and in particular does the property hold beyond solvable transitive configurations?

Every finite spherical configuration whose isometry group has a solvable transitive action is nearcircumsphere-Ramsey: for every number of colors and every $\varepsilon>0$ , a sufficiently high-dimensional sphere of radius at most $\rho+\varepsilon$ contains a monochromatic congruent copy.

Details and sources

AI contribution

Pálvölgyi reports heavy ChatGPT use for brainstorming, checking failed approaches, exposition, and references. The model contributed significantly to the Kneser-shift theorem and improved bounds, but the author explicitly says the main idea behind the nearcircumsphere-Ramsey theorem was human-origin.

Problem origin

The paper places the result within the human Euclidean-Ramsey literature and describes it as progress toward Graham's radius conjecture and the Leader–Russell–Walters characterization.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint with a scoped AI-use statement

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the long-running Euclidean Ramsey program and the restricted scope of the theorem.

This is a substantial restricted-case theorem, not a resolution of Graham's full radius conjecture. Removing solvability and extending from transitive to subtransitive configurations remain open.

Solvable-transitive case proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT
Verification: Author-checked proof preprint; external review pending
Open for: Restricted case of Graham's radius program
Research activity*: 3/5

13 Aug 2026Finite posets and random linear extensions

Expected-rank gap–width conjecture for finite posets

Problem statement

Must the largest gap between consecutive expected ranks of a finite poset be at most

2w(P)-1

Every finite nonempty poset $P$ satisfies $\operatorname{gap}(P)\leq2w(P)-1$ , where $w(P)$ is its width. A stronger weighted ideal inequality proves the Aires–Kahn formulation.

Details and sources

AI contribution

Haqi states that ChatGPT 5.6 Sol found the key ideas and theorems. The author wrote the manuscript with Codex assistance; the paper thanks Jeff Kahn and Max Aires for identifying an application of the main inequality.

Problem origin

The conjecture appears in progressively stronger forms in work of Brightwell–Trotter, Biró–Trotter, and Aires–Kahn, before the AI-assisted paper.

Verification

Author-written proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the repeated appearance of the conjecture in the linear-extension literature and the stronger ideal inequality proved.

This is the positive result in a paper that also gives separate negative resolutions of the maximal-chain and hereditary-entropy formulations.

Gap–width conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol / Codex
Verification: Author-written proof preprint; external review pending
Open for: Conjectural in several forms since at least 2002
Research activity*: 4/5

13 Aug 2026Finite posets and random linear extensions

Maximal-chain expected-rank gap question

Problem statement

Does every finite poset contain a maximal chain whose consecutive expected-rank gaps are bounded by a function of the poset's width?

For every $L>0$ , there is a width-two finite poset in which every maximal chain has an expected-rank gap at least $L$ . Thus no bound depending only on width can always be witnessed along a maximal chain.

Details and sources

AI contribution

Problem origin

The maximal-chain question was posed by Aires and Kahn before the AI-assisted construction.

Verification

Author-written proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on a recent specialist question and a sharp width-two obstruction.

Endpoint spacings are included in the chain-gap convention used by the paper. The global gap–width bound remains true and is proved separately in the same manuscript.

Chain-gap question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol / Codex
Verification: Author-written proof preprint; external review pending
Open for: Aires–Kahn Question 12.2
Research activity*: 3/5

13 Aug 2026Finite posets, entropy, and random linear extensions

Expected-rank gaps versus hereditary order entropy

Problem statement

Must every sequence of finite posets with unbounded expected-rank gaps have unbounded hereditary entropy per point?

A sequence of finite posets $P_r$ has hereditary order entropy below $3|X|$ on every nonempty selected set $X$ , while $\operatorname{gap}(P_r)\geq(3/2)^r$ . Hence unbounded gaps do not force unbounded hereditary entropy per point.

Details and sources

AI contribution

Problem origin

The implication from unbounded expected-rank gaps to unbounded hereditary entropy per point was conjectured by Aires and Kahn.

Verification

Author-written proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

TeX source containing the Lean formalization supplement

Activity evidence

A documented editorial estimate based on a recent specialist conjecture connecting order entropy and expected ranks.

The counterexample concerns hereditary entropy of induced relative orders, not the classical $1/3$ – $2/3$ conjecture for balanced pairs.

Hereditary-entropy conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol / Codex
Verification: Author-written proof preprint; external review pending
Open for: Aires–Kahn Conjecture 2.5
Research activity*: 3/5

13 Aug 2026Latin squares and combinatorial design

Minimum imbalance of Latin squares when $n\equiv1\pmod3$

Problem statement

What is the minimum possible imbalance of an

n\times n

Latin square when

n\equiv1\pmod3

Every Latin square with $n\equiv1\pmod3$ has imbalance at least $4n(n-1)/9$ . Near-perfect permutations attain the bound, and their existence is computationally verified for every eligible order $4\leq n\leq52$ .

Details and sources

AI contribution

A Claude Opus 4.5 agent used SageMath, a Rust enumerator, simulated annealing, persistent memory, and multi-model criticism. It noticed the parity structure and drafted the proof after a human-directed change of research question.

Problem origin

Minimum Latin-square imbalance in the residue class $n\equiv1\pmod3$ was left open by prior human work; the decisive reframing in this project was also made by the human researcher.

Verification

Lean-checked universal lower bound; finite matching constructions verified

Publication

Revised public preprint with process logs and formal theorem statement

Preprint / manuscript

Supplementary code, verifier, and data bundle

Activity evidence

A documented editorial estimate based on a general formal lower bound, verified sharp instances through order 52, and the remaining construction problem.

The Lean development covers the universal lower bound. The paper proves tightness through order $52$ and conditionally from a near-perfect permutation, but it does not prove that matching constructions exist for every eligible $n$ ; the unrestricted exact minimum therefore remains open in general.

Universal lower bound formalized; general tightness open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Opus 4.5 with symbolic tools
Verification: Lean-checked universal lower bound; finite matching constructions verified
Open for: Open in the combinatorial-design literature
Research activity*: 4/5

13 Jul 2026Strongly regular graphs and constraint solving

Conway's 99-graph — forced-structure reduction and bounds

Problem statement

Does a strongly regular graph with parameters

\operatorname{srg}(99,14,1,2)

exist?

An autonomous Claude research agent helped prove an exhaustive $68.0\%$ ceiling for circulant candidates, derive a reduction to a $12$ -regular graph on $84$ vertices, and build validated prescribed-automorphism encodings. The full existence question remains unresolved.

Details and sources

AI contribution

The paper says Claude operated as an autonomous research-and-coding agent with shell access, CP-SAT, Z3, NumPy, and SciPy. It derived the reduction, wrote solver and verification code, and repeatedly narrowed overclaims; the human author audited and reported the results.

Problem origin

The existence of a strongly regular graph with parameters $\operatorname{srg}(99,14,1,2)$ is a human-origin problem identified by Biggs and popularized by Conway.

Verification

Conference-accepted; exhaustive computation and public verifier bundle

Publication

CAISc 2026 paper with reproducibility checklist and supplementary artifacts

Preprint / manuscript

Author page and acceptance record CAISc verifiable-problem specification

Activity evidence

A documented editorial estimate based on the long-standing prize problem, the reproducible exact subcase bound, and the remaining unresolved global existence question.

The paper explicitly does not construct the graph or prove non-existence. Its $69.43\%$ best artifact is empirical; the rigorous claims are the exhaustive circulant bound and the validated structural reduction.

Rigorous partial advance; existence remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Anthropic Claude via Claude Code CLI
Verification: Conference-accepted; exhaustive computation and public verifier bundle
Open for: Conway prize problem; full existence question still open
Research activity*: 4/5

12 Aug 2026Hadamard matrices and combinatorial designs

Hadamard matrix of order 668

Problem statement

Does there exist a matrix

H\in\{\pm1\}^{668\times668}

satisfying

HH^{\mathsf T}=668I

A three-person Anthropic research team and Claude reportedly produced a Hadamard matrix of order $668$ as part of a construction covering every previously unknown admissible order below $2000$ .

Details and sources

AI contribution

The public report credits Claude alongside three human researchers. A detailed contribution account has not yet been released, so the division between model-generated construction ideas, search, and human direction remains provisional.

Problem origin

The existence problem is a long-standing human case of the Hadamard conjecture and was independently selected for FrontierMath: Open Problems.

Verification

Provisional FrontierMath classification; public matrix not located

Claim audit

The result is currently announcement-level: the matrix, construction, and contribution breakdown have not been released for independent inspection.

Publication

Public benchmark update and researcher announcement

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A documented editorial estimate based on order 668 being the smallest previously unknown Hadamard order, its inclusion in FrontierMath, and the immediate specialist attention to the reported construction.

This concerns the finite order- $668$ existence case, not the full Hadamard conjecture. Epoch says its AI attribution may be revised, and the announced matrix or an independently runnable certificate was not public in the sources inspected on 12 August.

FrontierMath problem and provisional solution update Researcher announcement FrontierMath changelog

Provisionally reported solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude (version not disclosed)
Verification: Provisional FrontierMath classification; public matrix not located
Claim audit: Issue documented
Open for: 21 years
Research activity*: 5/5

6 Aug 2026Triangle packing and covering

Tuza's conjecture for maximum degree seven

Problem statement

Does every graph of maximum degree at most seven satisfy Tuza's inequality

\tau(G)\le2\nu(G)

Every graph of maximum degree at most seven satisfies $\tau(G)\le2\nu(G)$ , extending the previously covered maximum-degree-at-most-six range.

Details and sources

AI contribution

The authors report extensive model use for proof exploration, exact verification, and software. They developed the discharging and reducible-configuration argument and take responsibility for it.

Problem origin

Tuza's triangle packing-covering conjecture is a human conjecture from extremal graph theory.

Verification

Author proof plus public exact-search code; external review pending

Exact-search and verification code

Publication

Public proof preprint and computational repository

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture's long history and the extensive triangle packing-covering literature.

This is a substantial bounded-degree case, not a proof of Tuza's conjecture for arbitrary graphs.

Maximum-degree-seven case proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Code / OpenAI Codex
Verification: Author proof plus public exact-search code; external review pending
Research activity*: 5/5

4 Aug 2026Phylogenetic tree metrics

Bandelt–Dress maximum quartet-distance conjecture

Problem statement

Is the maximum quartet distance between two binary phylogenetic trees asymptotic to

\frac23\binom n4

The maximum quartet distance between two binary phylogenetic trees on $n$ leaves is $(2/3+o(1))\binom n4$ .

Details and sources

AI contribution

The author states that GPT-5.6 proved the theorem and drafted the initial manuscript. The author assisted the strategy, verified every argument, edited the paper, and takes responsibility.

Problem origin

Bandelt and Dress posed the asymptotic maximum-quartet-distance conjecture in 1986.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture's age and the use of quartet distance in phylogenetic tree comparison.

The theorem determines the asymptotic maximum; it does not give an exact finite- $n$ formula.

1986 conjecture resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6
Verification: Author-checked proof preprint; external review pending
Open for: 40 years
Research activity*: 4/5

4 Aug 2026Degree–diameter problem

Bollobás's asymptotic degree–diameter conjecture

Problem statement

For fixed diameter

k

, can graphs asymptotically attain the Moore bound

d^k

as the maximum degree

d

grows?

For every fixed diameter $k$ , the maximum order $n_k(d)$ of a graph of maximum degree $d$ satisfies $n_k(d)/d^k\to1$ as $d\to\infty$ ; the corresponding edge version is also resolved asymptotically.

Details and sources

AI contribution

The authors directed exploration toward higher-rank buildings. The model suggested splitting complete flags into odd- and even-rank subflags; the authors developed, verified, and wrote the halved-flag construction. Generative tools also assisted the Lean development.

Problem origin

Bollobás posed the asymptotic fixed-diameter form of the degree–diameter problem in the human graph-theory literature.

Verification

Lean-checked central theorem chain and author proof preprint

Lean 4 formalization

Publication

Public proof preprint and Lean 4 formalization

Preprint / manuscript

Frozen candidate repository and Lean development

Activity evidence

A documented editorial estimate based on the conjecture's long history and central role in extremal graph construction.

The Lean repository exports the main theorem, the principal finite construction, and the edge corollary with a documented proof-correspondence boundary.

Conjecture proved for every fixed diameter

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Lean-checked central theorem chain and author proof preprint
Research activity*: 5/5

6 Aug 2026Permanents and matrix inequalities

Dittert's conjecture in dimension five

Problem statement

For every nonnegative

5\times5

matrix with total entry sum five, is Dittert's functional maximized uniquely by the all-

1/5

matrix?

A public candidate certificate proves the sharp $n=5$ inequality and its equality case within an explicit Lean encoding, using a symmetry-reduced sum-of-squares witness.

Details and sources

AI contribution

The AI system selected the problem, recovered the certificate, wrote the exact verifiers and Lean formalization, ran internal audits, and prepared the manuscript package with minimal human mathematical supervision.

Problem origin

Dittert's conjecture is a human conjecture concerning the permanent and row/column sums of nonnegative matrices.

Verification

Lean-checked encoded theorem; no independent specialist validation

Claim audit

No qualified specialist has independently checked novelty, the historical-conjecture-to-Lean encoding, or the paper-to-artifact correspondence.

Publication

Immutable GitHub commit with manuscript, certificate, verifiers, and Lean code

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A documented editorial estimate based on the long-standing permanent inequality and the unusually complete but still unaudited formal artifact.

The repository explicitly says the artifact must not yet be cited as an established solution. Lean uses standard native_decide for large finite checks, and a separate FLINT verifier is public.

Candidate review boundary

Lean-checked candidate; specialist review required

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra via Codex
Verification: Lean-checked encoded theorem; no independent specialist validation
Claim audit: Issue documented
Research activity*: 4/5

5 Aug 2026Extremal and number-theoretic graph theory

Staton's conjecture on the Havel--Hakimi residue of common-divisor graphs

Problem statement

G_n

is the common-divisor graph on

\{2,\ldots,n\}

, is

R(G_n)\sim(\zeta(2)-1)n/\log n

For the graph $G_n$ on $\{2,\ldots,n\}$ joining integers with a nontrivial common divisor, the Havel--Hakimi residue satisfies $R(G_n)=(\zeta(2)-1)n/\log n+(\zeta(2)-1-A)n/\log^2 n+O(n/\log^3 n)$ , proving Staton's conjectured leading term and determining the next term.

Details and sources

AI contribution

The paper describes an advisor-supervised AI loop that turned registry experiments and counterexamples into the theorem and a traceable proof strategy.

Problem origin

The problem is Conjecture 448 of Fajtlowicz's Written on the Wall; the stronger asymptotic was attributed to Staton in the historical notes.

Verification

Author-presented proof preprint; external review pending

Publication

Public proof preprint with provenance discussion

Preprint / manuscript

Readable abstract and provenance

Activity evidence

A documented editorial estimate based on the Graffiti registry history, the Erdős--Staton strengthening, and the connection between graph degree sequences and prime-number asymptotics.

The 31 August v2 determines the complete asymptotic expansion and recovers prime counts from the unlabeled graph. Its Riemann-hypothesis equivalences do not prove RH. The stronger AI-loop conjecture $R(G_n)\leq\lceil\operatorname{CW}(G_n)\rceil+2$ still remains open.

Staton asymptotic conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Theo-Conjecture advisor-supervised AI loop
Verification: Author-presented proof preprint; external review pending
Open for: Conjecture 448 in Written on the Wall; stronger form attributed to Staton
Research activity*: 3/5

5 Aug 2026Graph dynamics and chip-firing

Period two on the middle stair of parallel chip-firing

Problem statement

Must every parallel chip-firing game with

2|E|-|V|<|\sigma|<2|E|

have period

2

and activity

1/2

Every parallel chip-firing game on a graph $G=(V,E)$ with $2|E|-|V|<|\sigma|<2|E|$ has period $2$ , giving activity $1/2$ throughout the middle stair of the devil's staircase.

Details and sources

AI contribution

The authors explicitly report using GPT-5.6 Sol to prove the general conjecture, unifying the previously known graph classes.

Problem origin

Ji, Li, and Wang stated the conjecture in their 2024 paper after proving special cases and citing earlier cases for trees, cycles, and complete graphs.

Verification

Author-presented proof preprint; external review pending

Publication

Public seven-page proof preprint

Preprint / manuscript

Original 2024 conjecture paper

Activity evidence

A documented editorial estimate based on the earlier special-case literature and the conjecture's role as the general middle rung of the chip-firing devil's staircase.

This resolves the 2024 general-graph conjecture. Earlier work had already established trees, cycles, complete graphs, and complete bipartite graphs.

Ji--Li--Wang conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-presented proof preprint; external review pending
Open for: Conjectured by Ji, Li, and Wang in 2024
Research activity*: 3/5

4 Aug 2026Ehrhart theory and lattice polytopes

Large-width lattice polytopes with non-unimodal $h^*$ -vectors

Problem statement

Can a fixed dimension contain an infinite family of lattice polytopes with arbitrarily large width whose Ehrhart

h^*

-vectors are all non-unimodal?

In every fixed dimension, sufficiently large lattice width forces the Ehrhart $h^*$ -polynomial to have positive coefficients and distinct negative real roots, hence a strictly log-concave and unimodal coefficient vector.

Details and sources

AI contribution

The author reports that ChatGPT 5.6 Sol found both short proofs from a lattice-point approximation theorem of Basu and Oertel and produced the first draft. The author is responsible for the final paper.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-presented proof; independent review pending

Publication

Public five-page proof preprint

Preprint / manuscript

Original published Question 5

Activity evidence

A documented editorial estimate based on the active Ehrhart-unimodality literature and the question's links to lattice width, IDP polytopes, and real-rootedness.

This rules out the 2023 question asking for an infinite fixed-dimensional family of non-unimodal $h^*$ -vectors with unbounded width. The analogous local- $h^*$ theorem is proved for simplices; extending it to arbitrary lattice polytopes remains conjectural.

Averkov–Hofscheier–Nill question answered negatively

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Sol
Verification: Author-presented proof; independent review pending
Open for: Question 5 of Averkov–Hofscheier–Nill
Research activity*: 3/5

31 Jul 2026Algebraic and enumerative combinatorics

Han–Xiong $1/2$ -conjecture for fractional Gaussian binomial coefficients

Problem statement

Is the integer trace of the fractional Gaussian binomial coefficient coefficientwise largest at

r=1/2

for every positive rational

r

and every

k\geq1

A support-dominance theorem proves the conjecture for every rational $r\geq1/2$ , reduces the full problem to the unit fractions $r=1/(2m)$ , and leaves only finitely many nontrivial values of $m$ for each fixed $k$ . Exact computation verifies all positive rational $r$ for $k\leq200$ .

Details and sources

AI contribution

Given a natural-language problem statement but no proof, AxiomProver autonomously produced the formal statements and Lean proofs of the support-dominance theorem, its half-line corollary, and the canonical reduction. Ken Ono then wrote the human exposition; the finite $k\leq200$ computation was separate from the Lean development.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Core theoretical results Lean checked; finite computation author reported

AxiomProver Lean development

Publication

Public proof manuscript, rerunnable Lean repository, and exact finite computation

Preprint / manuscript

Original Han–Xiong conjecture

Activity evidence

A documented editorial estimate based on the conjecture's recent origin, its connections with partitions and $q$ -series, and the complete public formal-proof artifact.

This is a substantial reduction, not a full resolution. The uniform all- $k$ primitive comparisons remain open; in particular, the $r=1/4$ family is still unresolved beyond the verified range. The Lean repository checks Theorem 1.3, Corollary 1.4, and Theorem 1.5, but not the finite-computation theorem.

Half-line proved; full conjecture reduced to primitive cases

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: AxiomProver
Verification: Core theoretical results Lean checked; finite computation author reported
Open for: Recent Han–Xiong coefficientwise-positivity conjecture
Research activity*: 3/5

3 Aug 2026Ramsey theory

Superexponential multicolor Ramsey numbers for every fixed odd cycle

Problem statement

Do the

k

-color Ramsey numbers of fixed odd cycles grow faster than exponentially in

k

For every fixed $p\geq1$ , the family $\mathcal O_p=\{C_3,C_5,\ldots,C_{2p+1}\}$ satisfies $R_k(\mathcal O_p)\geq(\log^{(p-1)}k)^{k/3-o(k)}$ . Consequently, the multicolor Ramsey number of every fixed odd cycle grows superexponentially in the number of colors.

Details and sources

AI contribution

Raphael Steiner states that ChatGPT 5.6 Pro/Sol autonomously found the proof and supplied substantial parts of the initial draft. The author edited and rewrote the manuscript and takes responsibility for its correctness.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; independent review pending

Publication

Public 12-page proof preprint

Preprint / manuscript

Odd-cycle Ramsey proof

Activity evidence

A documented editorial estimate based on the foundational role of Ramsey-number growth and the active literature on multicolor odd-cycle bounds.

The construction extends OpenAI's recent triangle result to all fixed odd cycles. It settles the qualitative exponential-versus-superexponential question, but the large gap to the best upper bounds remains, so the broader asymptotic problem is classified as partial. A separate human-authored consequence now proves the 2018 Mohar–Wu odd-girth conjecture and sharply improves Rödl's bound; that note explicitly reports no AI use beyond proofreading, so it is linked here as downstream evidence rather than counted as another AI-solved record.

Human downstream proof of the Mohar–Wu conjecture

Exponential barrier broken; sharp growth still open

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro/Sol
Verification: Author checked; independent review pending
Open for: Longstanding exponential barrier in multicolor Ramsey theory
Research activity*: 5/5

3 Aug 2026Extremal graph theory

Simonovits product-structure conjectures for extremal graphs

Problem statement

Must a finite forbidden family with superlinear extremal surplus have the conjectured join-product structure, or at least a forest in its decomposition family?

A fixed finite forbidden family $\mathcal L$ with $p(\mathcal L)=2$ has superlinear surplus above the Turán graph, yet has an extremal graph with connected complement at every sufficiently large order. Its decomposition family also contains no forest.

Details and sources

AI contribution

The author reports that GPT-5.6 Sol found the counterexample during a Codex project devoted to the Product Conjecture. The public manuscript gives the finite construction, exact extremal analysis, and the deductions refuting the two stated conjectures.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-presented proof; peer review pending

Publication

Public ten-page counterexample preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjectures' role in structural extremal graph theory and their repeated appearance in surveys and problem lists.

The 24 August v2 makes the surplus quantitative: it exceeds $c n^{3/2}$ above the bipartite Turán number at every sufficiently large order, while a non-join extremal graph exists at each such order. The weaker conjecture asking only for some product-form extremal graph remains open.

Two product-structure conjectures disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-presented proof; peer review pending
Open for: Longstanding product-structure program in extremal graph theory
Research activity*: 4/5

1 Aug 2026Algebraic and enumerative combinatorics

Ehrhart positivity of Schubitopes

Problem statement

Does every Schubitope have an Ehrhart polynomial with all coefficients positive?

There is a Schubitope whose Ehrhart polynomial has a negative quadratic coefficient, disproving the Monical--Tokcan--Yong conjecture that every Schubitope is Ehrhart positive.

Details and sources

AI contribution

The authors state that GPT-5.6 Sol Pro in ChatGPT found the counterexample. Their public paper records the explicit diagram, computes its Ehrhart polynomial, and explains the discovery process.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; independent review pending

Publication

Public five-page counterexample preprint

Preprint / manuscript

Lean certificate for the binary-code bound Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Activity evidence

A documented editorial estimate based on current work on Schubitopes, generalized permutahedra, and Ehrhart positivity.

This refutes Ehrhart positivity for Schubitopes. It is distinct from the separately indexed saturated-Newton-polytope conjecture for Schubert polynomials, despite the shared Monical--Tokcan--Yong attribution and Schubert-combinatorics setting.

Ehrhart-positivity conjecture disproved

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: GPT-5.6 Sol Pro
Verification: Author checked; independent review pending
Open for: Monical--Tokcan--Yong Conjecture 5.19
Research activity*: 3/5

1 Aug 2026Coding theory

Binary-code exponent beyond the MRRW bound

Problem statement

Can the optimized MRRW upper bound for asymptotic binary-code rates be improved at every fixed relative distance?

For every fixed relative distance $0<\delta<1/2$ , a harmonic-subspace construction gives a binary-code rate bound strictly below the fully optimized second McEliece--Rodemich--Rumsey--Welch exponent.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on the central role of asymptotic code bounds and the absence of a general leading-exponent improvement since the MRRW work of the 1970s.

This is the first general exponential improvement to the binary-code bound since 1977. It is a substantial advance rather than a complete determination of the optimal asymptotic code rate. Binary and spherical codes are indexed separately because they are distinct extremal problems, although their certificates share one Lean module.

General asymptotic exponent improved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: Classical coding-theory exponent barrier
Research activity*: 5/5

1 Aug 2026Spherical codes

Spherical-code exponent beyond Kabatianskii–Levenshtein

Problem statement

Can the optimized Kabatianskii--Levenshtein exponent for high-dimensional spherical codes be improved for every fixed inner-product threshold?

For every fixed maximum inner product $0<s<1$ , a hierarchy using harmonics orthogonal to each code point yields an exponent strictly below the optimized Kabatianskii--Levenshtein bound.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for the spherical-code bound Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on the foundational role of spherical-code bounds in discrete geometry and coding theory and the longevity of the Kabatianskii--Levenshtein exponent.

This is the first general exponential improvement to the spherical-code bound since 1978. It does not determine the optimal spherical-code rate. The result also supplies the upper-bound direction of the companion Cohn--Elkies asymptotic through a small-angle limit.

General asymptotic exponent improved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: Classical spherical-code exponent barrier
Research activity*: 5/5

1 Aug 2026Ramsey theory and Shannon capacity

Erdős Problem #183 — multicolor triangle Ramsey growth

Problem statement

\lim_{k\to\infty}R_k(3)^{1/k}

finite, where

R_k(3)

is the

k

-color triangle Ramsey number?

There is an absolute $c>0$ such that $R_k(3)\ge (ck/(\log k)^{1/3})^k$ . Together with the classical factorial upper bound this gives $R_k(3)=k^{\Theta(k)}$ , so $R_k(3)^{1/k}\to\infty$ and the corresponding Shannon capacities are unbounded.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for Erdős Problem #183 Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions Merged canonical Lean-status review Status-review discussion and independent rebuild

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion Community Erdős Problem #183 record Erdős Problem #183 discussion

Activity evidence

A documented editorial estimate based on decades of work on multicolor Ramsey numbers, an Erdős prize, and the equivalent unbounded-Shannon-capacity question.

This resolves the question whether the exponential-growth limit is finite. On 2 Aug 2026, canonical status-review PR #373 was merged after a successful rebuild of the pinned Lean module. The community database now records informal status open and formal status Lean: the machine-checked artifact is recognized, while independent human mathematical review remains pending.

Finite exponential-growth expectation disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: Erdős prize problem
Research activity*: 5/5

1 Aug 2026Extremal graph theory

Corrected Erdős–Simonovits compactness conjecture

Problem statement

For a finite family of connected bipartite graphs that all contain cycles, must the family extremal number be within a constant factor of one member’s extremal number?

A finite family $\mathcal F$ of connected bipartite graphs, each containing a cycle, satisfies $\operatorname{ex}(n,\mathcal F)=O(n^{4/3-1/48})$ while $\operatorname{ex}(n,F)=\Omega(n^{4/3})$ for every $F\in\mathcal F$ .

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for the compactness counterexample Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions Canonical Lean-status merge for Problems #146 and #180

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion Community Erdős Problem #180 record Erdős Problem #180 discussion

Activity evidence

A documented editorial estimate based on the 1982 Erdős--Simonovits compactness program and continued work on extremal numbers of graph families.

The literal original formulation already has a simple folklore star-and-matching counterexample. This new result disproves the meaningful repaired version in which every forbidden graph is connected, bipartite, and contains a cycle; it is not presented as the first counterexample to the literal statement. The community database now records informal status open and formal status Lean, recognizing the certificate while leaving human mathematical review pending.

Corrected cyclic version disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: 44 years
Research activity*: 5/5

1 Aug 2026Extremal graph theory

Erdős Problem #146 — 2-degenerate extremal bound

Problem statement

Must every fixed bipartite

r

-degenerate graph

H

satisfy

\operatorname{ex}(n,H)=O(n^{2-1/r})

, in particular

O(n^{3/2})

when

r=2

There is a fixed connected bipartite $2$ -degenerate graph $H$ and constants $c,\varepsilon>0$ such that $\operatorname{ex}(n,H)\ge c n^{3/2+\varepsilon}$ for all sufficiently large $n$ , contradicting the predicted $O(n^{3/2})$ bound.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for Erdős Problem #146 Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions Canonical Lean-status merge for Problems #146 and #180

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion Community Erdős Problem #146 record Erdős Problem #146 discussion

Activity evidence

A documented editorial estimate based on Erdős’s 1967 conjecture and sustained research on extremal numbers of sparse bipartite graphs.

The construction disproves the $r=2$ case of Erdős’s broader conjecture that every fixed bipartite $r$ -degenerate graph has extremal number $O(n^{2-1/r})$ . The exact exponent gap is positive but is not claimed to be optimal. The community database now records informal status open and formal status Lean, recognizing the certificate while leaving human mathematical review pending.

Degeneracy conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: 59 years
Research activity*: 5/5

26 Jul 2026Algebraic inequalities and finite exchangeability

Elementary symmetric-polynomial bounds for centered vectors and matrices

Problem statement

How sharply can elementary symmetric polynomials be bounded under vector or matrix centering constraints, and what quantitative consequences follow for permutation mixtures and finite exchangeability?

New bounds control elementary symmetric polynomials of zero-sum vectors and zero-row-and-column-sum matrices. Applications include a unified mean-field bound for permutation mixtures and a sharp $\chi^2$ finite-alphabet de Finetti theorem.

Details and sources

AI contribution

The authors report that GPT-5.5 Pro developed the core ideas behind both inequalities. They refined the arguments, developed the applications, checked the proofs, and wrote the paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; independent review pending

Publication

Public author-reviewed preprint

Preprint / manuscript

Inequalities and applications preprint

Activity evidence

A documented editorial estimate based on recent work on symmetric-polynomial inequalities, permutation mixtures, and finite de Finetti bounds.

The paper presents new research theorems and applications, not a named longstanding conjecture. It is indexed as an AI-assisted mathematical advance and not as a full resolution of the broader geometry of attainable elementary symmetric polynomials.

New inequalities and sharp application proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author checked; independent review pending
Open for: Research direction extending classical ESP inequalities
Research activity*: 2/5

22 Jul 2026Distance spectra of graphs

Graffiti Conjecture 284

Problem statement

If a finite graph

G

has girth at least five, must its minimum dual degree satisfy

\delta^*(G)\leq-\partial_n(G)

, where

\partial_n(G)

is the smallest eigenvalue of its distance matrix?

The Hoffman–Singleton graph is an exact counterexample: its minimum dual degree is 7 while the negative of its smallest distance-matrix eigenvalue is 4, so the conjectured inequality would require 7 ≤ 4.

Details and sources

AI contribution

A Capy research agent running Grok 4.5 Medium selected the conjecture, connected it to the Hoffman–Singleton graph, and produced the counterexample during an eight-minute agent run.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Exact certificate independently reproduced

Independent exact verification

Publication

Public agent transcript, independent integer-only verification, code, and artifact ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A 1996 Graffiti conjecture recorded as open in a 2014 distance-spectra survey and again in a 2024 computational study.

The result is not yet a peer-reviewed journal publication. Its finite certificate has nevertheless been independently reproduced using exact integer matrix arithmetic and a second construction of the Hoffman–Singleton graph.

Capy agent transcript Public announcement

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Grok 4.5 Medium / Capy Build
Verification: Exact certificate independently reproduced
Open for: 30 years
Research activity*: 3/5

2 Feb 2026Graph polynomials and statistical physics

Multivariate independence-polynomial lower bound

Problem statement

Can the known degree-sensitive lower bound for the hard-core partition function be extended from one common activity to arbitrary nonnegative vertex fugacities?

Lee and Seo prove a degree-sensitive lower bound for the multivariate independence polynomial at arbitrary nonnegative vertex fugacities, extending the known univariate theorem and yielding a stronger two-color inequality.

Details and sources

AI contribution

The authors report that key technical steps were obtained with a custom Gemini Deep Think research agent, Aletheia, before the argument was completed and checked as a conventional paper.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked preprint

Publication

Complete public arXiv proof

Preprint / manuscript

Activity evidence

Independence polynomials and hard-core partition functions are central objects in an active combinatorics and statistical-physics literature.

The result generalizes the Sah–Sawhney–Stoner–Zhao lower bound from a common activity to vertex-dependent fugacities.

General lower bound proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Gemini Deep Think / Aletheia
Verification: Author-checked preprint
Research activity*: 4/5

25 Mar 2026Hamiltonian graph decompositions

Hamilton decompositions of the directed 3-torus

Problem statement

For

D_3(m)=\vec C_m\square\vec C_m\square\vec C_m

, can the full arc set be partitioned into three directed Hamilton cycles for every integer

m\geq3

The directed product of three $m$ -cycles is decomposed into three arc-disjoint directed Hamilton cycles for every integer $m\geq3$ .

Details and sources

AI contribution

Claude first supplied the odd-order construction; later human–AI work handled even orders, unified the proof, and produced a Lean development.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean 4 formalization

Knuth’s discovery account

Publication

Public arXiv proof, certificates, and Lean source

Preprint / manuscript

Unified research paper

Activity evidence

The question prompted several independent constructions, a long unified proof, computational certificates, and a complete formalization in a short period.

This is the full directed 3-torus theorem. It is distinct from the narrower LEAP/Knuth subproblem already indexed elsewhere.

All dimensions in the family proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Claude Opus 4.6 / GPT-5.3 Codex / GPT-5.4 Pro
Verification: Lean 4 formalization
Open for: Newly posed in 2026
Research activity*: 4/5

31 Jul 2026Ramsey theory

Ramsey numbers for book graphs

Problem statement

Does every

n

admit a graph on

4n-2

vertices containing no book

B_{n-1}

whose complement contains no

B_n

, and hence prove

R(B_{n-1},B_n)=4n-1

An AI scaffold found two additional infinite families and constructions for every remaining case through $n=56$ . A later GPT-5.6 Sol Pro run supplied the $n=70$ construction and derivation, proving $R(B_{69},B_{70})=279$ . The all- $n$ equality remains open.

Details and sources

AI contribution

GPT models supplied the core mathematical search while Claude Code implemented the search program; Epoch records the results and subsequent review.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Exact construction checks

GPT-5.6 Sol Pro construction for n = 70

Publication

Epoch open-problem page, write-up, and verifier

Preprint / manuscript

Derivation of the n = 70 construction Result manuscript

Activity evidence

A specialist Ramsey problem with a 1978 bound, modern infinite families, and several serious AI-assisted construction searches.

The $n=70$ construction is public as executable code with a separate derivation. A Dualverse system also proposed another infinite family, but Epoch describes that output as under review; it is not counted here as a verified full result.

Epoch problem and progress record

New families and finite cases

Problem origin: Origin not yet traced
System: GPT-5.2 Pro / GPT-5.4 Pro / GPT-5.6 Sol Pro / Claude Code
Verification: Exact construction checks
Open for: Upper bound known since 1978
Research activity*: 3/5

16 Mar 2026Ramsey theory

Diagonal Ramsey upper-bound constant

Problem statement

Within the stated diagonal-Ramsey ansatz, choose a correction polynomial and auxiliary functions satisfying all sufficient inequalities while minimizing the resulting constant

c=e^{F(1)}

Within the specified Gupta–Ndiaye–Norin–Wei ansatz, GPT-5.4 Pro proposed a correction reducing the automatically validated constant from about $3.7992$ to $3.6961$ .

Details and sources

AI contribution

The model proposed a quintic correction and piecewise auxiliary functions; the benchmark checked the sufficient inequalities using interval arithmetic.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Interval checked; expert review pending

Publication

HorizonMath paper and open verification framework

Preprint / manuscript

HorizonMath paper

Activity evidence

Diagonal Ramsey numbers are a long-running international topic, though this record concerns one narrow optimization within a recent upper-bound framework.

This improves a constant inside a particular proof framework. HorizonMath labels the contribution as pending expert review, so the entry remains partial.

Certified constant improved

Problem origin: Origin not yet traced
System: GPT-5.4 Pro
Verification: Interval checked; expert review pending
Open for: Current optimization frontier
Research activity*: 4/5

21 May 2026Ramsey theory

Erdős Problem #138 — gaps between van der Waerden numbers

Problem statement

W(k)

is the least

N

such that every two-colouring of

[1,N]

has a monochromatic

k

-term arithmetic progression, must

W(k+1)-W(k)\to\infty

AlphaProof Nexus proves $W(k+1)-W(k)\to\infty$ for the two-colour van der Waerden numbers. The stronger parent question $W(k)^{1/k}\to\infty$ remains open.

Details and sources

AI contribution

The system found a greedy extension argument for colourings and formalized the complete proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Formal statement Lean proof

Publication

AP Nexus preprint and public Lean source

Preprint / manuscript

Activity evidence

Van der Waerden-number growth has a broad and long-running literature; this particular asymptotic gap variant is narrower.

The theorem is a separately recorded difference variant on the official problem page, not a full solution to Erdős #138.

Formal statement and proof Lean source

Long-standing variant proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AlphaProof Nexus
Verification: Lean checked
Open for: 45 years
Research activity*: 4/5

21 May 2026Extremal graph theory

Written on the Wall II, Graph Conjecture 2

Problem statement

For a finite simple connected graph

G

, let

L_s(G)

be the maximum number of leaves in a spanning tree and

\ell(G)=|V(G)|^{-1}\sum_v\alpha(G[N(v)])

. Must

L_s(G)\geq2(\ell(G)-1)

For every finite connected graph, the maximum number of leaves in a spanning tree is at least twice the average local independence number minus two.

Details and sources

AI contribution

The system generated the proof and compiled a complete Lean formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Public natural-language proof and Lean source

Preprint / manuscript

AP Nexus paper

Activity evidence

Part of the long-running Graffiti conjecture program, with a specialist literature on spanning-tree leaves and local independence.

This is Conjecture 2 from the 1996 Written on the Wall II / Graffiti collection.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AlphaProof Nexus
Verification: Lean checked
Open for: 30 years
Research activity*: 3/5

21 May 2026Higher-order Fourier analysis

Ben Green’s Open Problem 57

Problem statement

For a finite abelian group

G

, let

\Phi(G)

be the absolutely convex hull of the specified trilinear kernels and let

\Phi'(G)

restrict the third factor to depend only on

x_1+x_2

. Is

\Phi(G)=\Phi'(G)

A counterexample over $G=\mathbb Z/3\mathbb Z$ separates the two absolutely convex hulls in Green’s question, with the strict support-function gap certified in Lean.

Details and sources

AI contribution

After an initial real-valued result and clarification of the intended complex formulation, numerical search found a candidate and the agent produced its rigorous proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Formal statement Lean proof

Publication

AP Nexus preprint and public formal proof

Preprint / manuscript

Green’s open-problem list

Activity evidence

A young but technically serious item on a prominent 2024 open-problem list in additive combinatorics.

The clarification step matters: the final certificate addresses the intended complex-valued question, not only the easier real variant.

Intended complex form disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: AlphaProof Nexus
Verification: Lean checked
Open for: 2 years
Research activity*: 3/5

14 Jul 2026Enumerative combinatorics

Record compositions of alternating permutations

Problem statement

Can the record-partition enumeration for alternating permutations be refined to ordered record compositions, and does it admit a natural lift to noncommutative symmetric functions?

The ordered refinement is counted by an explicit product formula, and a canonical noncommutative-symmetric-function lift is constructed together with a broader exponential-seed generalization.

Details and sources

AI contribution

AxiomProver autonomously produced and Lean-verified the three main theorems answering the questions of Amdeberhan, Shareshian, and Stanley.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Formal source

Publication

Public arXiv paper and formal source

Preprint / manuscript

Solution paper Original problem paper

Activity evidence

A new specialist enumerative-combinatorics problem from an active group of researchers, without a long independent literature trail.

The problem was newly posed in 2026, so its importance is assessed by mathematical content rather than age.

Open problem solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AxiomProver
Verification: Lean checked
Open for: Open for less than one year
Research activity*: 2/5

20 May 2026Partition theory

Reciprocals of partition polynomials

Problem statement

For

\operatorname{sp}(\lambda,x)=\prod_i(1+x^{\lambda_i})

, what divisibility, coprimality, recurrence, irreducibility, and coefficient-shape properties hold after reducing sums of

1/\operatorname{sp}(\lambda,x)

over the standard partition families?

AxiomProver proves six of ten conjectures about reduced reciprocal sums over ordinary, binary, odd, and ternary partitions, and finds a counterexample to the printed binary log-concavity statement.

Details and sources

AI contribution

The system generated the proofs and counterexample; the response paper’s authors checked and organized the results.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked

Formal artifacts

Publication

Public response paper and Lean repository

Preprint / manuscript

Response paper Original conjectures

Activity evidence

A newly published specialist family of ten concrete conjectures, with rapid formal follow-up but no long history.

Several irreducibility and shape questions remain open. This is therefore an umbrella partial result, not a full settlement of the ten-conjecture family.

Six conjectures proved; one corrected

Claimed outcome: ProvedDisproved
Problem origin: Origin not yet traced
System: AxiomProver
Verification: Lean checked
Open for: Resolved within days of publication
Research activity*: 2/5

21 May 2026Graph reconstruction

Weak bipartite reconstruction with distinct vertex types

Problem statement

If a suitably 2-connected bipartite graph has distinct vertex types

\tau(v)=(\deg v,\{\!\{\deg u:u\sim v\}\!\})

, is it determined up to bipartite isomorphism by the multiset of incidence-deletion cards?

A 2-connected bipartite graph is reconstructed from its incidence-deletion deck under an explicit strong condition that all degree-and-neighbour-degree vertex types are distinct.

Details and sources

AI contribution

AlphaEvolve helped formulate the reconstruction algorithm; AlphaProof Nexus proved its correctness under the distinct-type hypothesis.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked

Official problem record Research discussion and artifacts

Publication

AP Nexus paper and public Lean source

Preprint / manuscript

Exact theorem and proof

Activity evidence

The parent reconstruction program has more than sixty years of international attention, while this distinct-type theorem is narrowly scoped and new.

This is a strong-hypothesis variant. It does not solve the general graph reconstruction conjecture or the unrestricted bipartite problem.

Restricted reconstruction theorem

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: AlphaEvolve / AlphaProof Nexus
Verification: Lean checked
Open for: New restricted variant
Research activity*: 4/5

7 May 2026Critical graph theory

Erdős Problem #1032 — minimum degree in 4-critical graphs

Problem statement

Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?

A new density–degree inequality implies $\delta(G)\leq(3/10+o(1))|V(G)|$ for 4-critical graphs, improving the previous coefficient $0.328$ . The existence of a positive linear lower construction remains open.

Details and sources

AI contribution

GPT-5.5 Pro with a research harness generated the note; Codex with the same harness produced the Lean development.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + expert screening

Publication

Public note, formal proof, and official discussion

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A long critical-graph literature involving Simonovits, Toft, and a substantial 2023 advance.

The new inequality narrows the feasible range but does not answer Erdős’s existence question.

Upper obstruction improved

Problem origin: Human-source problem
System: GPT-5.5 Pro / Codex
Verification: Lean checked + expert screening
Open for: At least 53 years
Research activity*: 4/5

21 Jun 2026Discrepancy theory

Erdős Problem #176 — discrepancy on arithmetic progressions

Problem statement

Let

N(k,\ell)

be the least

N

such that every

f:[N]\to\{-1,1\}

has a

k

-term arithmetic progression

P

with

|\sum_{n\in P}f(n)|\geq\ell

. In particular, is

N(k,2)\leq C^k

A formal proof gives a polynomial upper bound for $N(k,2)$ , stronger than Erdős’s requested exponential bound. The broader two-parameter discrepancy problem remains open.

Details and sources

AI contribution

The repository author reports assistance from Codex 5.5 and ChatGPT 5.5 Pro while preparing the construction and its Lean formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public Lean proof

Official problem record Research discussion

Publication

Official discussion and Lean repository

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Repeated in many Erdős sources since 1965 and closely tied to van der Waerden numbers and arithmetic-progression discrepancy.

The AI community ledger retains a cautious candidate-partial label, so the site does not promote the multipart parent problem to resolved.

The N(k,2) clause proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Codex 5.5 / ChatGPT 5.5 Pro
Verification: Public Lean proof
Open for: 61 years
Research activity*: 4/5

26 May 2026Extremal graph theory

Pentagons in triangle-free graphs

Problem statement

Does every triangle-free graph on

5n

vertices contain at most

n^5

copies of the five-cycle

C_5

The known affirmative theorem now has an AI-assisted, kernel-checked Lean proof.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

5 Feb 2026Combinatorial number theory

Distinct consecutive sums of permutations

Problem statement

For a permutation of

1,\ldots,n

, must the number of distinct sums of consecutive terms always be

o(n^2)

A counterexample to the conjecture was encoded and verified in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

24 May 2026Graph cycles

Cycle lengths in arithmetic progressions

Problem statement

Must every graph of sufficiently large average degree contain a cycle whose length lies in any prescribed infinite arithmetic progression containing even integers?

The affirmative theorem was reconstructed and checked in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Canonical Lean status; durable source not located

Publication

Official problem and discussion record

Preprint / manuscript

No public preprint or manuscript located.

The canonical record labels the theorem Lean-verified, but this audit did not locate a durable public source file. The discussion thread is retained for provenance.

Problem, history, and proof record Discussion thread Community AI ledger

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / Claude Opus 4.7 / GPT-5.5
Verification: Canonical Lean status; durable source not located

7 Feb 2026Extremal graph theory

Triangle-free graph completion to diameter two

Problem statement

For every

\epsilon,\delta>0

and all sufficiently large

n

, can every triangle-free

n

-vertex graph with maximum degree

<n^{1/2-\epsilon}

be made triangle-free of diameter

2

by adding at most

\delta n^2

edges?

The known affirmative result now has a checked Lean formalization.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

31 Mar 2026Graph enumeration

Growth rate of minimal graph cuts

Problem statement

Does the exponential growth limit for the maximum number of minimal cuts in an

n

-vertex graph exist, and is it strictly below

2

The affirmative theorem and strict upper bound were formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

6 Feb 2026Additive combinatorics

Equidistribution of dense Sidon sumsets

Problem statement

A\subseteq\{1,\ldots,N\}

is a Sidon set with

|A|\sim N^{1/2}

, must

A+A

be well-distributed over all small moduli?

A proof exposition produced with ChatGPT was converted into a checked Lean development.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: ChatGPT / Aristotle
Verification: Lean checked

21 Apr 2026Discrepancy theory

Countable simultaneous discrepancy

Problem statement

Given infinite sets

A_i=\{a_{i1}<a_{i2}<\cdots\}

, is there

f:\mathbb N\to\{-1,1\}

such that

\max_{m,1\leq i\leq d}|\sum_{j\leq m}f(a_{ij})|\ll_d1

for every

d\geq1

Beck's affirmative theorem was autoformalized and kernel checked.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

27 Feb 2026Euclidean Ramsey theory

Monochromatic collinear sets

Problem statement

For every

k\geq3

, is there a finite

A\subset\mathbb R^2

such that every two-coloring of

A

has a line containing at least

k

points of

A

, with all points of

A

on that line having the same color?

A construction based on Hales–Jewett was given a checked Lean proof.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / Gemini 3 Flash
Verification: Lean checked

10 May 2026Arithmetic progressions in words

Abelian-square-free lattice walks

Problem statement

For which dimensions must every positive unit-coordinate lattice walk contain three vertices in arithmetic progression? The threshold is true through dimension three and false from dimension four onward.

Keränen's 1992 construction and dimension threshold were formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

15 Apr 2026Arithmetic progressions

Monotone progressions in an ordering of the reals

Problem statement

Must every linear ordering of

\mathbb R

contain a monotone

k

-term arithmetic progression? The conjecture already fails for

k=3

The Ardal–Brown–Jungić counterexample was reconstructed and checked in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

24 Feb 2026Arithmetic progressions

Progression-free sets and their complements

Problem statement

A\subset\mathbb R

contains no three-term arithmetic progression, must its complement contain an infinite arithmetic progression?

Baumgartner's negative construction was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

19 Jun 2026Additive combinatorics

Partitioning bounded-representation sets

Problem statement

Can every set with uniformly bounded additive representation count be partitioned into finitely many sets with a strictly smaller bound?

The Nešetřil–Rödl negative theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: AxiomProver
Verification: Lean checked

31 Jan 2026Additive combinatorics

Difference collisions at square-root density

Problem statement

Must two sets with counting functions of order at least

\sqrt N

have infinitely many common nonzero differences?

Ruzsa's binary-digit counterexample was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

1 Jun 2026Set theory and infinitary combinatorics

Erdős Problem #501 — relative independence under a measure extension

Problem statement

If every

A_x\subset\mathbb R

is bounded with outer measure below

1

, must there be an infinite

X\subseteq\mathbb R

such that

x\notin A_y

whenever

x\neq y

are in

X

A short AI-assisted note proves a positive answer to the infinite independent-set question assuming a countably additive extension of Lebesgue measure to all subsets of the real line.

Details and sources

AI contribution

GPT-5.5 Pro assisted the author in finding and revising the relative consistency argument and its section inequality.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; community screening

Publication

Public repository and Erdős Problems discussion

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A long-running Erdős–Hajnal set-theoretic problem with classical independence results and modern discussion.

This is a conditional or relative-independence result, not a ZFC solution. The official problem page remains open.

Official problem and literature record AI-assisted note and discussion

Conditional independence result

Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author checked; community screening
Open for: 65 years
Research activity*: 3/5

27 Jun 2026Extremal graph theory

Erdős Problem #1033 — triangle degree-sum bound

Problem statement

If every graph on

n

vertices with more than

n^2/4

edges contains a triangle whose vertex degrees sum to at least

h(n)

, is

h(n)\geq(2(\sqrt3-1)-o(1))n

A public GPT-5.5 Pro conversation claims an explicit construction disproving the proposed asymptotic lower bound for the maximum guaranteed degree sum of a triangle.

Details and sources

AI contribution

The model generated the claimed construction in a public discussion linked from the problem page.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Single community claim; not incorporated

Claim audit

The official page remains OPEN and says that no partial or complete solution has been incorporated from the comments.

Publication

Official open-problem page and one-comment discussion

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A Bollobás–Erdős extremal graph problem with classical upper and lower bounds but only one recent AI-claim comment.

This is included as a claim-audit record so that the public evidence is searchable. It is not counted as a confirmed disproof.

Official problem and known bounds Claim and public discussion

Counterexample claimed

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Single community claim; not incorporated
Claim audit: Issue documented
Open for: 44 years
Research activity*: 3/5

23 Jul 2026Infinite hypergraph theory

Erdős Problem #593 — obligatory triple systems

Problem statement

Which finite triple systems occur in every triple system of uncountable chromatic number?

A finite triple system is claimed to occur in every uncountably chromatic triple system exactly when, after isolated vertices are removed, it is linear, every hyperedge-node of its Levi graph meets a bridge, and every Berge cycle is even.

Details and sources

AI contribution

ChatGPT assisted ideation, proof exploration, refinement, programming, and orchestration. Aristotle assisted the author in building the accompanying Lean development.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two public Lean artifacts; draft status PR proposes open (Lean); human review pending

Lean development Pinned self-contained Lean closure Draft canonical status-review PR

Publication

Revised arXiv manuscript and public Lean 4 development

Preprint / manuscript

Revised proof manuscript

Activity evidence

The classification builds on decades of work on obligatory subsystems, uncountable chromatic number, and finite hypergraph structure.

The revised manuscript says the imported interfaces are proved inside Lean and packages the result in a hypothesis-free theorem using only standard Mathlib axioms. A separate pinned, self-contained Lean closure was rebuilt for draft canonical PR #366, which proposes open (Lean) but explicitly does not claim independent mathematical review. The canonical database remains open and unformalized until that review is accepted.

Claimed resolution; Lean checked end to end

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT + Aristotle
Verification: Two public Lean artifacts; draft status PR proposes open (Lean); human review pending
Open for: Public Erdős problem
Research activity*: 4/5

23 Jul 2026Infinite hypergraph theory

Erdős Problem #1177 — exact avoidance spectra

Problem statement

For a finite forbidden triple system

G

, what exact uncountable chromatic cardinalities can occur among

G

-free triple systems, and how do those spectra interact?

The revised manuscript gives truth values yes, no, and yes to the problem’s three exact-cardinal avoidance questions and proves a complete spectrum dichotomy for every finite forbidden triple system.

Details and sources

AI contribution

ChatGPT assisted the mathematical search and manuscript development. Aristotle supported the full Lean formalization, including the transfinite calibration argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked end to end; not yet conventionally refereed

Publication

Revised arXiv manuscript and public Lean 4 development

Preprint / manuscript

Revised proof manuscript

Activity evidence

The exact-cardinal formulation connects classical infinite combinatorics with modern obligatory-subsystem results.

This shares a proof core with Problem #593 but answers a distinct three-part Erdős problem. The kernel certificate verifies the encoded theorem; novelty and exposition still await normal scholarly review.

Proof note and verification summary Certificates and checkers Verification catalog and runnable checks Verification method and limitations

Claimed resolution; Lean checked end to end

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT + Aristotle
Verification: Lean checked end to end; not yet conventionally refereed
Open for: Public Erdős problem
Research activity*: 4/5

12 Jun 2026Spectral graph theory

Graffiti Conjecture 143

Problem statement

For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance?

Exact dumbbell-graph certificates refute the claimed bound on the variance of positive eigenvalues under both standard conventions for average distance.

Details and sources

AI contribution

The pipeline found counterexamples beyond earlier search horizons and produced exact spectral-isolation checks, an independent rebuild, and mutation tests.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Dual exact checker routes; not externally refereed

Publication

Public note, certificates, source provenance, and runnable checks

Preprint / manuscript

Certificates and checkers Verification catalog and runnable checks Verification method and limitations

Activity evidence

The conjecture survived an early computational attack and a 2025 eight-algorithm search to order 100.

The smallest certified witness under both common average-distance conventions has 39 vertices. No general minimality below 37 vertices is claimed.

Conjecture refuted

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Dual exact checker routes; not externally refereed
Open for: Survived computational attacks since 1990–91
Research activity*: 3/5

12 Jun 2026Spectral graph theory

Graffiti Conjecture 154 under the standard-deviation reading

Problem statement

For every connected graph, is the deviation of its adjacency eigenvalues at most its order divided by its average distance?

Exact lollipop-graph certificates refute the inequality when “deviation” means population standard deviation, under both common average-distance conventions.

Details and sources

AI contribution

The pipeline discovered large witnesses and supplied independently written exact checkers, an independent rebuild, and mutation tests.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Dual exact checkers; reading-sensitive; not externally refereed

Claim audit

The original word “deviation” is ambiguous. Only the standard-deviation interpretation is claimed refuted; the alternative mean-absolute-deviation reading survives these witnesses.

Publication

Public note, frozen source, certificates, and runnable checks

Preprint / manuscript

Proof note and exact scope

Activity evidence

The conjecture survived old and modern computational searches, with the first certified lollipop-family violations appearing above 100 vertices.

The result is deliberately reading-specific. These examples do not refute the mean-absolute-deviation interpretation.

One standard reading refuted

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Dual exact checkers; reading-sensitive; not externally refereed
Claim audit: Issue documented
Open for: Reading-sensitive Graffiti conjecture
Research activity*: 3/5

12 Jun 2026Graph domination and zero forcing

TxGraffiti–Davila Conjecture 9

Problem statement

G

is connected, cubic, and diamond-free, must

Z(G)\leq\gamma(G)+2

A connected, cubic, triangle-free fourteen-vertex graph has zero-forcing number $Z=7$ and domination number $\gamma=4$ , violating $Z(G)\leq\gamma(G)+2$ .

Details and sources

AI contribution

The system produced an explicit graph and independent Python and Rust checkers that exhaustively certify both graph parameters.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Dual independent checkers; not externally refereed

Certificate and independent checkers Verification catalog and runnable checks Verification method and limitations

Publication

Public certificate, checkers, frozen source, and write-up

Preprint / manuscript

Original conjecture Result write-up

Activity evidence

The conjecture connects two active graph invariants and complements a proved claw-free sibling theorem.

The data suggest an unbounded gap on a chain family, but the general lower bound needed for that stronger claim is not proved.

Conjecture refuted

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Dual independent checkers; not externally refereed
Open for: Central open question in a 2024 preprint
Research activity*: 3/5

12 Jun 2026Independence polynomials

Pandey parity conjecture for generalized Petersen graphs

Problem statement

For every

n\geq2k+1

, is the independence polynomial of

GP(n,k)

real-rooted if and only if

k

is even?

Exact Sturm counts and independent brute force show an even- $k$ example that is not real-rooted and odd- $k$ examples that are real-rooted.

Details and sources

AI contribution

The pipeline produced exact polynomial certificates, a clean-room Python checker, an independent Rust enumeration, and mutation tests.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Dual independent exact checks; not externally refereed

Certificates and independent checkers Verification catalog and runnable checks Verification method and limitations

Publication

Public certificates, checkers, and frozen source

Preprint / manuscript

Original conjecture Result write-up

Activity evidence

A recent exact conjecture about real-rootedness and generalized Petersen graphs, with a small computational search space.

The counterexamples lie outside the numerical window tested in the original paper. Other log-concavity claims in that paper are not addressed.

Both directions refuted

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Dual independent exact checks; not externally refereed
Open for: Uncorrected 2026 preprint conjecture
Research activity*: 2/5

12 Jun 2026Extremal graph theory

Koch–Narayan Conjecture 1

Problem statement

For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function

m(n,\gamma)

upper-bound the number of edges whenever

\gamma\geq2

and

n\geq3\gamma

A thirteen-vertex bipartite graph with domination number $4$ , a unique minimum dominating set, and $22$ edges exceeds the conjectured maximum of $21$ .

Details and sources

AI contribution

The system produced explicit graph certificates, two clean-room checkers, an exhaustive filter, and mutation tests.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Dual independent checkers; not externally refereed

Certificates and independent checkers Verification catalog and runnable checks Verification method and limitations

Publication

Public certificates, checkers, source snapshot, and write-up

Preprint / manuscript

Original conjecture Result write-up

Activity evidence

A recent extremal graph conjecture with a finite certificate and a natural parameter family.

The $\gamma=3$ strip is not refuted, and the claimed smallest-order classification relies on one exhaustive C implementation even though the concrete witness has independent checks.

Conjectured upper bound refuted

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Dual independent checkers; not externally refereed
Open for: Open conjecture in a 2025 preprint
Research activity*: 2/5

12 Jul 2026Enumerative combinatorics

Elizalde–Luo $\{1132,3312\}$ pattern-avoidance conjecture

Problem statement

Is the number of nonnesting multiset permutations avoiding

1132

and

3312

equal to

3^n-3\cdot2^{n-1}+1

The number of nonnesting permutations of $\{1,1,\ldots,n,n\}$ avoiding both $1132$ and $3312$ is proved to be $3^n-3\cdot2^{n-1}+1$ for every $n\geq1$ .

Details and sources

AI contribution

The system developed several proof routes, independently audited two complete written proofs, checked ground truth in three implementations, and formalized the general theorem in Lean.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked end to end; not externally refereed

Proof and verification overview Lean proof and supporting artifacts Verification catalog and runnable checks

Publication

Public proof note, Lean project, enumerators, and audit logs

Preprint / manuscript

Original published conjecture Proof manuscript

Activity evidence

A concrete conjecture in modern permutation-pattern enumeration, supported by substantial finite data before the proof.

The published source had checked the formula only through $n=8$ . The Lean theorem covers all $n$ with the project’s pinned containment conventions.

Conjecture proved and Lean checked

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Lean checked end to end; not externally refereed
Open for: Published 2025 conjecture
Research activity*: 3/5

8 Jul 2026Enumerative combinatorics

Kurkov’s Fubini-number sum conjecture

Problem statement

For the Fubini numbers

a(n)

, is

a(n)=\sum_{k=0}^{2^{n-1}-1}\mathrm{A284005}(k)

for every

n>0

A refined ordered-set-partition argument proves Kurkov’s 2018 identity expressing the Fubini number $a(n)$ as a sum of values of OEIS sequence A284005.

Details and sources

AI contribution

The system produced a self-contained combinatorial proof, exhaustive verification over all ordered set partitions through $n=8$ , and a mutation-tested checker through $n=20$ .

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Audited proof + exhaustive checker; not externally refereed

Proof note and checker Verification catalog and runnable checks Verification method and limitations

Publication

Public proof note, checker, and source record

Preprint / manuscript

Original OEIS conjecture record

Activity evidence

A precise sequence identity with a natural ordered-partition interpretation and a seven-year public record.

The general proof is human-readable and audit-panel checked rather than proof-assistant certified. The finite computations support but do not replace the argument.

OEIS conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Demonstrandum multi-agent pipeline
Verification: Audited proof + exhaustive checker; not externally refereed
Open for: OEIS conjecture posted in 2018
Research activity*: 2/5

8 Jul 2026Additive combinatorics

Erdős Problem #866 — eventual value of $h_4$

Problem statement

Estimate the least excess

g_k(N)

forcing

k

integers whose pairwise sums all lie in a dense subset of

\{1,\ldots,2N\}

; in particular, determine the corresponding positive variant

h_4(n)

A verification-first multi-agent paper proves $h_4(n)=4$ for every $n\geq331{,}777$ , improves global $h_4$ and $g_5$ bounds, and certifies 298 exact finite cells.

Details and sources

AI contribution

The workflow developed the mathematics, extended an upstream Lean formalization, and combined kernel proofs with two SAT engines and archived DRAT/LRAT certificates.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Headline theorems Lean checked; finite cells independently certified

Paper, Lean proof, and certificate guide Verification method and limitations

Publication

Public draft, complete Lean project, SAT certificates, and release freeze

Preprint / manuscript

Frozen proof manuscript

Activity evidence

The question goes back to Choi, Erdős, and Szemerédi in 1975 and connects density thresholds, additive configurations, formal proof, and exact SAT computation.

This determines the eventual constant for the positive $h_4$ variant and materially advances #866, but the database’s broader request to estimate $g_k(N)$ for general $k$ remains open.

Theorem, scope, and audit record Lean statements and rebuild instructions Verification method and limitations

Major parameter case resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Demonstrandum multi-agent pipeline
Verification: Headline theorems Lean checked; finite cells independently certified
Open for: Partial resolution of a broader Erdős problem
Research activity*: 4/5

13 Jul 2026Graph coloring

Strong-majority $4$ -edge-coloring with at most two degree- $3$ vertices

Problem statement

Does every admissible graph admit a strong-majority edge-coloring with four colors, and in particular can the five-color bound be improved on mixed degree-

\{3,4\}

graphs?

Every finite simple graph with all degrees in $\{3,4\}$ and at most two degree- $3$ vertices has a strong-majority edge-coloring with four colors.

Details and sources

AI contribution

A multi-model campaign built and falsified candidate approaches, assembled the Lean infrastructure, and completed the new mixed-class theorem under a verification-first acceptance gate.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked end to end; not externally refereed

Publication

Public theorem page, versioned archive, Lean statements, and falsification evidence

Preprint / manuscript

Prior five-color theorem Machine-checked manuscript

Activity evidence

A new graph-coloring program with contemporaneous five-color progress, extensive exact census evidence, and a remaining combinatorial descent lemma.

The theorem reaches genuinely mixed degree- $\{3,4\}$ graphs, where the previous general bound was five. The case of arbitrarily many degree- $3$ vertices and the universal four-color conjecture remain open.

New mixed-class theorem; full conjecture open

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: OpenAI reasoning model + Anthropic reasoning model + Aristotle
Verification: Lean checked end to end; not externally refereed
Open for: New partial theorem toward a 2026 conjecture
Research activity*: 3/5

27 Jul 2026Independence polynomials and graph theory

Levit–Mandrescu very-well-covered graph conjecture

Problem statement

Must the independence polynomial

I(G;x)

of every very well-covered graph be unimodal—or, in the stronger original formulation, log-concave?

The whiskered graph $H=W(E_{588}\vee(39K_{11}\sqcup85K_{12}))$ is very well-covered but has a nonunimodal independence polynomial, with the exact strict local valley $a_{1094}>a_{1095}<a_{1096}$ .

Details and sources

AI contribution

Lucas Baker credits GPT-5.6 Sol and Claude Fable 5, together with human reviewers, for the counterexample. The public announcement says the solution was found within a few hours of the question being posed.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Exact certificate; review pending

Publication

Public four-page preprint attached to the author announcement; no archival posting or peer review located

Preprint / manuscript

Counterexample preprint attached to author announcement Original Levit–Mandrescu paper

Activity evidence

The conjecture dates to Levit and Mandrescu’s 2004 work and remained a documented specialist question through 2026, with several positive range results and related work on well-covered graphs.

The note refutes both the stronger log-concavity formulation and the later unimodality formulation. It makes no minimality claim for the 4,074-vertex construction and does not settle the separate unimodality conjecture for trees.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol + Claude Fable 5
Verification: Exact certificate; review pending
Open for: 22 years
Research activity*: 3/5

26 Jul 2026Coding theory and quadratic forms

Antipodal cut code of $K_{11}$

Problem statement

For the antipodal cut code

\widehat C_m=C_{\mathrm{cut}}(K_m)\cup(\mathbf1+C_{\mathrm{cut}}(K_m))

, must

\rho(\widehat C_m)=\rho(C_{m-1})

hold for every

m\geq7

An exact computer-assisted certificate proves $M_{11}=17$ , equivalently $\rho(\widehat C_{11})=19$ . Esmaeili and Zaghian’s 2009 conjecture predicts radius $20$ at this first parameter beyond their verified range $7\leq m\leq10$ .

Details and sources

AI contribution

Curtis White selected and directed the problem. The model assisted with literature synthesis, reformulation, proof search, the computational construction, verification design, and preparation of the public artifacts.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Exact reproducible computation; review pending

Research repository and proof package Reproducibility instructions

Publication

Public research repository with exact witness, exhaustive C++ enumeration, independent MILP cross-check, logs, and integrity hashes; no archival preprint or peer review located

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A precise coding-theory conjecture from 2009 with a small published verification range and a related MathOverflow question; the new result is exact but currently awaits independent review.

The upper bound checks an explicit sign matrix over every spin configuration modulo global symmetry. The lower bound uses deterministic symmetry-reduced enumeration and an independent SciPy/HiGHS MILP check. No DRAT or VeriPB proof certificate is supplied, and the result does not settle the asymptotic behavior of $M_n/n^{3/2}$ .

Source and literature provenance Original 2009 journal record Related MathOverflow discussion Author announcement

Conjecture disproved at

m=11

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Exact reproducible computation; review pending
Open for: 17 years
Research activity*: 2/5

26 Jul 2026Extremal graph theory

Bondy’s minimum-degree longest-cycle conjecture

Problem statement

If a

k

-connected

n

-vertex graph satisfies

\delta(G)\geq[n+k(k-1)]/(k+1)

, must every path in

G-V(C)

have at most

k-1

vertices for every longest cycle

C

Mazur’s manuscript claims a proof for every connectivity $k$ , adapting the Ma–Ning–Zhao framework and importing a residual-cycle theorem of Nikoghosyan. A separate adversarial audit reported no mathematical error, but the claim is not yet an accepted resolution.

Details and sources

AI contribution

The author credits GPT-5.6 Pro with assistance in developing the candidate argument. The public page separates the manuscript, its proof architecture, and a later adversarial audit.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Adversarial audit; no formal or specialist review

Candidate proof page and audit boundary

Claim audit

Not formally verified or peer reviewed. The candidate proof depends on external theorems whose exact applicability remains a critical specialist-review question.

Publication

Public candidate manuscript and LaTeX source; no peer-reviewed publication or proof-assistant formalization

Preprint / manuscript

Candidate proof manuscript June 2026 sufficiently-large-graphs preprint

Activity evidence

Bondy posed the conjecture in 1980. It is known for $k\leq3$ , and a June 2026 paper proved it for all sufficiently large graphs, showing active expert progress immediately before this candidate appeared.

A June 2026 human preprint proves Bondy’s conjecture for all sufficiently large graphs but explicitly leaves the general $k\geq4$ case open. ProofAtlas continues to label this later manuscript a candidate and says its use of the Ma–Ning–Zhao and Nikoghosyan inputs, novelty, and historical relationship require specialist checking.

LaTeX manuscript source Author announcement

Candidate full proof; review pending

Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Adversarial audit; no formal or specialist review
Claim audit: Issue documented
Open for: 46 years
Research activity*: 4/5

6 Jul 2026Infinite combinatorics

Erdős Problem #603 — colouring infinite set families

Problem statement

If a family of countably infinite sets has no pair intersecting in exactly two elements, what is the fewest colours always sufficient to colour their union so that none of the sets is monochromatic?

GPT-5.4 Pro produced an argument showing that no fixed finite number of colours always suffices for a countable family of infinite sets with no pair intersecting in exactly two elements.

Details and sources

AI contribution

Chojecki prompted and checked the model’s construction before releasing a written proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked paper; official record updated

Publication

Public proof manuscript and official problem record

Preprint / manuscript

Official problem record Discussion and formalization notes

Activity evidence

One original source and a small public discussion were located; the score is a conservative editorial estimate.

A forum Lean experiment formalizes a conditional route using an Erdős–Rado-style input; it is not an unconditional end-to-end formal proof and is therefore not counted as such.

No finite uniform bound

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.4 Pro + Przemek Chojecki
Verification: Author-checked paper; official record updated
Open for: 39 years
Research activity*: 1/5

24 Jul 2026Matroid theory and algebraic combinatorics

Rota’s matroid-flat unimodality conjecture

Problem statement

For every matroid, is the sequence counting flats of each rank unimodal?

The paper constructs local valleys in the number of flats by rank using a $q$ -lift mechanism, producing explicit counterexamples to the conjectured unimodality.

Details and sources

AI contribution

At the authors’ suggestion the model found a seed example for a weaker reciprocal-convexity question. The human authors generalized the mechanism and wrote the mathematical paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked explicit construction

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Activity evidence

The conjecture is attributed to Rota and sits inside a central, highly active program on log-concavity and matroid invariants.

The explicit construction disproves the universal unimodality claim. Minimal counterexample questions and broader structural classifications remain open.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT-5.6 Pro
Verification: Author-checked explicit construction
Open for: 56 years
Research activity*: 5/5

22 Jul 2026Spectral graph theory

Fajtlowicz’s graph-energy conjecture

Problem statement

Does every graph of order

n

and independence number

\alpha(G)

have energy at least

2(n-\alpha(G))

The authors prove $\mathcal E(G)\geq2(n-\alpha(G))$ for every graph $G$ , resolving the conjectured lower bound relating graph energy and independence number.

Details and sources

AI contribution

The manuscript acknowledges AI-assisted ideation but does not identify the model or isolate the model-generated step. The final proof is presented and owned by the authors.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two-author arXiv proof; AI role underspecified

Publication

Public arXiv manuscript and TeX source; peer review pending

Preprint / manuscript

Independent spectral proof

Activity evidence

The conjecture arose from Fajtlowicz’s Graffiti program and has a substantial spectral-graph-theory literature.

The theorem and proof are explicit, but the AI contribution cannot be reconstructed from the public record. No formalization or peer-review report was located.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AI tools (not specified)
Verification: Two-author arXiv proof; AI role underspecified
Open for: 39 years
Research activity*: 4/5

17 Jul 2026Extremal combinatorics and random CSPs

Feige’s hypergraph Moore-bound conjecture

Problem statement

At the conjectured density, must every

k

-uniform hypergraph contain a short nontrivial even cover, with constants free of superfluous polylogarithmic factors?

The paper proves the original hypergraph Moore bound for all $k\geq3$ without the extra polylogarithmic factors left by earlier work, bounding the size of the smallest nontrivial even cover at the conjectured density scale.

Details and sources

AI contribution

The authors state that GPT-5.6 Sol found the core technical innovation for even-uniform hypergraphs. They translated the argument, extended it to odd $k$ , checked it, and wrote the paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Five-author proof plus independent spectral proof

Publication

Public arXiv proof and an independent human-derived preprint

Preprint / manuscript

AI-assisted proof

Activity evidence

Feige posed the conjecture in 2008; it motivated sustained work on random-CSP refutation, Kikuchi matrices, and average-case complexity.

The AI-assisted manuscript proves the full conjecture. A separate July 28 spectral proof independently confirms the theorem and reports that its own proof ideas were human; neither manuscript has yet completed peer review.

Conjecture proved for every uniformity

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol / GPT-5.5 Pro / Claude Opus 4.8 / Claude Fable 5
Verification: Five-author proof plus independent spectral proof
Open for: 18 years
Research activity*: 5/5

26 Jul 2026Newton polytopes and symmetric polynomials

Powers of the Vandermonde determinant are eventually non-SNP

Problem statement

For each fixed positive power of the Vandermonde determinant, is its Newton polytope non-saturated in all sufficiently large dimensions?

For every fixed positive power of the Vandermonde determinant, sufficiently many variables produce a non-saturated Newton polytope. For every even $k\geq4$ , the proof gives an explicit lattice point whose coefficient vanishes.

Details and sources

AI contribution

Codex supplied the key even-power construction and proof strategy. Le and Weber checked and organized the argument, and included the complete model transcript in the paper appendix.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked with an independent assumption-free verifier

Research paper and full transcript

Publication

Public arXiv paper, complete transcript, and Lean repository

Preprint / manuscript

Activity evidence

The conjecture belongs to the active study of saturated Newton polytopes, symmetric polynomials, and combinatorial positivity.

The odd case follows from alternation and the quadratic case was already known. The repository’s Ward-recurrence route gives an independent assumption-free verification of the new even-power seed case.

Monical–Tokcan–Yong conjecture proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: OpenAI Codex / GPT-5.6 Sol Extra High
Verification: Lean checked with an independent assumption-free verifier
Research activity*: 3/5

100

27 Jul 2026Finite posets and sorting under partial information

Gold Partition Conjecture through fourteen elements

Problem statement

Does every finite non-chain poset satisfy the Gold Partition property, and hence the

1/3

–

2/3

balance bound?

Every non-chain poset on at most $14$ elements satisfies the Gold Partition Conjecture. At order $14$ , exact certificates cover all $1{,}338{,}193{,}159{,}770$ non-chain isomorphism classes, so the $1/3$ – $2/3$ conjecture also holds through order $14$ .

Details and sources

AI contribution

The systems were used extensively for implementation and optimization, experiment design, literature discovery, and manuscript drafting. Gupta selected the methods and checked the sources and reported computations.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Exact archived computation with independent small-order checks

Source, tests, and aggregate data Complete shard archive and manifest

Publication

Public arXiv paper, source and tests, and a complete Zenodo shard archive

Preprint / manuscript

Activity evidence

The $1/3$ – $2/3$ conjecture is a major open problem in poset theory and sorting; this is the first reported exhaustive Gold Partition extension beyond $11$ elements since 2006.

Neither conjecture is solved in general. The production run uses exact integer recurrences and complete shard accounting, but there is no independent per-poset implementation above order $9$ and no Lean formalization.

Computational frontier extended to order

14

Problem origin: Origin not yet traced
System: Claude Code 5 family / OpenAI Codex GPT-5.6 family
Verification: Exact archived computation with independent small-order checks
Research activity*: 4/5

101

28 Jul 2026Differential posets and enumerative combinatorics

Stanley’s rankwise lower-bound conjecture for differential posets

Problem statement

Must every

r

-differential poset have at least as many elements in each rank as the Cartesian power

Y^r

of Young’s lattice?

For every $r\geq3$ , an infinite $r$ -differential poset has fourth rank smaller than that of $Y^r$ by $\lfloor r/3\rfloor$ . At $r=3$ the initial ranks are $1,3,9,22,50$ , versus $51$ for $Y^3$ .

Details and sources

AI contribution

The paper states that TARS generated the counterexample autonomously; the five human authors subsequently examined and independently verified it.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human-verified explicit construction

Publication

Public revised arXiv manuscript; peer review pending

Preprint / manuscript

Activity evidence

Stanley posed the extremal rank problem in 1988; it belongs to a sustained literature on differential posets and rank growth.

The universal coefficientwise conjecture is disproved for every $r\geq3$ . The cases $r=1,2$ and the exact minimum fourth-rank size remain open.

Universal lower bound disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: TARS agent system (foundation model not disclosed)
Verification: Human-verified explicit construction
Open for: 38 years
Research activity*: 4/5

102

28 Jul 2026Differential posets and generating functions

Stanley’s weighted-multichain rationality problem

Problem statement

For every differential poset

P

and fixed

k

, must its weighted

k

-multichain series be a rational multiple of the

k

th power of its rank series?

A locally finite $1$ -differential poset has $M_{P,2}(q)/F_P(q)^2$ nonrational over every characteristic-zero field. The construction in fact yields continuum many nonrational quotient series.

Details and sources

AI contribution

The authors state that TARS found the counterexample autonomously; Xinan Dai reconstructed the proof, reviewed the literature, wrote the manuscript, and manually verified the mathematics.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human-reconstructed and manually verified proof

Publication

Public revised arXiv manuscript; peer review pending

Preprint / manuscript

Merged formal proof and disclosure

Activity evidence

A named 1988 problem in the established theory of differential posets and generating functions.

The counterexample already works for $k=2$ and directly refutes the rational-factor formula in Stanley’s 1988 Problem 4.

Problem answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: TARS agent system (foundation model not disclosed)
Verification: Human-reconstructed and manually verified proof
Open for: 38 years
Research activity*: 3/5

103

25 Jul 2026Graph invariants and eccentricity

Written on the Wall II, Graph Conjecture 103

Problem statement

For every connected graph

G

, is

\alpha(G)\leq\lfloor b(G)-\log(\operatorname{ecc}_{avg}(G))\rfloor

, where

b(G)

is the largest induced-bipartite-subgraph order?

An $11$ -vertex graph has independence number $9$ , largest induced-bipartite-subgraph size $10$ , and average eccentricity $30/11$ , while the proposed upper bound evaluates to $8$ .

Details and sources

AI contribution

ChatGPT assisted the counterexample search and mathematical write-up; Codex assisted independent verification, Lean formalization, and repository validation.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + exhaustive subset enumeration

Publication

Merged Formal Conjectures proof and reproducible finite check

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Part of the long-running Graffiti/Written on the Wall graph-conjecture collection.

The commit explicitly says that historical or worldwide priority for the counterexample has not been established. Native evaluation contributes the documented trusted-compiler axioms.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI ChatGPT + Codex (exact ChatGPT model undisclosed)
Verification: Lean checked + exhaustive subset enumeration
Open for: 1996 Written on the Wall II conjecture
Research activity*: 3/5

104

26 Jul 2026Extremal graph invariants

Written on the Wall II, Graph Conjecture 109

Problem statement

Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order?

The family $\overline K_{2r+1}\vee(K_r\sqcup K_r)$ violates the proposed bound for every $r\geq3$ . At $r=3$ , the $13$ -vertex graph has $\alpha=7$ , induced-bipartite size $9$ , residue $2$ , and claimed upper bound $6$ .

Details and sources

AI contribution

The merged record credits GPT-5.6 Sol Max with the search, infinite counterexample family, proof, Lean formalization, independent verifiers, and write-up under human direction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + independent Python/C++ enumeration

Merged strengthened result Lean proof and independent verifiers

Publication

Merged Formal Conjectures record, immutable Lean certificate, verifier repository, and CI

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A checked resolution in the long-running Written on the Wall graph-conjecture collection.

The authors report no earlier indexed resolution, while explicitly declining to rule out unpublished or unindexed priority. The repository also preserves the earlier $21$ -vertex certificate.

Initial 21-vertex formal proof

Conjecture disproved by an infinite family

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol Max in Codex
Verification: Lean checked + independent Python/C++ enumeration
Open for: 1996 Written on the Wall II conjecture
Research activity*: 3/5

105

21 Jul 2026Graph girth, induced trees, and degree

Written on the Wall II, Graph Conjecture 143

Problem statement

For every connected graph

G

, is

\operatorname{girth}(G)+1

at most the product of its largest induced-tree order and second-smallest degree?

For every finite connected graph, $\operatorname{girth}(G)+1\leq L_s(G)\,\sigma(G)$ , where $L_s$ is the largest induced-tree order and $\sigma$ is the second-smallest degree.

Details and sources

AI contribution

The model developed the argument and Lean artifact; the contribution was reviewed and merged into the Formal Conjectures repository.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; maintainer review completed

Merged Formal Conjectures proof Public proof archive

Publication

Merged formal proof and public proof archive

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Part of a long-running graph-conjecture corpus and now represented by a merged formal proof.

This Written on the Wall II Conjecture 143 is distinct from the unrelated spectral Graffiti Conjecture 143 already indexed.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Thinking
Verification: Lean checked; maintainer review completed
Open for: 1996 Written on the Wall II conjecture
Research activity*: 3/5

106

13 Jul 2026Arithmetic progressions and Ramsey multiplicity

Erdős Problem #1186 — minimum density of monochromatic $3$ -APs

Problem statement

What is the minimum asymptotic density

\delta_k

of monochromatic

k

-term arithmetic progressions in every two-coloring of

\{1,\ldots,n\}

The exact certificate gives $\delta_3=117/2192$ , matching the known $548$ -bead coloring and settling the Parrilo–Robertson–Saracino conjecture for $3$ -term progressions.

Details and sources

AI contribution

Star Fleet generated the proof architecture and exact certificate. The release includes two independent arithmetic checkers, Lean proofs of the certificate schemas, and a Lean-checked extremal census.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Dual exact checkers + partial Lean formalization

Official problem record Community discussion Star Fleet report

Publication

Public proof report, exact certificates, Rust/Python checkers, and Lean schema/census proofs

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

The discrete-to-continuum reduction is hand-verified, and the $548\times548$ PSD certificate is checked outside the Lean kernel. Full end-to-end Lean formalization remains in progress, and the parent problem asks about $\delta_k$ beyond $k=3$ .

Exact

k=3

constant claimed; general

k

open

Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Dual exact checkers + partial Lean formalization
Research activity*: 4/5

107

13 Jul 2026Ramsey theory

Erdős Problem #129 — printed Ramsey bound

Problem statement

For the printed definition of

R(n;3,r)

, is there a constant

C(r)>1

such that

R(n;3,r)<C(r)^{\sqrt n}

The literal printed bound is false: for $n\geq120$ , $2^{\lfloor n/120\rfloor}<R(n;3,2)\leq2^{2n}$ , ruling out every $C^{\sqrt n}$ upper bound.

Details and sources

AI contribution

Star Fleet independently reconstructed and formalized the disproof after Antonio Girão had already identified the issue; the release explicitly assigns him priority.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; prior human result credited

Official problem record Community discussion Star Fleet report

Publication

Public source audit and complete Lean verification bundle

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

The community record suspects the historical source intended a different, unstated parameter. This artifact answers only the definite printed proposition and does not invent a replacement.

Known disproof independently formalized

Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked; prior human result credited
Research activity*: 3/5

108

13 Jul 2026Extremal graph theory

Erdős Problem #584 — dense-graph short-cycle subgraphs

Problem statement

Must every graph with

n

vertices and

\delta n^2

edges contain large subgraphs in which every two edges lie on specified short cycles?

A sorry-free Lean construction of sufficiently dense high-girth graphs refutes the statement when the density parameter is allowed to shrink with the graph order.

Details and sources

AI contribution

Star Fleet reconstructed and formalized the literal counterexample after Przemek Chojecki had identified the issue first.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked for the literal statement

Official problem record Community discussion Star Fleet report

Publication

Public fidelity audit and complete Lean project

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

The historically intended fixed-density formulation plausibly remains open. The site therefore classifies this as a statement variant rather than a resolution of that intended problem.

Literal wording disproved; intended variant open

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked for the literal statement
Research activity*: 3/5

109

13 Jul 2026Infinite Ramsey theory

Erdős Problem #638 — finite and infinite triangle-Ramsey families

Problem statement

Does finite triangle-Ramsey richness of a hereditary family force an infinite-cardinal triangle-Ramsey graph whose finite subgraphs all lie in that family?

Even for an isomorphism- and subgraph-closed family containing finite $n$ -color triangle-Ramsey graphs for every $n$ , no corresponding graph need be triangle-Ramsey for infinitely many colors.

Details and sources

AI contribution

Star Fleet independently reconstructed and formalized both the literal and hereditary readings after InfiniteInsights had solved the problem first.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked for both natural readings

Official problem record Community discussion Star Fleet report

Publication

Public report, pinned statements, and complete Lean bundle

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

The report gives the prior solver full credit. The AI contribution indexed here is the complete machine-checked reconstruction and source-fidelity analysis.

Known negative answer independently formalized

Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked for both natural readings
Research activity*: 3/5

110

30 Jul 2026Zero-error information theory and graph products

Record Shannon-capacity lower bounds for odd cycles

Problem statement

Determine the Shannon capacities of odd cycles beyond

C_5

, or improve the best explicit independent-set bounds in their strong graph powers.

Explicit independent sets improve the known bounds to $\Theta(C_7)>3.258020$ , $\Theta(C_{11})>5.289773$ , $\Theta(C_{13})>6.300109$ , and $\Theta(C_{15})>7.301399$ . The $C_{15}$ construction was added in the 30 July revision.

Details and sources

AI contribution

The model generated and executed search programs from repeated prompts and returned the explicit constructions. The four authors checked that every reported set is independent.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked constructions; public data and checking code

Constructions and verification code

Publication

Public arXiv manuscript, prompts, construction files, and verification scripts

Preprint / manuscript

Research paper, revised 30 July

Activity evidence

Shannon introduced zero-error capacity in 1956; exact capacities of odd cycles beyond $C_5$ remain notorious problems in information theory.

These are record-setting advances, not exact evaluations of the Shannon capacities. For every odd cycle $C_{2r+1}$ with $r>2$ , the exact value remains unknown.

Four record lower bounds; exact capacities remain open

Problem origin: Human-source problem
System: ChatGPT-5.6 Sol Pro
Verification: Author-checked constructions; public data and checking code
Open for: 70 years
Research activity*: 5/5

111

29 Jul 2026Extremal graph theory

Erdős Problem #608 — pentagonal edges

Problem statement

Must every

n

-vertex graph with more than

n^2/4

edges have at least

2n^2/9

edges contained in a

5

-cycle?

A Lean 4 development machine-checks a rational Füredi–Maleki construction with an explicit gap $\varepsilon=47/7056$ below the proposed $2n^2/9$ bound. It also proves the statement-to-cycle bridge and handles the literal small- $n$ reading separately.

Details and sources

AI contribution

Claude agents wrote the Lean proofs in one overnight Claude Code session from a frozen, human-approved statement. The mathematical construction predates this formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; zero sorrys; axiom-audited and clean-room rebuilt

Complete Lean development and audit

Publication

Public Lean repository, CI configuration, audit notes, and original mathematics source

Preprint / manuscript

Original mathematics source

Activity evidence

A documented editorial estimate based on the official problem record, the extremal-graph literature, and the new formal audit.

This is an AI-produced formalization of known mathematics, not a newly discovered disproof. It formalizes the asymptotic counterexample but not the deeper matching lower bound of Grzesik–Hu–Volec.

Official Erdős problem record Claude Code proof session Official formal-status review, merged 31 July

Known disproof formalized end to end

Problem origin: Human-source problem
System: Claude Fable 5
Verification: Lean checked; zero sorrys; axiom-audited and clean-room rebuilt
Open for: Known disproof now completely machine checked
Research activity*: 3/5

112

29 Jul 2026Additive combinatorics

Optimal exponent relating sumsets and difference sets

Problem statement

Can the exponent

1/2

in the lower sum-difference inequality

\sigma(A)^{1/2}\leq\delta(A)

be increased uniformly over finite subsets of abelian groups?

For finite nonempty $A$ in an abelian group, write $\sigma(A)=|A+A|/|A|$ and $\delta(A)=|A-A|/|A|$ . An explicit family $A_K\subset\mathbb Z$ satisfies $\log\sigma(A_K)/\log\delta(A_K)\to2$ , proving that the classical exponent $1/2$ in $\sigma(A)^{1/2}\leq\delta(A)$ cannot be improved.

Details and sources

AI contribution

Haowei Lin and Shanda Li report that the open-weight Hyra research agent developed the construction and proof during a long autonomous run. GPT-5.6 Sol later translated the natural-language argument into Lean 4; the authors independently checked and corrected the mathematics.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked proof and public Lean development

Lean 4 formalization

Publication

Public arXiv manuscript, AI-provenance report, and Lean 4 repository

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the classical sum-difference inequalities, their extensive additive-combinatorics literature, and the new formal proof artifact.

The result settles the optimal universal power-law exponent. It does not claim a complete classification of finite sets near equality.

Sharp exponent settled

Claimed outcome: Proved
Problem origin: Human-source problem
System: Hyra (open-weight Hy3) / GPT-5.6 Sol
Verification: Author-checked proof and public Lean development
Open for: Long-standing sharp-exponent question
Research activity*: 5/5

113

27 Jul 2026Spectral and extremal graph theory

Nikiforov’s spectral consecutive-cycle constant

Problem statement

What is the largest constant

C

forcing cycles of every length up to

(C-o(1))n

from the spectral condition

\rho(G)>\sqrt{\lfloor n^2/4\rfloor}

For every $\varepsilon>0$ and all sufficiently large $n$ , an $n$ -vertex graph with $\rho(G)>\sqrt{\lfloor n^2/4\rfloor}$ contains every cycle $C_g$ for $3\leq g\leq((3-\sqrt5)/2-\varepsilon)n$ . The constant $(3-\sqrt5)/2$ is best possible.

Details and sources

AI contribution

The second author and ChatGPT independently found gaps in an earlier embedding lemma. The present lemma and its proof were first suggested by the model, then rechecked, rewritten, and integrated by Bo Ning and Mingqing Zhai.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two-author proof; peer review pending

Publication

Public arXiv manuscript with a precise AI-use declaration

Preprint / manuscript

Exact certificate verifier Full independent audit script

Activity evidence

A documented editorial estimate based on repeated improvements since Nikiforov’s 2008 problem and the active spectral-extremal graph literature.

The theorem settles Nikiforov’s 2008 asymptotic problem for sufficiently large graphs. It does not supply an explicit optimal finite threshold $n_0(\varepsilon)$ .

Sharp asymptotic constant determined

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6
Verification: Two-author proof; peer review pending
Open for: 18 years
Research activity*: 4/5

114

30 Jul 2026Spectral graph theory

Signature of connected line graphs

Problem statement

Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one?

A $14$ -vertex graph has a connected line graph with adjacency signature $2$ , refuting the proposed upper bound $1$ . Chaining the witness gives connected line graphs of signature $k+1$ for every $k\geq1$ , so the signature is unbounded.

Details and sources

AI contribution

The $14$ -vertex witness came from a ChatGPT-assisted search. Claude assisted an independent $48$ -vertex search and the proof development leading to the unbounded family. Luke Francis and Trevor Uptain reproduced the computations with separately coded exact-arithmetic audits.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Exact finite certificates and independent audit implementations

Publication

Public revised arXiv manuscript; peer review pending

Preprint / manuscript

Author’s LinkedIn discussion

Activity evidence

A documented editorial estimate based on a recent spectral-graph conjecture, its computational evidence, and the new infinite counterexample family.

The conjecture was recent, but the result is stronger than a single finite disproof: it rules out every universal constant bound on the signature of connected line graphs. Version 2 (29 Jul) corrected the description of the 48-vertex mechanism and added Paone’s independent work; the 14-vertex witness and unbounded-family conclusion are unchanged.

Conjecture disproved; no constant repair possible

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Pro / Claude Fable 5
Verification: Exact finite certificates and independent audit implementations
Open for: Conjecture published in 2026 and rapidly disproved
Research activity*: 3/5

115

29 Jul 2026Distance spectra and extremal graph theory

WOW-284 distance-spectrum conjecture

Problem statement

Must every connected graph

G

of order at least three and girth at least five satisfy

\delta^*(G)\leq-\lambda_{\min}(D(G))

Exact connected graphs of orders $38,39,40,42$ , and $50$ violate the proposed inequality between minimum dual degree and the least distance eigenvalue. The paper also proves structural obstructions and deletion-stability results for the Hoffman--Singleton example.

Details and sources

AI contribution

The author reports using the model for adversarial proof checking, proof exploration, and Lean formalization. The mathematical paper distinguishes the portions checked in Lean from the remaining handwritten arguments.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked in stated scope; exact certificates; author preprint

Lean source and exact certificates

Publication

Public arXiv manuscript and versioned Lean 4 verification repository

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the WOW graph-conjecture record, the exact counterexample family, and the unusually complete verification package.

Lean 4 kernel-checks the $50$ -vertex graph-level counterexample, the finite spectral certificates at orders $38,39,40,42$ , and the analytic linear-program optimum and rigidity. The paper explicitly identifies arguments outside that formalized scope.

Conjecture disproved with exact certificates

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: OpenAI ChatGPT-5.6 Sol Pro
Verification: Lean checked in stated scope; exact certificates; author preprint
Research activity*: 3/5

116

27 Jul 2026Matroid Kazhdan–Lusztig polynomials

Kazhdan–Lusztig polynomial shape conjectures for matroids

Problem statement

Are Kazhdan--Lusztig polynomials of matroids always unimodal, log-concave, or real-rooted?

Over every finite field there are representable matroids whose Kazhdan--Lusztig polynomials are not unimodal. Consequently the proposed log-concavity and real-rootedness properties are also false.

Details and sources

AI contribution

Rethlas discovered the initial rank-seven binary-matroid counterexample to real-rootedness. Cheng and Liu checked the example and developed it into the broader non-unimodality theorem over every finite field.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked proof with exact AI discovery record

Publication

Public arXiv manuscript whose appendix reproduces the accepted Rethlas output and discovery trajectory

Preprint / manuscript

Readable proof and Rethlas appendix Community discussion and independent-announcement note

Activity evidence

A documented editorial estimate based on sustained work on positivity and shape properties of matroid Kazhdan--Lusztig polynomials.

The AI contribution was the initial real-rootedness counterexample; the authors supplied the general construction and the stronger non-unimodality result. A MathOverflow community update reports an independent human counterexample announced at ICM 2026, but no separate public manuscript was located, so it is recorded as discussion rather than a second claim. No formal proof was located.

Unimodality, log-concavity, and real-rootedness disproved

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Rethlas (GPT-5.6 Sol max)
Verification: Author-checked proof with exact AI discovery record
Research activity*: 4/5

117

6 Jul 2026Matroid Chow rings and wonderful compactifications

Integral tangent classes for matroid wonderful models

Problem statement

Can the tangent bundle class of a realizable wonderful compactification be lifted to a canonical integral combinatorial tangent class for every matroid and building set?

Danus constructs an integral combinatorial tangent class with the expected realizable specialization and the $P^K=\mathrm{Hilb}$ identity. A local justification used only for one Chern-- $\alpha$ lower bound is incomplete as written.

Details and sources

AI contribution

Seven Danus workers built the rational and then integral construction. Human experts corrected the initial scope and later identified the local Lemma 8.7 gap; the underlying lemma is independently known to be true.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Core construction author checked; local gap disclosed

Claim audit

The manuscript’s proof of Lemma 8.7 is incomplete. The system paper says this affects the Chern–alpha lower-bound justification but not the tangent-class construction or the $P^K=\mathrm{Hilb}$ identity.

Publication

Public preprint, detailed system report, and independent related paper

Preprint / manuscript

Result preprint Independent related result

Activity evidence

A documented editorial estimate based on current work connecting matroid K-theory, Chow rings, and wonderful compactifications.

The record is marked partial so the verified core is not conflated with the one component whose written proof has a documented gap.

Danus audit and gap disclosure

Construction proved; one local proof gap documented

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Danus / Claude Opus 4.8 / GPT-5.5 workers
Verification: Core construction author checked; local gap disclosed
Claim audit: Issue documented
Research activity*: 4/5

118

23 Jul 2026Algebraic combinatorics and graph colorings

Stanley’s claw-free Schur-positivity conjecture

Problem statement

Is the chromatic symmetric function

X_G(\mathbf x)

Schur positive for every claw-free graph

G

Two explicit $12$ -vertex line graphs are claw-free but have negative $s_{(3,3,3,3)}$ coefficients, respectively $-64$ and $-40$ , in their chromatic symmetric functions.

Details and sources

AI contribution

The model found both counterexamples after an 83-minute run using SAT-assisted search. Matherne and Morales extracted, checked, and presented the finite witnesses and their Schur expansions.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author preprint + independent reproduction + exhaustive minimality check

Independent witness and verifier scripts Independent exact counterexample and minimality paper

Publication

Public arXiv note with exact graphs, Sage computations, model session, and an independent community verifier

Preprint / manuscript

Research paper Infinite-family follow-up

Activity evidence

Stanley posed the conjecture in 1995. It connects a heavily studied graph invariant to symmetric-function positivity and had accumulated substantial positive evidence in important graph classes.

Line graphs are automatically claw-free, so either negative coefficient is a finite counterexample. Prajapati's 22 August v2 independently reports a census of all 216,777 connected claw-free graphs through eleven vertices and 1,728,404 at twelve vertices, finding exactly two non-Schur-positive twelve-vertex classes. Twelve is therefore the reported minimum order. A separate follow-up gives infinite families.

Discovery session MathOverflow verification discussion

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT-5.6 Sol Pro
Verification: Author preprint + independent reproduction + exhaustive minimality check
Open for: 31 years
Research activity*: 4/5

119

23 Jul 2026Newton polytopes and chromatic symmetric functions

Monical’s SNP conjecture for Schur-positive chromatic functions

Problem statement

X_G

is Schur positive, must

X_G(x_1,\ldots,x_k)

have saturated Newton polytope for every finite

k

A $12$ -vertex bipartite graph has a Schur-positive chromatic symmetric function whose three-variable specialization contains weights $(6,6,0)$ and $(8,2,2)$ but omits their midpoint $(7,4,1)$ , so its Newton polytope is not saturated.

Details and sources

AI contribution

The model found the graph on the first prompt in 37 minutes. The authors supplied a direct coloring obstruction for the missing midpoint and checked Schur positivity with SageMath.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked finite witness and combinatorial proof

Publication

Public arXiv note with the explicit graph, proof, computation, and discovery session

Preprint / manuscript

Research paper Readable full paper

Activity evidence

Monical posed the conjecture in a 2018 thesis. Later work verified meaningful special cases and showed that any counterexample requires at least twelve vertices.

The witness disproves the implication from Schur positivity to the saturated-Newton-polytope property. It contains claws, so the more restricted search for a claw-free counterexample remains open.

Discovery session

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT-5.6 Sol Pro
Verification: Author-checked finite witness and combinatorial proof
Open for: 8 years
Research activity*: 3/5

120

27 Apr 2026Sidon sets and additive combinatorics

Erdős Problem #43 — paired Sidon sets

Problem statement

If Sidon sets

A,B\subseteq\{1,\ldots,N\}

satisfy

(A-A)\cap(B-B)=\{0\}

, must

\binom{|A|}{2}+\binom{|B|}{2}\leq\binom{f(N)}{2}+O(1),

where

f(N)

is the largest Sidon-set size in

[N]

? If

|A|=|B|

, can the right side be improved by a fixed positive proportion?

Both questions have negative answers. A Barreto construction disproves the equal-size bound; the unrestricted bound fails as a consequence of GPT-5.5 Pro’s full solution of Erdős Problem #42.

Details and sources

AI contribution

GPT-5.5 Pro supplied the decisive #42 theorem. Aristotle and Claude helped formalize the separate Barreto construction used for the equal-size clause.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Site confirmed; component Lean proofs

Publication

Official Erdős Problems resolution record, discussion, and formal artifacts

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A prize-backed Sidon problem with multiple Erdős sources, classical extremal constructions, and active modern discussion and formalization.

The official #43 page marks the full problem disproved, but does not badge the combined implication itself as wholly Lean-verified. The formal status here therefore remains mixed.

Official problem record Discussion and formalization Decisive Problem #42 record

Both proposed bounds disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 Pro / Aristotle / Claude
Verification: Site confirmed; component Lean proofs
Open for: 44 years
Research activity*: 4/5

121

24 Mar 2026Extremal combinatorics

Ramsey-style hypergraph construction

Problem statement

Let

H(n)

be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than

n

. If

k_1=1

and

k_n=\lfloor n/2\rfloor+k_{\lfloor n/2\rfloor}+k_{\lceil n/2\rceil}

, prove that

H(n)\geq c\,k_n

for some constant

c>1

, already for

n=15

, and give a constructive algorithm.

GPT-5.4 Pro found a four-way frame construction proving a uniform constant-factor improvement over the known recurrence for H(n), beginning at n = 15. The contributor confirmed the argument and is preparing it for publication.

Details and sources

AI contribution

Kevin Barreto and Liam Price elicited the first solution. The model supplied the construction, proof, recurrence, and executable algorithm.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Contributor verified

Publication

Full transcript and proof write-up; journal paper in preparation

Preprint / manuscript

Proof write-up

Activity evidence

The question arose from a 2019 research line. Epoch reports roughly 5–10 serious attempts and rates the result as moderately interesting.

This is the full general challenge, not merely the finite warm-up. Epoch later obtained independent solutions from Claude Opus 4.6, Gemini 3.1 Pro, and GPT-5.4.

Epoch solution record Full GPT-5.4 transcript

Problem solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Contributor verified
Open for: 7 years
Research activity*: 2/5

122

10 Jul 2026Graph theory

Cycle Double Cover Conjecture

Problem statement

Does every finite bridgeless graph have a collection of cycles in which every edge appears exactly twice?

A short argument claims that every finite bridgeless loopless multigraph has a cycle double cover. Independent graph theorists have since published expositions of the proof.

Details and sources

AI contribution

OpenAI reports that a 64-agent run produced the proof in under an hour; Codex assisted with the writeup.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked end to end + specialist expositions

Sang-il Oum exposition Jim Geelen exposition

Publication

OpenAI proof note, multiple specialist expositions, and a public Lean development; not yet journal reviewed

Preprint / manuscript

Activity evidence

A central graph-theory conjecture with sustained work, many equivalent formulations, and broad expert attention.

The proof has moved beyond a bare announcement: expositions by Jim Geelen and Sang-il Oum treat the argument as a proof, and the public Lean development checks the complete finite-graph statement.

Proof note Full prompt

Proof released; archival review ongoing

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra
Verification: Lean checked end to end + specialist expositions
Open for: 53 years
Research activity*: 5/5

123

14 Jul 2026Graph theory

Sabidussi compatibility conjecture

A proof establishes the compatibility conjecture for graph products attributed to Sabidussi, closing the problem in the formulation studied by the authors.

Details and sources

AI contribution

The authors report that Pro found the proof and Sol helped turn the argument into a polished manuscript.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked + author review

Publication

Public arXiv preprint with a companion Lean repository

Preprint / manuscript

Formalization post-mortem Current problem record Discussion thread

The machine-checked development is strong evidence for the encoded theorem. As always, the accompanying paper is needed to audit the correspondence between the Lean statement and the historical conjecture.

Resolved in preprint

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.6 Pro + GPT-5.6 Sol
Verification: Lean checked + author review

124

5 Dec 2025Ramsey theory

Erdős Problem #124

Problem statement

For bases

3\leq d_1<\cdots<d_r

satisfying the stated reciprocal-sum condition, can every sufficiently large integer be represented as a sum of distinct powers of the

d_i

? Under a coprimality assumption, does the same remain true after forbidding all low powers?

Aristotle proved and formalized the literal formulation then displayed in the database, but historical evidence indicates that Erdős intended a stronger statement that remains open.

Details and sources

AI contribution

Harmonic’s theorem-proving system generated a Lean proof of the supplied formal statement.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Reported Lean proof; linked statement contains sorry

Claim audit

Statement fidelity: the Lean target captured a weaker formulation than the historically intended conjecture, and the public linked statement is not a complete proof artifact.

Publication

Community post-mortem and official problem record

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Several source references exist, but the AI proof addressed a weaker literal formulation.

The AI work addressed a weaker literal formulation, while the intended Burr–Erdős–Graham–Li problem remains open. The linked Formal Conjectures statement contains `sorry`, and no complete source file for the solved variant was located.

Weaker written variant only

Problem origin: Human-source problem
System: Aristotle
Verification: Reported Lean proof; linked statement contains sorry
Claim audit: Issue documented
Research activity*: 2/5

125

14 May 2025Extremal combinatorics

AlphaEvolve mathematics portfolio

Evolution over LLM-generated programs improved best-known constructions across a portfolio of open problems, including a 593-point lower bound for the 11-dimensional kissing number.

Details and sources

AI contribution

Language models proposed executable constructions while an evolutionary loop and problem-specific evaluators selected improvements.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Executable checks + expert review

Publication

Public technical report and repository covering 67 problems

Preprint / manuscript

Mathematics portfolio

These are genuine record improvements, not complete solutions of the surrounding open problems. Reproducible programs offer a stronger audit trail than an announcement alone, but they are not proof-assistant certificates.

DeepMind announcement Public problem repository

New bounds; parent problems open

Problem origin: Origin not yet traced
System: AlphaEvolve / Gemini
Verification: Executable checks + expert review

126

17 Oct 2025Claim audit

GPT-5 “ten Erdős problems” claim

A public claim that GPT-5 had solved ten previously open Erdős problems was walked back after most examples were traced to existing literature or misclassified database entries.

Details and sources

AI contribution

The model retrieved or reconstructed relevant arguments, but the announcement overstated their novelty as new mathematical solutions.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Community audit contradicted claim

Claim audit

Novelty failure: the announcement treated known or misclassified results as newly solved open problems.

Publication

Original social post deleted; contemporary reporting and the community audit are public

Preprint / manuscript

Primary GPT-5 research report Independent Erdős-case audit

The episode is retained as a negative control. Literature search can be mathematically useful, but reproducing a known solution is not the same as settling an open problem.

Contemporary reporting Community Erdős AI ledger

Novelty claim withdrawn

Problem origin: Origin not yet traced
System: GPT-5
Verification: Community audit contradicted claim
Claim audit: Issue documented

127

14 Dec 2023Additive combinatorics

FunSearch cap-set constructions

LLM-guided program search found larger cap sets in several dimensions and new constructions for related combinatorial problems; it did not solve the general cap-set problem.

Details and sources

AI contribution

The model proposed short programs inside an evolutionary search loop, with deterministic evaluators scoring candidate constructions.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Peer reviewed + executable checks

Publication

Peer-reviewed Nature paper with public code and constructions

Preprint / manuscript

No public preprint or manuscript located.

FunSearch is included because it established the modern pattern of AI-assisted mathematical discovery with machine-checkable outputs. Its achievements are improved examples and bounds, not a theorem resolving the full asymptotic question.

Nature paper DeepMind account Code and constructions

Construction records improved

Problem origin: Origin not yet traced
System: FunSearch / Codey
Verification: Peer reviewed + executable checks

128

9 Jul 2026Additive combinatorics

Erdős–Szemerédi sum–product conjecture over $\mathbb R$

An autonomous pipeline produced seven correct and structurally different disproofs of the real sum–product conjecture in eight independent trials.

Details and sources

AI contribution

A three-stage agent asked the model for proof plans, complete arguments, and adversarial review without problem-specific mathematical hints.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked + reproducible trials

Publication

Public arXiv preprint; journal review pending

Preprint / manuscript

Preprint and released runs

The result concerns the conjecture over the real numbers. It should not be conflated with every finite-field or quantitative sum–product problem that shares the same name.

Real-field conjecture disproved

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: GPT-5.5 Pro
Verification: Author checked + reproducible trials

129

2 Jun 2026Graph theory

Knuth Hamiltonian-decomposition subproblem

A verified result handles a key subproblem in Knuth’s challenge on Hamiltonian decompositions of even-order Cayley graphs; the full research program remains broader.

Details and sources

AI contribution

An agent decomposed the informal argument, searched Mathlib, and iteratively discharged the resulting Lean goals.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Hamiltonian-cycle formalization

Publication

Public research paper describing the formal development

Preprint / manuscript

LEAP paper

The entry is deliberately marked partial: it is a verified research-level component, not a claim that every Hamiltonian decomposition question posed by Knuth is solved.

Released LEAP artifacts

Key subproblem formalized

Problem origin: Human-source problem
System: LEAP + foundation models
Verification: Lean checked

130

8 Dec 2025Extremal combinatorics

Erdős Problem #1026

Problem statement

For distinct real numbers

x_1,\ldots,x_n

, determine the maximum possible sum along a monotone subsequence.

A weighted monotone-subsequence problem was resolved through a combination of computation, pattern discovery, literature search, packing reformulation, and proof.

Details and sources

AI contribution

Different systems contributed numerical exploration, formal assistance, literature retrieval, and conjecture generation within a human-led collaboration.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + expert synthesis

Tao’s full account Problem record

Publication

Detailed public exposition and direct Lean proof

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A classic extremal-sequence setting with substantial modern discussion around the precise formulation.

The final theorem emerged from many small AI and human contributions plus previously disconnected 2016–17 literature. The direct Lean proof is public.

Resolved with stronger conclusion

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle + GPT + Gemini + AlphaEvolve
Verification: Lean checked + expert synthesis
Open for: 54 years
Research activity*: 3/5

131

27 Apr 2026Sidon sets

Erdős Problem #42

Problem statement

Let

M\geq 1

and

N

be sufficiently large in terms of

M

. Is it true that for every Sidon set

A\subset \{1,\ldots,N\}

there is another Sidon set

B\subset \{1,\ldots,N\}

of size

M

such that

(A-A)\cap(B-B)=\{0\}

A question on the extremal behavior of Sidon-type sequences was resolved in a human–AI collaboration and subsequently encoded in Lean.

Details and sources

AI contribution

GPT-5.5 Pro supplied the central proof with Harjas Sandhu; Codex and GPT systems later assisted formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + community review

Publication

Problem-site record and public formalization

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 6 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The index follows the community ledger’s full-solution classification. The dedicated problem page should be consulted for the exact quantifiers and attribution.

Underlying theorem preprint

Resolved and later formalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro + Codex
Verification: Lean checked + community review
Open for: 31 years
Research activity*: 1/5

132

21 Apr 2026Graph theory

Erdős Problem #610

Problem statement

How large can the clique-transversal number

\tau(G)

be for an

n

-vertex graph? In particular, is

\tau(G)\leq n-\omega(n)\sqrt n

, or even

n-c\sqrt{n\log n}

A graph-theoretic problem of Erdős, Gallai, and Tuza follows from a 2021 theorem of Joret, Micek, Reed, and Smid; an AI-produced note made the implication explicit.

Details and sources

AI contribution

GPT-5.4 Pro wrote out the short deduction from the published theorem. Aristotle encoded a related Lean argument, but treated the external theorem as an assumption.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Published theorem + expert site record

Claim audit

Incomplete formalization: the decisive external theorem is assumed, so the Lean artifact is conditional rather than end to end.

Publication

Official problem record, AI note, and discussion of the incomplete formalization

Preprint / manuscript

Activity evidence

3 cited source records and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The mathematical credit belongs to the 2021 literature. The available Lean file is not end-to-end verification because its key external input is left as an assumption.

Problem record Discussion and provenance Community AI ledger

Problem resolved from prior literature

Problem origin: Human-source problem
System: Aristotle + GPT-5.4 Pro
Verification: Published theorem + expert site record
Claim audit: Issue documented
Open for: 29 years
Research activity*: 2/5

133

9 Jun 2026Graph theory

Erdős Problem #619

Problem statement

For a connected triangle-free graph

G

n

vertices, is there a constant

c>0

such that fewer than

(1-c)n

added edges always suffice to make the diameter 4 while keeping the graph triangle-free?

A counterexample settled a graph-theoretic conjecture of Erdős, Gyárfás, and Ruszinkó.

Details and sources

AI contribution

Fable generated the counterexample and additional systems assisted with checking and Lean formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Independent tracker entry

Publication

Public problem record and formal proof

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

2 cited source records and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The formal artifact certifies the encoded counterexample. The problem page remains the source of truth for the historical formulation.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Claude Fable 5 + Codex + GPT-5.5
Verification: Lean checked
Open for: 28 years
Research activity*: 2/5

134

3 Apr 2026Sidon Sets

Erdős Problem #152

Problem statement

For any

M\geq 1

, if

A\subset \mathbb{N}

is a sufficiently large finite Sidon set, must there be at least

M

sums

a\in A+A

for which neither

a-1

nor

a+1

lies in

A+A

The problem was proved after remaining open for 32 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

DeepMind prover agent is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: DeepMind prover agent
Verification: Lean checked
Open for: 32 years
Research activity*: 1/5

135

17 Jan 2026Number Theory, Covering Systems

Erdős Problem #281

Problem statement

Let

n_1<n_2<\cdots

be such that, for any choice of classes

a_i\pmod{n_i}

, the uncovered integers have density zero. For every

\epsilon>0

, must some

k

make the uncovered density below

\epsilon

for every choice of the first

k

classes?

The conclusion already follows from results of Davenport–Erdős and Rogers. The 2026 AI work supplied an independent proof that was later formalized.

Details and sources

AI contribution

GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 28 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The AI work recovered and formalized an implication of older literature; it did not establish the first historical solution.

Known theorem reconstructed

Problem origin: Human-source problem
System: GPT-5.2 Pro
Verification: Lean checked
Open for: Known from prior literature
Research activity*: 3/5

136

16 Apr 2026Additive Combinatorics

Erdős Problem #741

Problem statement

A+A

has positive upper density, can

A

be split into

A_1\sqcup A_2

so that both

A_1+A_1

and

A_2+A_2

have positive upper density? Is there a basis

A

of order

2

such that every partition prevents both sumsets from having bounded gaps?

The problem was resolved after remaining open for 32 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

DeepMind prover agent is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Formal-proof search preprint Independent resolution preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: DeepMind prover agent
Verification: Lean checked
Open for: 32 years
Research activity*: 1/5

137

3 May 2026Graph Theory, Chromatic Number

Erdős Problem #750

Problem statement

Does there exist an infinite-chromatic graph in which every

m

-vertex subgraph has an independent set of size at least

m/2-f(m)

for some

f(m)\to\infty

GPT-5.5 Pro supplied a proof of the infinite-chromatic graph construction, and the canonical Erdős database accepted the result as proved in Lean on 15 July 2026.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

2 cited source records and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

The claim itself circulated earlier; 15 July is the later community status update confirming the Lean-verified resolution.

Problem proved; official Lean status accepted

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 32 years
Research activity*: 2/5

138

22 Apr 2026Number Theory, Sidon Sets, Additive Combinatorics

Erdős Problem #863

Problem statement

Compare maximal finite sets with at most

r

representations of each sum to maximal sets with at most

r

representations of each difference. Are their asymptotic constants unequal, and is the difference-set constant smaller?

GPT-5.4 Pro recognized that existing bounds imply the answer; the relevant construction was already known by 2002.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The AI contribution is literature synthesis rather than the first proof.

Known bounds recognized

Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Erdős Problems site confirmed
Open for: Known by 2002
Research activity*: 1/5

139

22 Jun 2026Number Theory, Additive Combinatorics

Erdős Problem #865

Problem statement

Is there a constant

C

such that every sufficiently large

A\subseteq[1,N]

of size at least

5N/8+C

contains distinct

a,b,c

for which

a+b

a+c

, and

b+c

also lie in

A

The problem was proved after remaining open for 54 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

3 cited source records and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 54 years
Research activity*: 2/5

140

25 Feb 2026Number Theory, Additive Combinatorics, Ramsey Theory

Erdős Problem #966

Problem statement

For

k,r\geq2

, does there exist a set of integers with no nontrivial

(k+1)

-term arithmetic progression but whose every

r

-colouring contains a monochromatic

k

-term progression?

The 1975 source already reports Spencer’s proof. Aristotle supplied a new Lean proof of the known result.

Details and sources

AI contribution

Aristotle is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This is a formalization milestone rather than a new solution.

Known theorem formalized

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked
Open for: Known by 1975
Research activity*: 1/5

141

16 Jun 2026Graph Theory, Ramsey Theory

Erdős Problem #986

Problem statement

For every fixed

k\geq3

, is the off-diagonal Ramsey number

R(k,n)

bounded below by

n^{k-1}/(\log n)^{c(k)}

The problem was proved after remaining open for 79 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Claude, OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Off-diagonal Ramsey preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude, OpenAI internal model
Verification: Erdős Problems site confirmed
Open for: 79 years
Research activity*: 1/5

142

23 Apr 2026Graph Theory, Ramsey Theory

Erdős Problem #1014

Problem statement

For every fixed

k\geq3

, does

R(k,l+1)/R(k,l)\to1

l\to\infty

The problem was proved after remaining open for 55 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

OpenAI resolution manuscript

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Lean checked
Open for: 55 years
Research activity*: 1/5

143

9 Apr 2026Graph Theory, Chromatic Number

Erdős Problem #1091

Problem statement

Must every

K_4

-free 4-chromatic graph contain an odd cycle with at least two diagonals? More generally, can local 3-colourability force odd cycles with arbitrarily many diagonals?

Voss proved the first question in 1982; an OpenAI internal model disproved the second question in 2026, completing the two-part record.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 2 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The official status is solved rather than proved because the two clauses have different outcomes.

Two-part problem resolved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Erdős Problems site confirmed
Open for: 50 years
Research activity*: 1/5

144

28 Apr 2026Graph Theory, Chromatic Number

Erdős Problem #1092

Problem statement

If every

m

-vertex subgraph is the union of an

r

-colourable graph and a graph with at most

f_r(m)

edges, how large can

f_r

be while forcing the whole graph to be

(r+1)

-colourable? Is

f_r(n)\gg_r n

Rödl’s 1982 construction already disproved the conjecture. GPT-5.5 Pro supplied a later application and write-up.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 2 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The 2026 contribution is a new exposition/application, not the first counterexample.

Known construction applied

Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Erdős Problems site confirmed
Open for: Known by 1982
Research activity*: 1/5

145

23 Apr 2026Number Theory, Covering Systems

Erdős Problem #1190

Problem statement

For a finite family of distinct moduli

m<n_1<\cdots<n_k

whose residue classes can be chosen pairwise disjoint, determine the largest possible reciprocal sum

\sum_i1/n_i

m\to\infty

The problem was resolved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Dedicated proof building on several earlier covering-system results.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 3/5

Differential equations

8 Sep 2026Incompressible porous-media equation

Incompressible porous-media blowup with space–time smooth forcing

On $\mathbb T^2$ there are smooth odd initial data and a force smooth jointly in space and time for which the incompressible porous-media solution develops unbounded density gradient in finite time.

Details and sources

AI contribution

Levent Alpöge and Tristan Buckmaster report using Claude and OpenAI Codex, especially GPT-5.6 Sol, to extend the Córdoba–Martínez-Zoroa rough-forcing program to smooth forcing. They checked and organized the resulting argument.

Problem origin

The authors place these questions in the Córdoba–Martínez-Zoroa program on forced fluid blowup and explicitly say that neither they nor an LLM originated the program.

Verification

Public author-checked manuscript; independent review pending

Publication

Public 57-page proof manuscript and author statement

Preprint / manuscript

Authors' announcement and AI disclosure Earlier rough-time forcing result and open regularity remark Tao's preliminary expert discussion Community scope audit

This closes the previously anticipated regularity upgrade from spatially smooth, time-rough forcing to forcing smooth in space–time. It is a forced equation result; it does not prove finite-time blowup for the unforced IPM equation.

Smooth-forcing version proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude + OpenAI Codex (especially GPT-5.6 Sol)
Verification: Public author-checked manuscript; independent review pending

8 Sep 2026Inviscid Boussinesq equations

Planar inviscid Boussinesq blowup with smooth forcing

For every positive buoyancy constant, smooth compactly supported initial data and smooth compactly supported forces produce a unique finite-energy Lipschitz-class solution on each shorter interval whose temperature gradient blows up at a finite time and whose vorticity is unbounded near that time.

Details and sources

AI contribution

Alpöge and Buckmaster report that LLMs supplied substantial proof work under their direction. The repository says Claude wrote all Lean definitions, statements and proofs; Alpöge directed the work and checked the trusted statement against the mathematical theorem.

Problem origin

The authors place these questions in the Córdoba–Martínez-Zoroa program on forced fluid blowup and explicitly say that neither they nor an LLM originated the program.

Verification

Comparator-checked Lean proof; specialist review still in progress

Pinned Lean proof and comparator artifact

Claim audit

The project is unusually large and was not rebuilt during this audit. The repository records a clean kernel build and only standard axioms, but independent line-by-line mathematical review remains pending.

Publication

Public 76-page manuscript and pinned Lean 4 development

Preprint / manuscript

Authors' announcement and AI disclosure Tao's preliminary expert discussion Community statement and artifact audit

The theorem uses smooth forcing in both equations and proves blowup within a precisely stated uniqueness class. It does not establish blowup for the unforced two-dimensional Boussinesq equations.

Smooth-forcing version proved and formalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude + OpenAI Codex (especially GPT-5.6 Sol)
Verification: Comparator-checked Lean proof; specialist review still in progress
Claim audit: Issue documented

8 Sep 2026Incompressible Euler equations

Three-dimensional incompressible Euler blowup with smooth forcing

Smooth compactly supported initial velocity and smooth compactly supported forcing on $\mathbb R^3$ produce a unique finite-energy Lipschitz-class Euler solution on each shorter interval whose vorticity becomes unbounded and violates the Beale–Kato–Majda integral criterion at a finite time.

Details and sources

AI contribution

Alpöge and Buckmaster report that LLMs supplied substantial proof work under their direction, with GPT-5.6 Sol especially important. The accompanying Lean development was written by Claude under Alpöge's direction.

Problem origin

The authors place these questions in the Córdoba–Martínez-Zoroa program on forced fluid blowup and explicitly say that neither they nor an LLM originated the program.

Verification

Comparator-checked Lean proof; specialist review still in progress

Pinned Lean proof and comparator artifact

Claim audit

The project is unusually large and was not rebuilt during this audit. The repository records a clean kernel build and only standard axioms, while experts have begun reading rather than certifying the full argument.

Publication

Public 112-page manuscript and pinned Lean 4 development

Preprint / manuscript

Authors' announcement and AI disclosure Tao's preliminary expert discussion Community statement and artifact audit

This is a smooth-forcing theorem. The classical finite-time blowup question for unforced three-dimensional Euler remains open, and no released Navier–Stokes proof is part of this record.

Smooth-forcing version proved and formalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude + OpenAI Codex (especially GPT-5.6 Sol)
Verification: Comparator-checked Lean proof; specialist review still in progress
Claim audit: Issue documented

2 Sep 2026Differential equations

Interior curvature estimates for the graphical scalar-curvature equation

Admissible constant-scalar-curvature graphs with bounded height and slope satisfy an interior $C^2$ estimate in every dimension $n\geq3$ .

Details and sources

AI contribution

Qiu and Yan used ChatGPT to test comparison and cutoff functions, perform preliminary calculations, and find counterexamples to failed inequalities; they corrected and rewrote all AI-assisted work.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

Author-linked code (not retrieved during audit)

The authors identify this as the unrestricted quantitative estimate missing above dimension three. This is an author-checked preprint without formalization or referee report.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT (OpenAI; version unspecified)
Verification: Author-checked proof; independent review pending

7 Apr 2026Differential equations

De Giorgi–Nash–Moser elliptic regularity — Lean formalization

The development formalizes local boundedness, weak and full Harnack inequalities, and interior Hölder regularity for uniformly elliptic divergence-form equations with bounded measurable coefficients in dimensions at least three.

Details and sources

AI contribution

Scott Armstrong and Julia Kempe supplied mathematical blueprints and guided AI agents through the formal proof development.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Public formal development; project-reported checking, not replayed in this audit

Public Lean code

Publication

Public formalization project

Preprint / manuscript

Formalization paper

These are classical regularity theorems. The project is a substantial new formalization, not a new resolution of Hilbert's nineteenth problem. The repository reports complete Lean proofs; the present audit inspected source and scope without rebuilding the dependency graph.

Author's project account

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: Subscription language models; versions not identified in the inspected project
Verification: Public formal development; project-reported checking, not replayed in this audit

24 Aug 2026Differential equations

Anisotropic conductivity inverse problem — relative Isabelle formalization

Cătălin I. Cârstea reports an Isabelle/HOL formalization of the main result of his piecewise-polynomial anisotropic-conductivity preprint, relative to explicit imported literature statements.

Details and sources

AI contribution

The author directed model-generated Isabelle development and used separate model passes to examine the translations. Model review is not independent human validation.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Author-reported Isabelle check with declared literature assumptions

Publication

Public formalization project

Preprint / manuscript

Primary report, formal statements, and translation audit

The paper includes formal statements and back-translations. Literature theorems remain assumptions in Isabelle locales; semantic fidelity is explicitly provisional, including a corrected Lipschitz-domain translation. The author-linked repository could not be retrieved during this audit, and no proof replay was performed. This is one relative formalization, not an additional independently established open-problem resolution.

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: Codex and GPT-5.6 Sol; GPT-5.6 Sol Pro and Claude Opus 5 translation review
Verification: Author-reported Isabelle check with declared literature assumptions

20 Aug 2026Magnetohydrodynamics and induction equations

Smooth random fast dynamo on $\mathbb T^3$

Problem statement

Can a smooth random incompressible velocity field on the three-torus exhibit a resistivity-uniform positive fast-dynamo growth rate?

A smooth, uniformly bounded random time-dependent velocity field on $\mathbb T^3$ gives, for every fixed sufficiently small resistivity $\kappa$ , almost-sure magnetic-field growth at rate at least $1/2$ , with a $\kappa$ -dependent random prefactor having uniform inverse-moment control.

Details and sources

AI contribution

The model autonomously found the central tridiagonal Fourier-space idea and drafted proofs. The author rederived the argument, corrected minor errors, simplified and rewrote the proof, and carefully checked the final manuscript.

Problem origin

The author supplied the random fast-dynamo task to the model as a human-specified variant of Arnold's Problem 1994-28. The exact random formulation is not presented as the full classical conjecture.

Verification

Author carefully hand-checked the complete proof; independent review pending

Publication

Public proof preprint with source-contained AI drafts

Preprint / manuscript

Source package with original and revised AI drafts

Activity evidence

A documented editorial estimate based on the first smooth random construction in a long-running fast-dynamo program, while preserving the remaining deterministic and generic-random questions.

The exceptional probability-one event may depend on $\kappa$ , so the proof does not produce one deterministic smooth velocity field that works simultaneously for every small resistivity. Arnold's smooth autonomous fast-dynamo problem and a general theory for nondegenerate random flows remain open.

First smooth random fast dynamo; deterministic smooth problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol Ultra (OpenAI)
Verification: Author carefully hand-checked the complete proof; independent review pending
Open for: Human-specified random variant of Arnold's Problem 1994-28
Research activity*: 4/5

13 Aug 2026Elliptic PDE and low-regularity estimates

Interior $W^{1,1}$ estimates for uniformly elliptic nondivergence equations

Problem statement

Does uniform ellipticity alone imply an interior

W^{1,1}

estimate for solutions of three-dimensional nondivergence-form elliptic equations?

Smooth uniformly elliptic coefficient fields with one fixed ellipticity ratio admit smooth solutions with common boundary data and bounded $L^\infty$ norm but unbounded interior $W^{1,1}$ norm. Consequently, no ellipticity-only interior $W^{1,p}$ estimate exists for any $p\geq1$ in dimensions at least three.

Details and sources

AI contribution

Le, Sun, and Tran state that the main results and key strategies arose through chats with ChatGPT 5.6 Sol. They reworked and rewrote the paper, checked and simplified every argument, and accept responsibility for it.

Problem origin

The three-dimensional estimate was posed by Nadirashvili, Tkachev, and Vlăduţ and remained open in the PDE literature.

Verification

Three-author checked proof preprint; external review pending

Publication

Public proof preprint with explicit AI-use statement

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the foundational regularity question, its extension to all dimensions at least three, and the explicit counterexample mechanism.

The counterexample uses a large but fixed ellipticity ratio; positive near-isotropic regimes remain. The paper also constructs a measurable-coefficient limit outside local $BV$ .

Ellipticity-only estimate disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Three-author checked proof preprint; external review pending
Open for: Question of Nadirashvili–Tkachev–Vlăduţ
Research activity*: 5/5

16 Mar 2026Kinetic partial differential equations

Equilibria of the Vlasov–Maxwell–Landau system

Problem statement

Under the paper’s smoothness, positivity, Schwartz-decay, and score assumptions, are all steady solutions of the Coulomb Vlasov–Maxwell–Landau system necessarily spatially uniform Maxwellians?

Every smooth positive Coulomb Vlasov–Maxwell–Landau steady state on $\mathbb T^3\times\mathbb R^3$ satisfying the stated decay and score bounds is a spatially uniform Maxwellian, with zero electric field and constant magnetic field.

Details and sources

AI contribution

Gemini supplied the proof blueprint, Claude Code built more than ten thousand lines of Lean, and Aristotle closed the remaining lemmas; humans audited the definitions.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked end to end

Publication

Public preprint, Lean repository, and complete interaction logs

Preprint / manuscript

Activity evidence

The exact characterization is new, although it lies inside the large and active kinetic-PDE and plasma-physics literature.

The theorem has explicit regularity, positivity, velocity-decay, and score assumptions. The paper also records human corrections to definition alignment during formalization.

Public interaction logs

Conjecture proved under explicit hypotheses

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Gemini Deep Think / Claude Code / Aristotle
Verification: Lean checked end to end
Open for: Precise conjecture posed in 2026
Research activity*: 2/5

10 Jun 2026Stochastic partial differential equations

Unique invariant measure for the skew stochastic heat equation

Problem statement

For the stochastic heat equation

\partial_tu=\tfrac12\partial_x^2u+\delta_0(u)+\dot W

[0,1]

with Dirichlet boundary conditions, does its Markov semigroup have at most one invariant probability measure?

ProofCouncil gave a correct and novel uniqueness proof. All three expert referees rated it essentially flawless; the proof obtained a stronger finite-time absolute-continuity step than the human solution.

Details and sources

AI contribution

The harness found a stochastic-sewing drift estimate and combined it with Girsanov and ergodic arguments in a route different from the authors' proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three expert referees; essentially flawless

Publication

First Proof Second Batch report, complete submission, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

The authors built on several years of work on distributional-drift stochastic heat equations and reported needing four to five weeks for their proof.

The problem was solved but unpublished when given to the benchmark. This is one of the strongest independently reviewed cases in the index because the report explicitly identifies both correctness and novelty.

Research problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ProofCouncil (primarily GPT-5.5 Pro; auxiliary Gemini and Claude)
Verification: Three expert referees; essentially flawless
Open for: Unpublished SPDE research problem
Research activity*: 4/5

19 May 2026Partial differential equations

Lower bounds for advection–diffusion

Problem statement

For mean-zero solutions of advection–diffusion equations on the two-dimensional torus, derive explicit constructive lower bounds that rule out excessively fast mixing in three regimes: inviscid shear, diffusive shear, and rapidly oscillating time-periodic incompressible flows.

QED proved a polynomial lower bound for inviscid shears, a uniform positive mixing-scale lower bound for diffusive shears, and an exponential lower bound for rapidly oscillating time-periodic flows.

Details and sources

AI contribution

The system generated the complete chains of estimates without PDE-specific guidance; the most difficult case combined Floquet theory with techniques from other areas.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Domain expert verified

Proof and expert record

Publication

Dedicated arXiv paper, public proof records, and expert comments

Preprint / manuscript

Research paper QED system paper

Activity evidence

These were new questions from an expert’s active PDE research; one was estimated to require months or a year of human work and the resulting package was judged journal-level.

This record groups three closely linked questions from one research project. The expert judged the combined work suitable for journals such as JDE or SIAM Journal on Mathematical Analysis.

Three research questions proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: QED / GPT-5.4 and GPT-5.5
Verification: Domain expert verified
Open for: Under 1 year
Research activity*: 3/5

Geometry & topology

7 Sep 2026Convex polyhedra and equilibrium points

Explicit certified mono-monostatic polyhedron

Two rational half-space polytopes are exactly certified to have equilibrium signature $(S,H,U)=(1,0,1)$ with respect to their exact centroids; the smaller has $56{,}946$ faces.

Details and sources

AI contribution

The author posed and redirected the project; Claude agents designed and ran experiments, discovered and repaired obstructions, built the constructions, and implemented the verifier.

Problem origin

Lángi had proved existence of homogeneous mono-monostatic polyhedra, but no explicit published example was known; reducing the face count is an ongoing human problem.

Verification

Exact rational certificates and deterministic verifier

Pinned certificates, data, and verifier

Claim audit

The audit inspected the certificate architecture and pinned source but did not rerun the multi-minute, high-volume verification pipeline.

Publication

Public arXiv paper with pinned data, certificates, and code

Preprint / manuscript

Source package with Lean and replay code

This supplies an explicit certified object after an earlier existence theorem. It does not improve the known 14-face monostable record, prove minimality, or answer whether a mono-monostatic polyhedron with fewer than $1000$ faces exists.

First public explicit certificate; minimization open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Opus 4.8 + Claude Fable 5
Verification: Exact rational certificates and deterministic verifier
Claim audit: Issue documented

21 Aug 2026Symplectic linear algebra and stability conditions

Kontsevich's conjecture on products of positive primitive forms

There is a form in $\mathcal U^+(\mathbb R^6)\setminus\mathcal U_{\mathrm{ag}}(\mathbb R^6)$ whose every exterior self-product remains in $\mathcal U^+$ . Thus Kontsevich's conjecture fails already in complex dimension three.

Details and sources

AI contribution

Yuhang Liu states that ChatGPT 5.6 generated the main mathematical content and that he verified it; the revised version also records discussions with specialists.

Problem origin

Kontsevich proposed the conjecture in a 2012 lecture and private communication; Haiden later formulated the surrounding product question.

Verification

Author-checked revised preprint; external review pending

Publication

Public revised arXiv proof manuscript

Preprint / manuscript

Revised preprint

The same paper proves positivity when one factor has real dimension at most six and gives negative product examples in every complex dimension at least $26$ .

Conjecture disproved in complex dimension three

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6
Verification: Author-checked revised preprint; external review pending

4 Sep 2026Universal covers for unit-length arcs

Balanced triangular-cover bound for Moser's worm problem

A specified scaled isosceles triangle and an exact escape-factor certificate give a convex universal cover of area $0.257883595112076188\ldots$ .

Details and sources

AI contribution

Zhipeng Deng provided the methodological framework; GPT-5.6 Sol handled numerical calculations and proofs, assisted the Lean formalization, and polished the exposition.

Problem origin

Moser's 1966 worm problem asks for the minimum-area planar region containing a congruent copy of every rectifiable unit-length arc.

Verification

Author proof with Lean lemmas and exact symbolic replay

Claim audit

The formal and exact artifacts cover stated proof components and finite ledgers; independent specialist or peer review was not located.

Publication

Public arXiv preprint with Lean and Python source package

Preprint / manuscript

Preprint and exact bound

This is an explicit upper-bound construction for the convex-cover problem, not the minimum area, an optimality proof, or a full solution of Moser's worm problem.

New certified convex upper bound

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author proof with Lean lemmas and exact symbolic replay
Claim audit: Issue documented

30 Aug 2026Geometry & topology

Calabi's complex-structure question for $S^2\times S^4$

Assuming the validity of an anonymous unpublished 2026 construction of a compact complex threefold diffeomorphic to $S^6$ , the authors apply a blowup–quotient–flop construction to obtain a compact complex threefold diffeomorphic to $S^2\times S^4$ .

Details and sources

AI contribution

The authors report that GPT-5.6 Sol first suggested a fibration approach whose proof was incomplete, then helped motivate the final construction, introduced the relevant Wall classification, and drafted three lemmas or propositions that the authors checked and rewrote.

Problem origin

The preprint identifies Calabi's 1958 question asking whether the smooth six-manifold $S^2\times S^4$ admits a complex structure.

Verification

Author-checked conditional proof; foundational dependency unverified

Claimed Lean dependency for the source threefold

Claim audit

The construction depends on an anonymous, unpublished claimed complex structure on $S^6$ . The cited Lean repository does not by itself establish semantic correctness or independent acceptance of that source claim.

Publication

Public conditional research preprint

Preprint / manuscript

Conditional research preprint

This is indexed as a conditional advance, not as an unconditional settlement of Calabi's question. The paper explicitly conditions its theorem on the source threefold and describes that result as claimed. Until that dependency is publicly validated, the derived $S^2\times S^4$ conclusion remains provisional.

Community scope and dependency audit

Conditional claimed resolution; source threefold remains unverified

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked conditional proof; foundational dependency unverified
Claim audit: Issue documented

6 Sep 2026Geometry & topology

A 44-vertex triangulation of $\mathbb{RP}^{6}$

An explicit centrally symmetric simplicial seven-polytope with 88 integer vertices has antipodal boundary quotient a 44-vertex triangulation of $\mathbb{RP}^{6}$ , answering the cited fewer-than-45-vertices question affirmatively.

Details and sources

AI contribution

Alexey Kolosov reports using ChatGPT in the computational exploration, verifier development, and manuscript preparation.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Two exact author verifiers independently replayed by VibeMathed

Certificate repository

Publication

Public manuscript, coordinates, facets, and exact verification code

Preprint / manuscript

No public preprint or manuscript located.

This improves 45 to 44 vertices; it does not prove vertex minimality. The two verifiers were independently implemented but arose in the same AI-assisted workflow, so their agreement is not independent mathematical review.

Original Question 3.2 Independent verifier replay report

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT (OpenAI; version unspecified)
Verification: Two exact author verifiers independently replayed by VibeMathed

5 Sep 2026Geometry & topology

Wong's connectedness-preserving bijection question

For every $n\geq2$ , the preprint constructs a Borel bijection of $\mathbb R^n$ that preserves connected sets while its inverse does not, negatively answering the continuity and inverse-preservation question.

Details and sources

AI contribution

The author directed a two-system program: Codex produced the structural theory, construction, adversarial audits and draft, while Claude planned and reviewed stages and suggested the dimension-uniform coloring.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Public proof manuscript; no completed author or independent verification statement

Claim audit

The first public version still contains unfinished acknowledgements and a bracketed verification declaration; nobody outside the author's program is known to have checked it.

Publication

Zenodo preprint

Preprint / manuscript

Original MathOverflow question Community scope audit

This is retained as a provisional claimed resolution because a full proof artifact is public. It is not conflated with the companion closed-manifold theorem.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Codex Ultra and Claude Fable 5.1
Verification: Public proof manuscript; no completed author or independent verification statement
Claim audit: Issue documented

2 Sep 2026Geometry & topology

Trautman conjecture for smooth CR 3-manifolds

A smooth strongly pseudoconvex CR 3-manifold has a nowhere-zero closed section of its canonical bundle but is not locally embeddable in $\mathbb C^2$ .

Details and sources

AI contribution

Sean Curry reports discovering the counterexample through ChatGPT experimentation, which also produced an initial draft and later proofreading; he independently checked the mathematics.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked preliminary preprint; specialist review pending

Publication

Public research preprint

Preprint / manuscript

The author labels version 1 preliminary and invites comments. Three specialists are thanked, but the artifact does not say that they checked the complete proof.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT Plus (model unspecified)
Verification: Author-checked preliminary preprint; specialist review pending

13 Aug 2026Geometry & topology

Courtade's Minkowski-sum volume conjecture

The proposed volume inequality for Minkowski sums with the Euclidean ball fails even for zonoids in every dimension at least three, with explicit three-dimensional examples.

Details and sources

AI contribution

The authors used the model to explore examples, test determinant calculations, and assist preliminary proof development; they independently verified the final arguments and computations.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The counterexamples settle the general conjecture negatively because zonoids are a subclass of convex bodies. No formal certificate or independent referee report was found.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

5 Sep 2026 publicationGeometry & topology

Darboux injections from closed manifolds

Every connectedness-preserving injection from a connected closed $n$ -manifold into an $n$ -manifold is a homeomorphism onto a target component, answering Banakh--Banakh Problems 1.7 and 1.8 affirmatively.

Details and sources

AI contribution

The author directed the program and selected, checked, and assembled arguments produced with substantial model assistance; a later separate model run re-derived the theorem internally.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Public proof manuscript; independent expert review pending

Claim audit

No reviewer outside the author's AI research program is reported to have checked the argument; the second-model derivation is internal corroboration, not independent review.

Publication

Zenodo proof manuscript

Preprint / manuscript

Original Problems 1.7 and 1.8 Community scope audit

The result covers compact boundaryless sources and strengthens the earlier three-dimensional theorem. It does not cover noncompact Euclidean space, where the companion manuscript claims a counterexample.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro; later independent GPT-6 re-derivation
Verification: Public proof manuscript; independent expert review pending
Claim audit: Issue documented

3 Sep 2026Geometry & topology

Weak Chern conjecture with constant cubic trace

For each dimension, the squared second-fundamental-form values of closed embedded minimal hypersurfaces in the unit sphere are locally finite when both that value and the cubic trace are constant.

Details and sources

AI contribution

The authors used the model during preparation to interpret literature and tools such as Blaschke selection and real-analytic curve selection. They state that the mathematical arguments and proofs were independently developed and verified by the authors.

Problem origin

The manuscript identifies the classical human-origin Chern conjecture and its published history, including Yau's 1982 problem list.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This is the weak Chern conclusion only for embedded hypersurfaces with constant cubic trace. The original immersed problem remains open. The disclosed AI role is preparatory and substantially weaker than generating the proof.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

31 Aug 2026Geometry & topology

Degree-two multisections toward Iskovskikh's conic-bundle conjecture

A standard conic bundle over $\mathbb P^2_{\mathbb C}$ with very general discriminant of degree at least eighteen has no rational multisection of degree two.

Details and sources

AI contribution

After an initial human draft hit a technical obstacle, the model suggested the idea underlying Proposition 3.1. The authors independently checked all mathematics and wrote the paper.

Problem origin

The manuscript traces the result to Iskovskikh's conjecture that sufficiently general high-degree conic-bundle threefolds are not unirational.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This eliminates only the first possible even multisection degree. It does not prove non-unirationality, exclude all even degrees, or settle the broader existence of smooth rationally connected non-unirational varieties.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

2 Sep 2026Geometry & topology

Truncated Octahedron Conjecture for parallelohedra

Among all three-dimensional parallelohedra of a fixed volume, the regular truncated octahedron uniquely minimizes surface area, up to similarity.

Details and sources

AI contribution

Cesaroni and Novaga used ChatGPT to check identities and explore and verify proof arguments.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The admissible class is convex polytopes that tile by translation. This is not Kelvin's unrestricted foam problem. The authors acknowledge an independent proof by Thomas Hales and Lark Song; no exclusive priority is asserted.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-prepared proof; independent review pending

24 Aug 2026Geometry & topology

Escobar's first Steklov eigenvalue conjecture

In every dimension $n\geq3$ , conformal deformations of the unit ball have positive Ricci curvature and all boundary principal curvatures greater than one, yet first nonzero Steklov eigenvalue below one.

Details and sources

AI contribution

AI assisted candidate-polynomial searches in dimensions 3 through 100 and symbolic checking. The authors supplied and checked the mathematical proofs.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author proof with attached Mathematica computations

Publication

Public research preprint

Preprint / manuscript

Source and attached computation code

This refutes the 1999 Ricci-curvature formulation. Stronger hypotheses in other versions of Steklov inequalities must be assessed separately.

Claimed resolution in a public preprint

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol and Rethlas
Verification: Author proof with attached Mathematica computations

24 Aug 2026Geometry & topology

Three exact 604-point kissing configurations in dimension eleven

Three pairwise non-isometric exact configurations establish $K(11)\geq604$ , including two additional isometry classes beyond the concurrently reported construction.

Details and sources

AI contribution

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public proofs and exact verification artifacts; independent review pending

Publication

Public research preprint

Preprint / manuscript

The paper credits EinsteinArena with a concurrent earlier report of one configuration. The exact kissing number remains undetermined; three configurations are not counted as three separate problem resolutions.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Station agents using OpenAI, Anthropic, and Google models; see per-run artifacts
Verification: Public proofs and exact verification artifacts; independent review pending

24 Aug 2026Geometry & topology

Discrete Kakeya needle: exact small cases and improved finite bounds

The triangle-union problem has exact values $C_T(3)=5/18$ and $C_T(4)=1/4$ . New larger constructions give $C_T(128)\leq0.107067$ , and an asymmetric five-triangle example beats every symmetric one.

Details and sources

AI contribution

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public proofs and exact verification artifacts; independent review pending

Publication

Public research preprint

Preprint / manuscript

The five-triangle optimum and the general sharp finite values remain open. These finite improvements do not prove a new continuum Kakeya conjecture.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Station agents using OpenAI, Anthropic, and Google models; see per-run artifacts
Verification: Public proofs and exact verification artifacts; independent review pending

18 Aug 2026Toric algebraic geometry

Sato's weak $F$ -equivalence conjecture

Problem statement

Is every nonsingular projective toric weak Fano

d

-fold weakly

F

-equivalent to projective space

\mathbb P^d

For every $d\geq3$ , the authors construct a nonsingular projective toric weak Fano $d$ -fold that is not weakly $F$ -equivalent to $\mathbb P^d$ .

Details and sources

AI contribution

The authors submitted the human conjecture through MathDB to GPT-5.6 Sol, which supplied the key three-dimensional construction. They checked it, extended it to every $d\geq3$ , and revised the argument.

Problem origin

Hiroshi Sato conjectured in 2000 that every nonsingular projective toric weak Fano $d$ -fold is weakly $F$ -equivalent to $\mathbb P^d$ .

Verification

Authors independently checked and extended the counterexample; independent review pending

Publication

Public counterexample preprint

Preprint / manuscript

MathDB problem record used by the authors

Activity evidence

A documented editorial estimate based on refuting a 26-year-old classification conjecture in every dimension where it can fail.

The paper also proposes a Gorenstein refinement and proves it in low dimension and for the constructed family. That new follow-up is not counted here as a prior human open problem.

Conjecture disproved in every dimension

d\geq3

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol (OpenAI)
Verification: Authors independently checked and extended the counterexample; independent review pending
Open for: Sato's 2000 weak $F$ -equivalence conjecture
Research activity*: 4/5

16 Aug 2026Kähler geometry and comparison geometry

Volume comparison and rigidity under positive holomorphic sectional curvature

Problem statement

Does the Fubini--Study volume upper bound and its rigidity case follow from a positive lower bound on holomorphic sectional curvature alone?

If a compact connected Kähler $n$ -manifold has holomorphic sectional curvature at least $2$ , its volume is at most that of Fubini--Study $\mathbb{CP}^n$ ; equality holds exactly for manifolds biholomorphically isometric to $\mathbb{CP}^n$ .

Details and sources

AI contribution

The authors state that the proofs are due to ChatGPT 5.6 Sol Pro and that their paper is an exposition of those arguments. They verified the proofs and accept responsibility for them.

Problem origin

The paper explicitly presents its main theorem as an answer to a question of Xiong and Yang arising from their earlier volume-comparison result.

Verification

Authors verified the complete proof; independent review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on resolving a named rigidity question and removing a substantial auxiliary hypothesis from the prior theorem.

This removes the additional conjugate-radius hypothesis from the earlier comparison theorem and includes an extension to a mean RC-positivity condition. The record reflects the theorem proved in the manuscript, not every possible curvature normalization or non-Kähler analogue.

Xiong--Yang volume-rigidity question answered

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol Pro (OpenAI)
Verification: Authors verified the complete proof; independent review pending
Open for: Question of Xiong and Yang
Research activity*: 4/5

14 Aug 2026K-stability and toric geometry

Alpha-invariant spectrum of K-polystable toric Fano varieties

Problem statement

Which rational values occur as alpha invariants of fixed-dimensional K-polystable toric

\mathbb Q

-Fano varieties?

For every dimension $n$ , the alpha invariants of $n$ -dimensional K-polystable toric $\mathbb Q$ -Fano varieties are exactly $\mathbb Q\cap[1/(n+1),1/2]$ .

Details and sources

AI contribution

The author states that GPT-5.6 Sol suggested the initial construction. The author then refined the idea and supplied the rigorous argument determining the complete spectrum.

Problem origin

The paper positions the attainable alpha-invariant values as a strengthening and completion, in the toric subclass, of questions posed by Liu--Zhuang and Liu--Zhu.

Verification

Author-developed public proof; independent review pending

Publication

Public revised preprint

Preprint / manuscript

TeX source and embedded verification program

Activity evidence

A documented editorial estimate based on a complete spectrum theorem in an active, well-defined toric K-stability setting.

The theorem is a complete answer for K-polystable toric $\mathbb Q$ -Fano varieties. It does not classify alpha invariants for arbitrary non-toric K-polystable Fano varieties.

Complete toric spectrum determined

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol (OpenAI)
Verification: Author-developed public proof; independent review pending
Open for: Questions of Liu--Zhuang and Liu--Zhu
Research activity*: 3/5

12 Aug 2026Metric geometry and finite point configurations

Borsuk's partition problem in dimension 63

Problem statement

Can every bounded subset of

\mathbb R^{63}

be partitioned into

64

sets of strictly smaller diameter?

A $321$ -point set in $\mathbb R^{63}$ cannot be partitioned into $64$ subsets of smaller diameter, proving $b(63)\geq65$ and disproving the dimension-63 instance of Borsuk's conjecture.

Details and sources

AI contribution

The preprint states that ChatGPT generated the construction and proof entirely. Yibo Ji submitted the result, personally verified it, and explicitly declines credit for the construction's originality.

Problem origin

Borsuk's partition question is classical; the dimension-63 instance and the search for the lowest counterexample dimension predate the AI-generated construction.

Verification

Author-checked proof plus reproduced exact verification code

Publication

Public proof preprint with verification code in an appendix

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the classical problem's prominence and the improvement in the smallest dimension with a known counterexample.

This resolves the yes/no Borsuk question in dimension $63$ and improves the previously known dimension- $64$ construction. It does not determine the exact value of $b(63)$ or the smallest counterexample dimension.

Dimension-63 conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Author-checked proof plus reproduced exact verification code
Open for: Classical Borsuk problem; dimension 63 previously unresolved
Research activity*: 5/5

7 Aug 2026Triangulations of compact 2-manifolds

Chen–Lawrencenko cyclic-colouration conjectures

Problem statement

Are cyclic chromatic numbers of triangulations bounded in the two ways conjectured by Chen and Lawrencenko?

A surface-dependent constant bound for cyclic chromatic number is proved, while the proposed universal inequality is refuted by the minimal eight-vertex triangulation of the Klein bottle, whose cyclic chromatic number is nine.

Details and sources

AI contribution

The author used extensive model interaction for exploration and proof development, then corrected, revised, and verified the arguments.

Problem origin

The two statements were posed by Chen and Lawrencenko in the human triangulation literature.

Verification

Author-verified proof and finite counterexample; external review pending

Publication

Public proof and counterexample preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the triangulation and graph-colouring literature and the finite extremal example.

The paper addresses two of the four conjectures in the source article; two further conjectures remain open, so the parent research program is classified as partial.

One conjecture proved and one disproved

Claimed outcome: ProvedDisproved
Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Author-verified proof and finite counterexample; external review pending
Research activity*: 3/5

6 Aug 2026Convex geometry and lower-dimensional sections

Generalized Busemann–Petty problem for $m=2,3$

Problem statement

For origin-symmetric convex bodies, do inequalities for every

m

-dimensional section imply the corresponding total-volume inequality when

m=2

3

The volume comparison implication is proved for section dimensions $m=2$ and $m=3$ . Together with known negative results for $3<m<n$ , this completes the generalized problem's classification.

Details and sources

AI contribution

The authors report that a key integral-geometric theorem was found with ChatGPT 5.6 Sol and led them to the main result; they developed and verified the complete paper.

Problem origin

The generalized Busemann–Petty problem is a classical human problem in convex geometry; the $m=2,3$ cases had remained unresolved for decades.

Verification

Author-verified proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the problem's long history and its central place in geometric tomography and convex geometry.

The entry records the two remaining section dimensions. Previously known cases are not duplicated as new AI results.

Remaining lower-dimensional cases solved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Author-verified proof preprint; external review pending
Open for: 30 years
Research activity*: 5/5

4 Aug 2026Brauer groups and algebraic geometry

Period–index conjecture

Problem statement

Must the index of a Brauer class satisfy the conjectured dimension-dependent bound in terms of its period?

For every dimension $d\ge3$ over suitable algebraically closed characteristic-zero fields, explicit varieties carry Brauer classes violating the conjectured period–index bound; the threefold construction works over $\overline{\mathbb Q}$ .

Details and sources

AI contribution

ChatGPT first proposed a flawed quotient construction that nevertheless contained features of the final counterexample. Further author-directed dialogue led to the correct construction; the author wrote and verified the paper.

Problem origin

The period–index conjecture is a classical human conjecture in arithmetic and algebraic geometry.

Verification

Author-verified counterexample preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture's centrality in Brauer groups and arithmetic geometry.

The initial model example was wrong; the site records this explicitly because the useful contribution was the structural seed, not a correct one-shot proof.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT
Verification: Author-verified counterexample preprint; external review pending
Research activity*: 5/5

28 Apr 2026Discrete and metric geometry

Steinerberger's antipodal-pair versus neighboring-pair conjecture

Problem statement

For a sufficiently large finite set

X\subset\mathbb R^2

of diameter at most

1

, is the optimal lower bound for neighboring pairs forced by many nearly antipodal pairs of order

\varepsilon^{1/2}

For every sufficiently large planar point set of diameter at most $1$ , the number of pairs at distance at most $\varepsilon$ is at least a universal constant times $\varepsilon^{1/2}$ times the number at distance at least $1-\varepsilon$ . A sharp two-parameter extension is also proved.

Details and sources

AI contribution

The authors report that the main theorem was completed in close collaboration with ChatGPT 5.4. The model suggested one of the supporting lemmas; the authors supplied its proof and the final manuscript.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-presented proof; independent review pending

Publication

Public 19-page proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on recent follow-up work in discrete geometry, extremal distance problems, and the sharp two-parameter extension.

The lower-size hypothesis $|X|=\Omega(\varepsilon^{-2})$ matters. The paper corrects earlier statements that omitted it and proves that the $\varepsilon^{1/2}$ exponent is optimal under the corrected formulation.

Optimal exponent proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.4
Verification: Author-presented proof; independent review pending
Open for: Recent conjecture of Stefan Steinerberger
Research activity*: 3/5

4 Aug 2026Birational and algebraic geometry

Finite generation of canonical rings for klt generalized pairs

Problem statement

Is the generalized log canonical ring of every projective klt generalized pair over

\mathbb C

finitely generated?

A smooth projective $19$ -dimensional klt generalized pair is constructed whose generalized log canonical ring is not finitely generated. Its nef anticanonical ring is also infinitely generated.

Details and sources

AI contribution

The authors state that the main result was obtained by GPT-5.6 Sol Ultra, Fable 5, and Danus, a specialized agent built on Rethlas. Human verification and polishing followed.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human verified; independent review pending

Publication

Public 13-page proof preprint

Preprint / manuscript

Quantitative-gap manuscript

Activity evidence

A documented editorial estimate based on the centrality of finite generation in the minimal model program, multiple expert discussions, and recent positive special cases.

The paper traces the question to expert discussions and a 2023 research summary, while noting known positive special cases. The example has dimension $19$ ; determining the minimum possible dimension, known to be at least $4$ , remains open.

Finite-generation question answered negatively

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: GPT-5.6 Sol Ultra + Claude Fable 5 + Danus / Rethlas
Verification: Human verified; independent review pending
Open for: Question discussed in the birational-geometry community since at least 2023
Research activity*: 4/5

2 Aug 2026Metric geometry

Uniform Gromov–Hausdorff gap for finite homogeneous approximations of spheres

Problem statement

Is there a dimension-independent positive lower bound on the Gromov–Hausdorff distance from

S^n

to every finite homogeneous metric space?

For sufficiently large $n$ , every finite homogeneous metric space $X$ satisfies $d_{GH}(X,S^n)\geq c/(1+\log(n+1))^2$ . This quantifies the obstruction to approximating round spheres, while the sought dimension-independent positive lower bound remains open.

Details and sources

AI contribution

The manuscript states that ChatGPT combined approximate-action, stability, almost-representation, and finite-transitive-set width results to obtain the quantitative lower bound. The author supplied and published the argument.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author manuscript; independent review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the problem's links to metric geometry, approximate group actions, and quantitative stability theory.

The headline open problem asks whether $\inf_{n\geq2}\delta_n>0$ . A bound decaying like $1/(\log n)^2$ is meaningful progress but does not settle that question, so this record is classified as partial rather than resolved.

Dimension-dependent lower bound proved; uniform gap open

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT (version not disclosed)
Verification: Author manuscript; independent review pending
Open for: Uniform form of the finite-homogeneous approximation gap problem
Research activity*: 3/5

2 Aug 2026Convex and discrete geometry

Equality case of Ehrhart's volume conjecture

Problem statement

If a centered convex body attains equality in Ehrhart's sharp volume bound, must it be unimodularly equivalent to the centered simplex?

Every full-dimensional compact convex body in $\mathbb R^n$ whose barycenter is its unique interior lattice point and whose volume is $(n+1)^n/n!$ is a unimodular image of the centered simplex $(n+1)\Delta_n-(1,\ldots,1)$ .

Details and sources

AI contribution

The author reports that the main result was obtained with GPT-5.6 Sol, Fable 5, and Danus, with essential human strategic input. The author checked the proof and published the full argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human verified; independent review pending

Publication

Public 30-page proof preprint

Preprint / manuscript

Equality-case proof preprint

Activity evidence

A documented editorial estimate based on the conjecture's 1964 origin and the active literature connecting lattice points, convex geometry, and complex analysis.

This settles the equality-classification half of Ehrhart's 1964 conjecture after the sharp inequality was separately proved by OpenAI. It is kept as a separate record because the claim, proof, authorship, systems, and verification path differ from the inequality result.

Extremizers classified

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol + Fable 5 + Danus
Verification: Human verified; independent review pending
Open for: 62 years
Research activity*: 5/5

1 Aug 2026Sphere packing and Fourier analysis

Cohn–Elkies high-dimensional sphere-packing rate

Problem statement

What is the exponential rate of the Cohn--Elkies linear-programming bound for sphere packing as

d\to\infty

The Cohn--Elkies linear-programming bound satisfies $\operatorname{LP}_d^{1/d}\to\sqrt{e/(2\pi)}$ . Consequently, the general sphere-packing density is at most $2^{-(0.6044\ldots+o(1))d}$ , the first improvement to its leading exponent since 1978.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for the sphere-packing rate Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on sustained work on high-dimensional packing bounds, the 2020 rate conjecture, and the first improvement to the classical leading exponent since 1978.

This resolves the conjectured asymptotic rate of the Cohn--Elkies linear program, not the full high-dimensional sphere-packing problem. The same chapter also settles the asymptotic positive and negative Fourier sign-uncertainty radii. The released solution module is sorry-free under the repository’s stated standard axioms, but independent mathematical review has not yet been located.

Conjectured linear-programming rate proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: 6 years
Research activity*: 5/5

1 Aug 2026Convex and discrete geometry

Ehrhart’s sharp volume conjecture

Problem statement

Does the centered simplex maximize volume among convex bodies whose barycenter is their unique interior lattice point?

Every full-dimensional compact convex body $K\subset\mathbb R^n$ with barycenter $0$ and $\operatorname{int}(K)\cap\mathbb Z^n=\{0\}$ satisfies $\operatorname{vol}(K)\le (n+1)^n/n!$ .

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for Ehrhart’s inequality Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on Ehrhart’s 1964 formulation and sustained work connecting the conjecture to convex geometry, lattice points, and Kähler geometry.

This record covers the sharp inequality in every dimension. The equality-classification half was subsequently resolved in a separate AI-assisted proof on 2 Aug 2026 and is indexed independently because its proof, authorship, systems, and verification path differ.

Sharp inequality proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: 62 years
Research activity*: 5/5

29 Jul 2026Algebraic geometry

Positivity for arbitrary coherent quotients on Deligne–Mumford stacks

Problem statement

Can the torsion-free hypothesis be removed from the positivity theorem for coherent quotients of tensor powers of

\Omega^1_{\mathcal X}(\log\Delta)

Under the Casalaina-Martin--Zhjeqi hypotheses, the first Chern class of every coherent quotient of every positive tensor power of the logarithmic cotangent sheaf is pseudo-effective; the quotient need not be torsion-free.

Details and sources

AI contribution

The authors report that ChatGPT 5.5 Pro and the Danus agent obtained the main result, followed by human verification and polishing.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; independent review pending

Publication

Two-author public preprint

Preprint / manuscript

Source theorem and open remark Danus system repository

Activity evidence

A documented editorial estimate based on a recent technical question in logarithmic and stack-theoretic positivity.

The proof answers the technical question in Remark 4.5 of the source paper. The authors explicitly caution that an AI-assisted literature search may have missed related references, so novelty beyond the stated source question remains open to specialist checking.

Torsion-free hypothesis removed

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.5 Pro + Danus
Verification: Author checked; independent review pending
Open for: Question raised in a 2026 source paper
Research activity*: 2/5

29 Jul 2026Algebraic geometry

Twelve common flex lines in a general pencil of cubics

Problem statement

Does a general pencil of plane cubics over

\mathbb C

have exactly

12

common flex lines?

A general pencil of complex plane cubics has exactly $12$ common flex lines. The proof identifies them with the ordinary nodes accounting for the full genus defect of the degree- $9$ flex-line curve.

Details and sources

AI contribution

The authors report that ChatGPT 5.5 Pro and the Danus agent obtained the main result, followed by human verification and polishing.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; independent review pending

Publication

Two-author public preprint

Preprint / manuscript

Original question Danus system repository

Activity evidence

A documented editorial estimate based on a recent enumerative question about flex-line curves of plane pencils.

This answers the question in Remark 5.3 of Ciliberto--Miranda--Roé. The authors caution that their AI-assisted literature search may have missed related references, so the novelty claim is recorded at preprint level.

Question answered affirmatively

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.5 Pro + Danus
Verification: Author checked; independent review pending
Open for: Question posed in a 2026 preprint
Research activity*: 2/5

16 Mar 2026Geometric optimization

Thin-triangle Kakeya optimization at 128 slopes

Problem statement

For the fixed slopes

a_i=i/128

, choose rational intercepts

b_i

minimizing the area of the union of the prescribed thin triangular neighborhoods of the lines

y=a_ix+b_i

GPT-5.4 Pro proposed rational intercepts for the fixed slopes $a_i=i/128$ that reduce the verified union area by about 8.44% relative to the stated baseline.

Details and sources

AI contribution

The model searched the structured space of intercepts; HorizonMath’s public checker validates the resulting rational construction.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Automatically checked; expert review pending

Publication

HorizonMath paper and open verification framework

Preprint / manuscript

HorizonMath paper

Activity evidence

The parent Kakeya program is major, while this exact finite optimization instance is a narrower, automatically verifiable research target.

The authors explicitly call this a potential novel contribution pending expert review. It is an optimization advance, not a solution of the Kakeya conjecture.

Best-known construction improved

Problem origin: Origin not yet traced
System: GPT-5.4 Pro
Verification: Automatically checked; expert review pending
Open for: Current optimization frontier
Research activity*: 3/5

Feb 2026Combinatorial geometry

Erdős Problem #652 — distinct distances from selected points

Problem statement

For planar points

x_1,\ldots,x_n

, let

R(x_i)

count the distinct distances from

x_i

and order these counts increasingly. If

\alpha_k

is the least constant permitting

R(x_k)<\alpha_k\sqrt n

in arbitrarily large configurations, must

\alpha_k\to\infty

Aletheia connected the question to Mathialagan’s 2021 theorem, which implies the required divergence. The contribution is a literature-based resolution rather than a newly invented proof.

Details and sources

AI contribution

The research agent searched the literature, matched the problem to an existing theorem, and produced a public solution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Official problem record updated

Publication

DeepMind paper, released output, and Erdős Problems record

Preprint / manuscript

Erdős case-study paper Aletheia system paper

Activity evidence

A documented combinatorial-geometry problem with a modest literature trail; the decisive theorem had already appeared in 2021.

This corrects the open-status record by applying known literature. It should not be read as a new 2026 theorem.

Official problem record Released Aletheia outputs

Resolved through literature recovery

Problem origin: Human-source problem
System: Aletheia / Gemini Deep Think
Verification: Official problem record updated
Open for: 29 years
Research activity*: 2/5

Feb 2026Combinatorial geometry

Erdős Problem #654 — distinct distances with no four concyclic

Problem statement

n

planar points have no four concyclic, must some point determine

(1-o(1))n

distinct distances? Failing that, can one always force more than

(1/3+c)n

for some fixed

c>0

Aletheia constructed configurations in which every point sees at most about $3n/4$ distinct distances, refuting the proposed $(1-o(1))n$ lower bound. The weaker improvement beyond $n/3$ remains open.

Details and sources

AI contribution

The agent autonomously generated and revised the counterexample in DeepMind’s solver–verifier workflow.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert-reviewed project result

Publication

DeepMind paper, released output, and official problem discussion

Preprint / manuscript

Erdős case-study paper Aletheia system paper

Activity evidence

A decades-old distinct-distances problem connected to an active geometric-combinatorics literature, but narrower than the classical Erdős distinct-distances problem.

Only the strongest proposed asymptotic form is settled. The parent problem remains open and is therefore classified as partial.

Official problem record Released Aletheia outputs

Strongest form disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Aletheia / Gemini Deep Think
Verification: Expert-reviewed project result
Open for: 39 years
Research activity*: 3/5

28 Apr 2026Decision problems in topology

Kirby Problem 5.16 for noncommutative semifree DGAs

Problem statement

For semifree noncommutative differential graded algebras, are stable tame isomorphism, quasi-isomorphism, or derived Morita equivalence algorithmically decidable?

Over every nontrivial computable unital commutative ring, the released proofs show that stable tame isomorphism, quasi-isomorphism, and derived Morita equivalence are all undecidable for arbitrary semifree noncommutative DGAs.

Details and sources

AI contribution

Aletheia produced the public natural-language proof artifact; the repository identifies it as a resolution of Problem 5.16 in the stated noncommutative setting.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human-checked public proofs

Publication

Dedicated public preprint and released raw outputs

Preprint / manuscript

Dedicated research paper

Activity evidence

Kirby’s problem lists have broad standing in low-dimensional topology; this is a precise algebraic-decision subcase rather than the whole surrounding program.

The analogous graded-commutative formulations remain open. The scope qualifier is therefore essential.

Released Aletheia outputs

Specified case proved undecidable

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Aletheia / Gemini Deep Think
Verification: Human-checked public proofs
Open for: K3 / Kirby problem-list question
Research activity*: 3/5

3 Feb 2026Flat surfaces and moduli spaces

Spin parity for $k$ -differentials

Problem statement

For odd

k

and

\gcd(n,k)=\gcd(n+1,k)=1

, let

N_k(n)

count pairs

1\leq b_i\leq(k-1)/2

with

b_1+b_2\geq(k+1)/2

and

b_2\equiv nb_1\pmod k

. Is

N_k(n)\equiv\lfloor(k+1)/4\rfloor\pmod2

The parity identity conjectured by Chen and Gendron is proved for every odd $k$ under the stated coprimality assumptions, removing a conditional step in the genus-zero and genus-one spin-parity classification.

Details and sources

AI contribution

The system found a Jacobi-symbol reformulation and the key proof strategy; the central number-theoretic identity was then formalized in Lean.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert proof + Lean-checked core

Lean source

Publication

Public preprint and Lean repository

Preprint / manuscript

Activity evidence

A focused conjecture in a sustained specialist program on strata of differentials, flat surfaces, and spin parity.

Lean checks the number-theoretic identity itself, not every downstream statement about moduli spaces.

Number-theoretic conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AxiomProver
Verification: Expert proof + Lean-checked core
Open for: 4 years
Research activity*: 3/5

17 Feb 2026Discrete geometry

Distances in convex polygons

Problem statement

Must the vertices of every convex

n

-gon determine at least

\lfloor n/2\rfloor

distinct distances?

Altman's 1963 affirmative proof was formalized in Lean through a mixed agent workflow.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / Claude Opus 4.5 / Claude Opus 4.6 / Gemini 3 Flash / Gemini 3 Pro / Numina Lean Agent
Verification: Lean checked

15 Jan 2026Discrete geometry

Convex-distance multiplicities

Problem statement

f(d)

counts pairs of vertices of a convex

n

-gon at distance

d

, is

\sum_d f(d)^2=O(n^3)

A stronger known theorem was given a public Lean proof, with independent AI proof work also recorded.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Codex / GPT-5.2 Thinking / Seed Prover 1.5
Verification: Lean checked

17 Nov 2025Combinatorial geometry

Lines through one planar point set

Problem statement

For disjoint planar sets

A

and

B

of sizes

n

and

n-3

, respectively, with not all of

A

contained on one line, must some line contain at least two points of

A

and no point of

B

Xichuan's three explicit counterexamples were reconstructed with GPT Pro and verified by Aristotle in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: ChatGPT Pro / Aristotle
Verification: Lean checked

17 Dec 2025Euclidean Ramsey theory

Monochromatic rectangles of prescribed area

Problem statement

\mathbb R^2

is finitely colored, must there be one color class containing the vertices of a rectangle of every positive area?

A negative construction was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / Gemini 3 Pro
Verification: Lean checked

14 Jan 2026Discrete geometry

Obtuse angles among Euclidean points

Problem statement

Must every set of

2^d+1

points in

\mathbb R^d

contain three points forming an obtuse angle?

The classical affirmative theorem received a new AI-generated Lean proof.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Codex / GPT-5.2 Thinking
Verification: Lean checked

27 Dec 2025Discrete geometry

Distance multiplicities beyond lines and circles

Problem statement

Let

A\subset\mathbb R^2

have size

n

, let

d_1,\ldots,d_k

be its distinct distances, and let

f(d)

be the distance multiplicity. Is

k=n-1

and

\{f(d_i)\}=\{n-1,\ldots,1\}

equivalent to

A

being a set of equidistant points on a line or a circle?

Aristotle independently found the four-point counterexample and produced the public Lean proof.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

19 Jan 2026Geometric graph theory

Graphs of Euclidean dimension four

Problem statement

What is the minimum number of edges in a graph of Euclidean dimension four, and which graph attains it?

A Lean proof establishes the minimum as nine edges, uniquely attained by $K_{3,3}$ .

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

19 Jun 2026Combinatorial geometry

Ordinary triangles in line arrangements

Problem statement

For

d\geq4

, must every collection of

d

pairwise non-parallel lines in

\mathbb R^2

, with no point incident to four lines, contain three lines whose three intersection points are distinct and each incident to exactly two lines of the arrangement?

Escudero's counterexample to the conjecture was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: AxiomProver
Verification: Lean checked

2 Mar 2026Euclidean Ramsey theory

Unit-square Ramsey complement

Problem statement

S\subset\mathbb R^2

contains no pair of points at unit distance, must its complement contain the four vertices of a unit square?

Juhász's affirmative theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

10 Jun 2026Discrete geometry and low-dimensional topology

Piecewise-affine Möbius-band width

Problem statement

For the benchmark's

G_\beta

-invariant clean triangulations and squeeze maps of the strip, prove that

\sqrt 3

is the infimum of the realized values of

\beta

Three systems proved the sharp threshold and received minor-revision decisions. Reviewers found the mathematics correct, while documenting missing attribution and an incorrect historical citation in some submissions.

Details and sources

AI contribution

Three independent GPT-5.5-Pro-based systems translated ideas around the optimal paper Möbius band into the benchmark's piecewise-affine setting.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Double-blind expert review; minor revisions

Publication

First Proof Second Batch report, complete submissions, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

The formulation is new, but it is rooted in the nearly fifty-year history of the sharp paper Möbius-band problem.

The benchmark problem is a new affine variant of the Halpern–Weaver problem. Reviewers emphasized that several solutions closely followed Schwartz's prior work without adequate attribution.

Research problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro / ProofCouncil / UCLA Moonshot
Verification: Double-blind expert review; minor revisions
Open for: New variant of a 50-year problem
Research activity*: 4/5

10 Jun 2026Lattice theory and low-dimensional topology

Irreducible vertices in definite tree lattices

Problem statement

Let

T

be a positive-definite weighted tree with exactly one vertex

v

satisfying

w(v)<d(v)

. Must

T

contain a vertex that is irreducible in its lattice

L(T)

The UCLA Moonshot harness and ChatGPT 5.5 Pro produced complete proofs. All three referees assigned to each submission found the arguments mathematically correct and essentially flawless.

Details and sources

AI contribution

A one-shot harness and the base model independently reconstructed the tree-lattice argument used in work toward the Neumann–Zagier conjecture.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three expert referees; essentially flawless

Publication

First Proof Second Batch report, complete submissions, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

A specialized but technically serious problem connected to ongoing work on the Neumann–Zagier conjecture; the authors reported several months of effort.

The authors had a private proof after several months of work. Referees found the AI proofs complete but less conceptually organized than the human solution.

Research problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro / UCLA Moonshot
Verification: Three expert referees; essentially flawless
Open for: Unpublished research lemma
Research activity*: 3/5

10 Jun 2026Combinatorial topology

Contractibility of a crossing-matching complex

Problem statement

For a reducible quasi-reduced free-group word

w

, let

F_w

be the CW complex whose cells encode crossing matchings and their resolutions. Must

F_w

be contractible?

Every tested system found a counterexample. Two submissions were rated essentially flawless and the other two required only minor revisions.

Details and sources

AI contribution

Four independent systems recognized that the newly defined CW complex need not be contractible and supplied explicit topological counterexamples.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Double-blind expert review; all four passed

Publication

First Proof Second Batch report, complete submissions, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

A new technical object arising inside an active topology project, with little prior standalone literature.

The object was newly defined in work on a stable homotopy refinement of Legendrian contact homology. This was a short-lived research conjecture rather than a longstanding public problem.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ProofCouncil / UCLA Moonshot / ChatGPT 5.5 Pro / Momus with Gemini 3.1 Pro
Verification: Double-blind expert review; all four passed
Open for: Newly posed research conjecture
Research activity*: 2/5

11 Jun 2026Polyhedral combinatorics

IRIS Conjecture 6.1 on simple $3$ -polytopes

Problem statement

For a simple

3

-polytope with at least three faces of size at least

7

, must

p_6\geq\frac{39}{20}+\frac{p_3}{2}-\frac{p_5}{4}-\sum_{k\geq7}p_k

Five minimal ten-face counterexamples refute the printed face-vector inequality. The simplest has $p_3=4$ , $p_5=3$ , $p_7=3$ , and $p_6=0$ .

Details and sources

AI contribution

A verification-first multi-agent search produced exact certificates, independently implemented Python and Rust checkers, a mutation suite, and a clean-room census.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Dual independent checkers; not externally refereed

Certificates and independent checkers Verification catalog and runnable checks Verification method and limitations

Publication

Public certificates, checkers, frozen source, and write-up

Preprint / manuscript

Original conjecture record Result write-up

Activity evidence

A recent AI-for-mathematics workshop conjecture with a finite polyhedral search space and no prior public correction located.

The integer-rounded weakening survives the census through sixteen faces, and no claim is made about the paper’s other conjectures.

Conjecture refuted as printed

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Dual independent checkers; not externally refereed
Open for: Open workshop-paper conjecture
Research activity*: 2/5

11 Jun 2026Discrete geometry

Discrete Borsuk Conjecture 3

Problem statement

For bounded

S\subset\mathbb Z^d

, is

\beta_{\mathbb Z}(S)=2^d

if and only if

\operatorname{conv}(S)

is unimodularly equivalent to

[0,m]^d

A four-point set in $\mathbb Z^2$ has lattice Borsuk number $4=2^2$ but a convex hull containing seven lattice points, so it is unimodularly equivalent to no lattice square.

Details and sources

AI contribution

The pipeline found the witness, froze the source definitions, and built a complete Lean development with a statement-fidelity ledger and axiom audit.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked end to end; not externally refereed

Lean proof and fidelity audit Verification catalog and runnable checks Verification method and limitations

Claim audit

Statement-reading caveat: the checked counterexample kills the published all-bounded-sets biconditional, while a plausible full-set repair remains open.

Publication

Public proof note, Lean project, and reproducible audit

Preprint / manuscript

Original conjecture Counterexample manuscript

Activity evidence

The problem is a discrete analogue of the Borsuk partition problem, with both geometric and formal-definition subtleties.

This refutes the conjecture exactly as printed. The unstated repair requiring $S=\operatorname{conv}(S)\cap\mathbb Z^d$ is not refuted by this witness.

Printed biconditional disproved

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Lean checked end to end; not externally refereed
Claim audit: Issue documented
Open for: Published 2025 conjecture
Research activity*: 3/5

13 Jul 2026Discrete geometry

Erdős Problem #662 — triangular-lattice short-distance extremality

Problem statement

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold?

Explicit rational oblique lattices beat the triangular-lattice comparison under several closed- and strict-shell readings, with the finite certificates and variant statements checked in Lean.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

The maintained problem record calls the source statement ambiguous: its sample values, normalization, and final clause conflict. The artifact disproves multiple source-authenticated natural readings but does not claim to recover one uniquely intended formulation.

Ambiguous historical statement; natural variants disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 2/5

13 Jul 2026Discrete geometry

Erdős Problem #769 — homothetic cube-tiling counts

Problem statement

For the least cutoff

c(n)

after which every

k

occurs as the number of homothetic cubes in a decomposition of the unit

n

-cube, is

c(n)\gg n^n

A Lean proof shows $c(n)=o(n^n)$ along odd dimensions, disproving the specific conjectured lower bound $c(n)\gg n^n$ . The broader request for good bounds on $c(n)$ remains open.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

The artifact completely resolves the highlighted yes/no subquestion, but not the full open-ended request to determine sharp bounds.

Named lower-bound subquestion disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 3/5

13 Jul 2026Distinct distances

Erdős Problem #959 — gap between top distance multiplicities

Problem statement

How large can the difference between the largest and second-largest distance multiplicities be among

n

planar points?

A Lean-checked construction proves $M(n)\geq n^{1+1/(50000\log\log n)}$ for large $n$ , improving the previous $\Omega(n\log n)$ bound. The exact order remains open.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a formally verified advance on the modern lower-bound question, not a matching asymptotic estimate.

Superlinear lower bound proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 4/5

28 Jul 2026Algebraic geometry and pluripotential theory

Boucksom’s local analytic Bertini conjecture

Problem statement

Does the analytic Bertini restriction theorem for multiplier ideals hold locally outside a pluripolar exceptional set?

The paper proves the local analytic Bertini theorem in full generality, showing that multiplier ideals restrict correctly to fibers outside a locally pluripolar exceptional set.

Details and sources

AI contribution

The author had the proof idea before the AI era. Rethlas first carried out the details using GPT-5.6 Sol; Xia later simplified and largely rewrote the proof.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-rewritten AI proof; peer review pending

Publication

Public arXiv manuscript and public Rethlas agent

Preprint / manuscript

Rethlas proof-agent repository

Activity evidence

The problem is specialized but connects multiplier ideals, pluripotential theory, and deformation of singularities.

The manuscript states the full local conjecture and gives a conventional proof, but no Lean or other proof-assistant artifact was located.

Conjecture proved in full generality

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Rethlas / GPT-5.6 Sol
Verification: Author-rewritten AI proof; peer review pending
Research activity*: 3/5

28 Jul 2026Derived algebraic geometry

Bondal–Polishchuk transitivity conjecture for smooth projective varieties

Problem statement

Can the braid-group action on full exceptional collections fail to be transitive for

D^b(X)

when

X

is a smooth projective variety?

A smooth projective weak Fano threefold has two full exceptional collections in distinct braid-group-and-shift orbits, giving the first counterexample of the form $D^b(X)$ for a smooth projective variety.

Details and sources

AI contribution

The model did not produce the counterexample. It found numerous errors in earlier constructions, guided the successful modifications, and located a key reference; the author supplied the construction and wrote the paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human construction with disclosed AI audit assistance

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Exact certificate, cases, and verifier

Activity evidence

The smooth-projective case had remained open since the 1993 Bondal–Polishchuk conjecture despite a later categorical counterexample.

The general category-level transitivity conjecture had already been disproved by Chang, Haiden, and Schroll. This resolves the previously open smooth-projective-variety case; the AI role is error finding and research guidance, not authorship of the counterexample.

First smooth-projective counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT-5.6
Verification: Human construction with disclosed AI audit assistance
Open for: 33 years
Research activity*: 4/5

27 Jul 2026Discrete and computational geometry

Kuperberg’s six-cylinder conjecture

Problem statement

How many pairwise non-overlapping infinite unit cylinders can simultaneously touch a unit ball?

At most six pairwise non-overlapping infinite unit cylinders can touch a unit ball, and an arrangement of six exists. The upper bound reduces to $2{,}954{,}984$ exact rational-polynomial cases.

Details and sources

AI contribution

The authors report that some ideas originated in work with Claude. They reconstructed the initial correct but complicated argument into a simpler certificate-based proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Exact rational certificate and deterministic verifier

Publication

Public arXiv paper, complete case list, data, and standalone verifier

Preprint / manuscript

Lean certificates and independent audits Precise formalization-scope audit

Activity evidence

Kuperberg posed the question in 1990; successive work reduced the upper bound from eight to seven before this exact result.

The theorem concerns cylinders of radius exactly $1$ . The largest radius for which six cylinders can touch the ball remains open. The certificate is reproducible but is not a proof-assistant formalization; peer review is pending.

Exact maximum proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Anthropic Claude (version not specified)
Verification: Exact rational certificate and deterministic verifier
Open for: 36 years
Research activity*: 4/5

27 Jul 2026Metric and computational geometry

Bellman’s lost-in-a-forest problem for the golden gnomon

Problem statement

What is the exact shortest route guaranteed to escape the golden-gnomon triangle from an unknown starting point and heading?

For the isosceles triangle with equal sides $1$ and apex angle $108^\circ$ , the shortest guaranteed escape curve is an explicit symmetric seven-piece path of length $1.282676025459\ldots$ .

Details and sources

AI contribution

The models were used to search for the extremal curve, draft proof arguments, and write the accompanying Lean development. The authors packaged the conventional proof, certificates, and independent audits.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Partially Lean checked with explicit scope audit

Publication

Public arXiv paper, Lean certificates, Python audits, and source

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Bellman posed the general escape problem in 1956, but exact optima are known for very few regions; this is the first claimed exact isosceles-triangle solution below a $45^\circ$ base angle.

Lean verifies the finite algebraic certificate families and many reusable discrete identities, but not the entire theorem: compactness, parts of the planar combinatorics, support identification, and hull surgery remain prose. This solves one triangle, not Bellman’s general problem.

Golden-gnomon case solved exactly

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Claude Fable 5 / GPT-5.6 Sol / Claude Opus 5
Verification: Partially Lean checked with explicit scope audit
Research activity*: 4/5

29 Jul 2026Discrete geometry and packing

Erdős Problem #106 — packing $17$ squares

Problem statement

f(n)

is the maximum total side length of

n

interior-disjoint squares packed in the unit square, is

f(k^2+1)=k

An exact rational configuration packs $17$ interior-disjoint squares in the unit square with total side length greater than $4$ , refuting the proposed identity $f(k^2+1)=k$ at $k=4$ .

Details and sources

AI contribution

The public Lean source credits Codex as formal author. The available record does not establish that Codex discovered the packing, so this entry tracks the AI formalization rather than assigning mathematical priority.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status update pending

Lean proof commit Comparator integration

Publication

Public 613-line Lean proof and Comparator integration; no standalone manuscript located

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A concrete Erdős packing problem with a new exact construction and formal certificate; independent status review is still developing.

The Lean file is reported placeholder-free. The community problem page had not yet incorporated the result at the time of this audit.

Community problem record Community discussion

Lean-checked counterexample; status review pending

Problem origin: Human-source problem
System: OpenAI Codex (formal author; discovery provenance not established)
Verification: Lean checked; community status update pending
Research activity*: 3/5

13 Jul 2026Discrete geometry and graph coloring

Erdős Problem #130 — infinite-chromatic integer-distance graph

Problem statement

For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be; in particular, can the chromatic number be infinite?

There is an infinite $A\subset\mathbb R^2$ with no three collinear and no four concyclic points whose positive-integer-distance graph has infinite chromatic number.

Details and sources

AI contribution

Star Fleet’s GPT-5.6 agents built the construction and complete Lean proof; Claude Fable 5 and a human reviewer audited the submitted artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked end to end; community review pending

Official problem record Community discussion Star Fleet report

Publication

Public report and complete reproducible Lean bundle

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

This settles the explicit infinite-chromatic question. The clique-number side of the broader problem remains open, so the record is classified as partial.

Infinite-chromatic subquestion proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked end to end; community review pending
Research activity*: 4/5

21 Jul 2026Algebraic geometry and motivic integration

Batyrev’s nonnegativity conjecture for stringy Hodge numbers

Problem statement

Must every projective variety with Gorenstein canonical singularities and polynomial stringy

E

-function have nonnegative stringy Hodge numbers?

For a genus-three curve $C$ , let $M_0$ be the coarse moduli space of rank-two semistable bundles with trivial determinant. The seven-dimensional projective variety $X=M_0\times\mathbb P^1$ has Gorenstein terminal singularities and polynomial stringy $E$ -function, but $h_{\mathrm{st}}^{2,5}(X)=-1$ .

Details and sources

AI contribution

Matthew Satriano and Jeremy Usatine state that the counterexample was discovered with ChatGPT assistance. The manuscript does not assign a more specific model or release a transcript.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two-author proof; specialist review pending

Publication

Public arXiv counterexample manuscript; peer review pending

Preprint / manuscript

Activity evidence

Batyrev’s conjecture has guided work in motivic integration, mirror symmetry, and the McKay correspondence since 1998.

This refutes Batyrev’s 1998 conjecture in its stated generality. The example has terminal, hence canonical, singularities and satisfies the required polynomiality assumption.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI ChatGPT (model not disclosed)
Verification: Two-author proof; specialist review pending
Open for: 28 years
Research activity*: 5/5

30 Jul 2026Discrete geometry and geometry of numbers

Nearly linear lattice coverings of arbitrary convex bodies

Problem statement

What is the smallest universal order of lattice-covering density needed for an arbitrary convex body in

\mathbb R^n

Every $n$ -dimensional convex body admits a lattice covering with density at most $Cn\log n(\log\log n)^{10/3+o(1)}$ . This improves the prior universal $O(n^2)$ bound, determines the worst-case polynomial exponent, and strengthens the Schymura–Wang–Xue conjecture.

Details and sources

AI contribution

Heng Li and Xizhi Liu disclose using generative AI to discuss proof strategies, check proofs, and improve exposition. The manuscript does not identify the models or tie individual theorems to specific outputs.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two-author proof; peer review pending

Publication

Public arXiv manuscript with general AI-use disclosure

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the longstanding geometry-of-numbers covering problem and the substantial improvement from quadratic to nearly linear density.

The result determines the optimal polynomial growth exponent while leaving polylogarithmic factors. The AI provenance is genuine but less granular than records with a theorem-specific disclosure.

Conjectured polynomial exponent confirmed and strengthened

Claimed outcome: Proved
Problem origin: Human-source problem
System: Generative AI tools (models not disclosed)
Verification: Two-author proof; peer review pending
Open for: Universal lattice-covering growth conjecture
Research activity*: 5/5

30 Jul 2026Infinite-dimensional Lie groups

Complete model spaces do not force Lie-group regularity

Problem statement

Is every infinite-dimensional Lie group modelled on a complete locally convex space regular?

A contractible complex analytic BCH--Lie group modelled on the complete Silva space $\varphi_{\mathbb C}$ has a homeomorphic exponential map but is not even $C^0$ -semiregular, disproving the proposed completeness-to-regularity implication.

Details and sources

AI contribution

Han and Liu say their initial ideas were developed and completed by AI; GPT-based agents carried out constructions and generated the manuscript, which the authors then manually checked.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked preprint; independent review pending

Publication

Public arXiv manuscript; authors plan a human rewrite

Preprint / manuscript

Danus case-study paper Danus source

Activity evidence

A documented editorial estimate based on the long-running regularity program for infinite-dimensional Lie groups and continuous inverse algebras.

This is a fresh, author-checked preprint with no independent specialist review or formal proof located. The record therefore reports the claim without treating it as peer reviewed.

Regularity question answered negatively

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: DeepMath agents / GPT models
Verification: Author-checked preprint; independent review pending
Open for: Fundamental regularity question
Research activity*: 4/5

20 May 2026Birational geometry and foliations

Optimal bend-and-break constant for foliations

Problem statement

What is the optimal universal bend-and-break constant for rational curves tangent to a rank-

r

foliation?

The optimal bend-and-break constant for a rank- $r$ foliation is $r+1$ , improving the previously known bounds $2n$ and $2r$ and matching the classical constant when $r=n$ .

Details and sources

AI contribution

Human experts supplied a strategy and references. Five Danus workers carried out the synthesis and completed the proof; the experts checked every detail and suggested one simplification.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human experts checked every detail

Publication

Public theorem preprint and Danus case-study report

Preprint / manuscript

Result preprint

Activity evidence

A documented editorial estimate based on the central role of bend-and-break in birational geometry and the prior nonoptimal bounds.

This is a human-steered proof rather than an autonomous discovery: the strategy was supplied by experts, while Danus executed and repaired the long argument.

Optimal constant

r+1

proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Danus / Claude Opus 4.8 / GPT-5.5 workers
Verification: Human experts checked every detail
Research activity*: 4/5

21 May 2026Foliated birational geometry

Shokurov global index conjecture for threefold foliations

Problem statement

Is the index of a numerically trivial log-canonical foliated log Calabi--Yau triple uniformly bounded in dimension three?

Log-canonical foliated triples of dimension at most three with DCC boundary coefficients and numerically trivial $K_{\mathcal F}+B$ have uniformly bounded torsion index. This resolves the three-dimensional foliated version.

Details and sources

AI contribution

Danus independently decomposed the problem and found a Lie-theoretic route for three of five classes. Human experts supplied the missing minimal-model-program idea for two algebraically integrable classes and later guided a stronger generalization.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human-expert-checked preprint

Publication

Public result preprint and detailed Danus case study

Preprint / manuscript

Result preprint

Activity evidence

A documented editorial estimate based on the global index program and the active minimal-model theory of foliations.

The result is precisely the threefold foliated conjecture. Shokurov’s ordinary global index conjecture remains open in dimensions at least four.

Danus case-study paper Readable case study and scope

Threefold foliated conjecture proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Danus / Claude Opus 4.8 / GPT-5.5 workers
Verification: Human-expert-checked preprint
Research activity*: 5/5

21 May 2026Rational singularities and local class groups

Bounded total Cartier indices in families

Problem statement

Are total Cartier indices of rational singularities uniformly bounded in the bounded-family setting of Han--Jiang Problem 4.5?

For rational singularities occurring in the stated bounded family, the total Cartier indices are uniformly bounded, answering Han--Jiang Problem 4.5 affirmatively.

Details and sources

AI contribution

The task was posed as a counterexample hunt, but Danus found an affirmative proof via reductions to commutative algebra and real algebraic geometry. A human corrected one misread PDF hypothesis; Danus repaired and completed the proof.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human-expert-checked proof after documented repair

Publication

Public result preprint and Danus verification narrative

Preprint / manuscript

Result preprint

Activity evidence

A documented editorial estimate based on the focused birational-geometry literature around rational singularities and Cartier-index boundedness.

The first manuscript draft misstated a verified lemma and was rejected by the internal verifier. The released proof incorporates the repair and the authors’ final mathematical check.

Danus case-study paper Readable case study and repair record

Han–Jiang Problem 4.5 answered affirmatively

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Danus / Claude Opus 4.8 / GPT-5.5 workers
Verification: Human-expert-checked proof after documented repair
Research activity*: 3/5

29 Jun 2026Complex hypersurface singularities

Weighted homogeneity via logarithmic vector fields

Problem statement

Do the logarithmic-vector-field conditions proposed by da Silva Machado and Seade characterize weighted homogeneous isolated hypersurface singularities?

Two proposed geometric characterizations of weighted homogeneous isolated hypersurface singularities, phrased through transverse and ambient logarithmic vector fields, are equivalent to weighted homogeneity.

Details and sources

AI contribution

Danus found an independent proof without a suggested route. Experts caught an erroneous literature definition that had propagated into one step; Danus revoked the affected facts, replaced the reference, and rebuilt the proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert-checked AI proof plus independent human proof

Publication

Two independent companion preprints and a public Danus report

Preprint / manuscript

Danus-produced proof Independent human proof

Activity evidence

A documented editorial estimate based on the classical Saito criterion and current work on geometric characterizations of singularities.

The independent human companion is especially useful verification because it reaches the same theorem by a genuinely different method. Neither proof is a proof-assistant formalization.

Danus case-study and repair account

Two geometric characterizations proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Danus / Claude Opus 4.8 / GPT-5.5 workers
Verification: Expert-checked AI proof plus independent human proof
Research activity*: 4/5

27 Apr 2026Geometric measure theory and distance sets

Erdős Problem #953 — planar sets avoiding integer distances

Problem statement

What is the largest possible measure of a subset of a radius-

R

disk in

\mathbb R^2

containing no pair of points at a positive integer distance?

A Poisson–Bessel kernel argument gives $M(R)\ll R^{1/2}$ ; with Sárközy’s lower construction this yields $M(R)=R^{1/2+o(1)}$ .

Details and sources

AI contribution

Przemek Chojecki’s public research index credits GPT-5.5 Pro with the claimed solution. This record follows the scope and caveats stated in the manuscript and community discussion.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public proof; expert digestion ongoing

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Primary problem record Community discussion Ulam research index

Activity evidence

The official record has 17 comments and active specialist discussion comparing the new kernel to Delsarte-style methods.

The discussion contains constructive expert feedback and a streamlined rewrite, but also states that a complete independent digestion of the kernel argument was still in progress.

Claimed asymptotic resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Public proof; expert digestion ongoing
Open for: Asked by 1977
Research activity*: 4/5

27 Apr 2026Discrete geometry and convexity

Erdős Problem #956 — unit distances between convex translates

Problem statement

How many unit set-distances can occur among

n

pairwise disjoint translates of one planar compact convex set?

A disjoint-translate construction gives $h(n)\gg n^{4/3}$ ; combined with Erdős–Pach’s upper bound, the manuscript obtains $h(n)=\Theta(n^{4/3})$ .

Details and sources

AI contribution

GPT-5.5 Pro is credited with adapting Valtr’s parabolic construction, and Aristotle produced a Lean artifact. Human community members identified both the Valtr precursor and gaps in the formal coverage.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Manuscript checked; Lean coverage incomplete

Claim audit

Do not label this “Lean verified” end to end. The formal artifact is reported incomplete, and mathematical priority is qualified by Valtr’s earlier work.

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Claimed-solution manuscript Valtr precursor manuscript

Activity evidence

The official discussion includes specialist checking, a formalization audit, and comparison with Valtr’s 2005 manuscript.

The conventional note contains a self-contained disjointness conversion. Community review reports that the accompanying Lean development formalizes less than half of the note, and that Pavel Valtr had previously announced a closely related construction.

Primary problem record Community discussion Ulam research index

Claimed resolution; audit caveats

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro + Aristotle
Verification: Manuscript checked; Lean coverage incomplete
Claim audit: Issue documented
Open for: Asked by 1990; related precursor
Research activity*: 4/5

21 Jul 2026Knot theory

Conant’s mod- $4$ Kawauchi conjecture

Problem statement

For every amphicheiral knot

K

, does there exist

f(z)\in\mathbb{Z}[z]

such that

\nabla_K(z)\equiv f(z)f(-z)\pmod 4\,?

Jim Conant proved the mod-4 factorization of the Conway polynomial for every amphicheiral knot with help from Claude Fable 5.

Details and sources

AI contribution

The paper credits Claude Fable 5 with helping produce the proof; no more granular discovery record was released.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked preprint

Publication

Complete public arXiv proof

Preprint / manuscript

QED system paper Human comparison paper

Activity evidence

A meaningful specialist conjecture connected to a substantial literature on amphicheiral knots, Conway polynomials, and factorization obstructions.

This proves Conant’s 2006 congruence conjecture. It does not revive Kawauchi’s stronger integral factorization, which is false in general.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5
Verification: Author-checked preprint
Open for: 20 years
Research activity*: 3/5

17 May 2026Algebraic geometry

Integral local invariant cycles in degree one

Problem statement

Let

X\to B

be a semistable one-parameter family of complex projective varieties, let

X_t

be a smooth nearby fiber, and let

T

act on

H^1(X_t,\mathbb{Z})

. Is the natural map

H^1(X,\mathbb{Z})\longrightarrow H^1(X_t,\mathbb{Z})^T

surjective?

QED found an independent proof that the integral local invariant cycle map is surjective in degree one for a semistable one-parameter degeneration, even though the integral statement fails in higher degree.

Details and sources

AI contribution

The system received only the statement and produced a proof mathematically different from the human author’s unreleased proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Domain expert verified

Publication

Public full proof, expert assessment, and comparison paper

Preprint / manuscript

Activity evidence

The rational invariant cycle theorem dates to the 1970s, but the exact integral degree-one question in the QED record was newly contributed in 2026.

The expert called the residue-theoretic idea original, elementary, elegant, and moderately difficult. This resolves degree one, not the false all-degrees integral analogue.

AI-generated proof

Degree-one theorem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: QED / GPT-5.5
Verification: Domain expert verified
Open for: Newly posed in 2026
Research activity*: 3/5

20 Jul 2026Algebraic geometry

Jacobian Conjecture

Problem statement

Does every polynomial map F: ℂⁿ → ℂⁿ with constant nonzero Jacobian determinant have a polynomial inverse?

An explicit polynomial map in three variables has constant nonzero Jacobian determinant but maps three distinct points to the same value. The two-variable case remains open.

Details and sources

AI contribution

Levent Alpöge credited Fable with finding the counterexample after Akhil Mathew suggested the question.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two independent Lean checks

Lean formalization Independent Lean verification record Formal Conjectures determinant-one integration

Publication

Public counterexample, expert expositions, and two independent Lean checks; no journal paper yet

Preprint / manuscript

Independent verification preprint Counterexample manuscript

Activity evidence

A major named conjecture with decades of international work, surveys, reductions, and specialist programs.

This is a complete disproof of the dimension-independent conjecture, not a solution of the still-open n = 2 case. The short explicit certificate can also be checked directly by symbolic algebra.

Original announcement Tao’s mathematical digestion

General conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Claude Fable 5
Verification: Two independent Lean checks
Open for: 87 years
Research activity*: 5/5

11 Jul 2026Algebraic geometry

Grothendieck’s group-scheme question

Problem statement

Is every finite locally free group scheme of order n killed by n, without assuming that the group scheme is commutative?

A finite locally free group scheme of order four was constructed that is not killed by four, settling Grothendieck’s general noncommutative question by counterexample.

Details and sources

AI contribution

Under Akhil Mathew’s direction, Sol found the construction and Fable translated the argument into Lean.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Alexeev's public Lean counterexample project

Publication

Public, unmerged mathlib pull request and expert account; journal publication pending

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A specialist question of Grothendieck with a substantial surrounding group-scheme literature.

The 1,076-line Lean development was compiled and its statement audited against mathlib’s definitions, but the public pull request remained unmerged when checked. The commutative case, previously proved by Deligne, is not contradicted.

Expert account

Question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol + Claude Fable 5
Verification: Lean checked
Research activity*: 3/5

20 May 2026Discrete geometry

Erdős unit-distance conjecture

Problem statement

If u(n) is the maximum number of unit-distance pairs among n planar points, is u(n) = n^(1+o(1))?

An infinite family of planar point sets gives at least n^(1+δ) unit-distance pairs, disproving the expected n^(1+o(1)) upper bound. The exact extremal growth rate remains open.

Details and sources

AI contribution

OpenAI reports an autonomous proof from a general-purpose model, later digested and improved by human mathematicians.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert checked + Lean formalizations

Publication

Proof and nine-author companion remarks released publicly

Preprint / manuscript

Human-verified companion preprint

Activity evidence

One of the best-known problems in discrete geometry, with a large literature and repeated improvements over eight decades.

The conjectured asymptotic upper bound is false; the exact extremal growth rate remains open. Boris Alexeev's public Erdos90 Lean project formalizes the counterexample, and Kevin Buzzard's July 20 account describes the verification. This is distinct from the separate conditional formalization effort. The present audit inspected the sources but did not replay either development.

OpenAI announcement Proof Released reasoning record Formalization account

Central conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Internal OpenAI reasoning model
Verification: Expert checked + Lean formalizations
Open for: 80 years
Research activity*: 5/5

19 Mar 2026Algebraic geometry

Simplicity of the Hodge bundle

For the moduli space of genus g ≥ 2 curves, the Hodge bundle was shown to contain no nontrivial sub-bundles.

Details and sources

AI contribution

The author supplied a single prompt asking for a proof and reports that the mathematical content came from the agent’s output.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Expert-authored and checked preprint

Publication

Public arXiv paper; no proof-assistant formalization located

Preprint / manuscript

Preprint and interaction card

The paper distinguishes mathematical generation from expository editing, but the proof remains human-reviewed rather than kernel checked.

Research question proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Aletheia / Gemini Deep Think
Verification: Expert-authored and checked preprint

30 Jan 2026Arithmetic geometry

Eigenweights for arithmetic Hirzebruch proportionality

Previously unknown eigenweights in the higher arithmetic Hirzebruch proportionality formula were determined for all classical groups.

Details and sources

AI contribution

The agent connected arithmetic geometry to symmetric-group representation theory and derived the general formulas.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Domain-expert checked preprint

Publication

Public arXiv research paper

Preprint / manuscript

Research paper Agent evaluation paper

This is a computation of new structure constants within an existing theory, not a proof of the entire arithmetic proportionality principle.

Prompt and model output

General eigenweights determined

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Aletheia / Gemini Deep Think
Verification: Domain-expert checked preprint

7 May 2026Algebraic geometry

Minimal volume of rank-one stable surfaces

A human proof established the sharp lower bound 1/6351 and uniqueness of the minimizing surface; an AI chatbot re-derived a plurigenus inequality used as the decisive filter.

Details and sources

AI contribution

The model supplied a key inequality that the authors integrated with classification arguments and additional human mathematics.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human proof in public preprint

Publication

Public arXiv paper; publication review ongoing

Preprint / manuscript

The theorem is a genuine resolution, but the AI contribution is one decisive ingredient rather than the complete proof. It is therefore classified as partial AI contribution.

Decisive AI-derived step

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: AI chatbot, model not specified
Verification: Human proof in public preprint

12 Jan 2026Algebraic geometry

Motivic class of genus-zero maps to a flag variety

Under a mild positivity condition on the curve class, the motivic class of based genus-zero maps to the complete flag variety was computed through an iterative human–AI proof strategy.

Details and sources

AI contribution

AI systems solved scaffolded special cases; human analysis extracted the general mechanism, after which the systems completed remaining steps.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Multi-author mathematical proof

Publication

Public arXiv preprint

Preprint / manuscript

This case is included to avoid reducing AI mathematics to autonomous one-shot proofs: the result depended on an iterative division of labor.

Human–AI proof completed

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: FullProof workflow
Verification: Multi-author mathematical proof

25 Feb 2026Discrete geometry

Erdős Problem #846

Problem statement

If every

n

-point subset of an infinite planar set contains at least

\epsilon n

points with no three collinear, must the whole set be a finite union of sets with no three collinear?

Independent AI efforts produced a counterexample and Lean proof; a 2024 result implying a comparable counterexample was identified afterward.

Details and sources

AI contribution

The DeepMind system produced a Lean-verified solution while an OpenAI model independently found an informal one.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + independent derivation

Publication

Public problem record; comparable 2024 literature found afterward

Preprint / manuscript

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The entry records independent problem solving and formal verification, not the first historical proof of the negative answer.

Independent rediscovery formalized

Problem origin: Human-source problem
System: DeepMind prover agent + OpenAI internal model
Verification: Lean checked + independent derivation
Open for: Known from 2024 literature
Research activity*: 1/5

23 Feb 2026Discrete geometry

Sphere packing in dimensions 8 and 24

Viazovska’s dimension-8 proof and the dimension-24 Leech-lattice proof were completed as sorry-free Lean developments totaling roughly 200,000 lines.

Details and sources

AI contribution

Gauss worked from a substantial human blueprint and existing repository, then completed the remaining proof goals at scale.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked + expert project audit

Project site

Publication

Public repository, project site, and formalization paper

Preprint / manuscript

Formalization paper

The mathematical theorems were proved in 2016. The 2026 milestone is formal verification, not a new solution of sphere packing.

Gauss account

Fields Medal proofs formalized

Problem origin: Origin not yet traced
System: Gauss
Verification: Lean checked + expert project audit

13 Jan 2026Geometry, Distances

Erdős Problem #659

Problem statement

Can

n

planar points determine only

O(n/\sqrt{\log n})

distances while every four-point subset determines at least three distinct distances?

The problem was proved after remaining open for 29 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Gemini 3 is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Erdős case-study preprint Dedicated solution manuscript

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 21 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Gemini 3
Verification: Lean checked
Open for: 29 years
Research activity*: 2/5

9 Apr 2026Geometry

Erdős Problem #960

Problem statement

For planar point sets with no

k

collinear points, how many ordinary lines force an

r

-point subset whose every connecting line is ordinary? Is the threshold

o(n^2)

, or even

O(n)

The problem was disproved after remaining open for 42 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Erdős Problems site confirmed
Open for: 42 years
Research activity*: 1/5

1 Feb 2026Geometry, Distances

Erdős Problem #1089

Problem statement

Let

g_d(n)

be the fewest points in

\mathbb R^d

that always determine at least

n

distances. Estimate

g_d(n)

; in particular, does

g_d(n)/d^{n-1}

have a limit as

d\to\infty

The problem was resolved after remaining open for 51 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aletheia is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Erdős case-study preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aletheia
Verification: Erdős Problems site confirmed
Open for: 51 years
Research activity*: 1/5

Logic & foundations

3 Sep 2026 revisionLogic & foundations

Heyting algebras as subterminal lattices of elementary topoi

The free Heyting algebra on two generators, and consequently every free Heyting algebra on $N\geq2$ generators, cannot occur as the lattice of subterminal objects of an elementary topos.

Details and sources

AI contribution

Two independently steered model sessions converged on the free two-generator algebra and a Bellissima-reachability argument. The authors then disentangled, reorganized, simplified, and wrote the proof.

Problem origin

The paper identifies a long-standing categorical-logic problem, cites Pitts's 2025 summary, and distinguishes the earlier positive result for second-order intuitionistic propositional logic.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This gives a negative answer to the general realization question through explicit counterexamples. The authors report roughly three and twenty-one hours of human–AI interaction in the two searches, but no complete public transcripts or formal proof are supplied.

Claimed resolution in a public preprint

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

31 Aug 2026Logic & foundations

Stable forking conjecture

A simple theory built from an infinite-dimensional vector space over the division ring of fractions of a quantum graph algebra gives a counterexample to the stable forking conjecture.

Details and sources

AI contribution

The authors call the counterexample AI-generated: carefully constrained prompts directed the model toward the construction strategy. The authors wrote and checked the proofs and manuscript.

Problem origin

Hart, Kim, and Pillay introduced the conjecture in 1996; the paper identifies it as a long-standing human-origin problem in model theory.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This is a claimed full disproof in a public preprint, not a Lean formalization or peer-reviewed result. The paper provides a detailed qualitative AI-use account but no complete prompt transcript or external verification.

Claimed resolution in a public preprint

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

7 Aug 2026Reverse mathematics and reflection principles

Lévy–Montague reflection over $\mathsf{WKL}_0$

Problem statement

What is the

\Pi^1_1

strength of Lévy–Montague reflection when added to

\mathsf{WKL}_0

Adding Lévy–Montague reflection to $\mathsf{WKL}_0$ is $\Pi^1_1$ -conservative over $\mathsf{WKL}_0$ .

Details and sources

AI contribution

The paper describes an extended human–model collaboration. Fable proposed the chain-of-extensions construction with universal sets; the author recognized the stronger reflection argument, repaired the technical core, and wrote the final proof.

Problem origin

The theorem concerns a classical human-developed reflection scheme and its exact reverse-mathematical strength.

Verification

Author-verified proof preprint; external review pending

Publication

Public proof preprint with detailed contribution account

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the central role of conservation theorems in reverse mathematics and the specialized nature of this reflection scheme.

This is a new research theorem rather than a famous named conjecture. The entry records the exact conservativity statement and does not generalize it to stronger base theories.

\Pi^1_1

-conservativity proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5
Verification: Author-verified proof preprint; external review pending
Research activity*: 3/5

Mathematical physics

1 Sep 2026Mathematical physics

McKean entropy-production monotonicity conjecture

Smooth positive radial mixtures for the homogeneous Boltzmann equation have increasing entropy production at some time for Maxwell molecules and hard spheres, disproving monotonicity for standard collision kernels.

Details and sources

AI contribution

The manuscript credits the models with finding and testing the mixture construction and assisting the proof; the human authors checked and wrote the argument.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This is the 1966 monotonicity question for the specified three-dimensional collision kernels, not every entropy-production inequality associated with McKean. External review remains pending.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol and Claude (version unspecified)
Verification: Author-checked proof; independent review pending

2 Sep 2026Mathematical physics

Nonlattice stealthy configuration in dimension twenty

A periodic configuration in $\mathbb R^{20}$ , formally dual to Vardy's sphere packing, is stealthier at unit particle density than every currently known twenty-dimensional lattice.

Details and sources

AI contribution

The model found and simplified the twenty-dimensional configuration from Conway–Sloane's antipode construction and checked the formal-duality connection; the authors then verified the calculations.

Problem origin

The paper frames maximizing the stealth radius at fixed density as a prior human optimization problem and compares the construction with the established lattice record.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

Paper and version history

This is a record construction, not an exact twenty-dimensional optimum. The paper's main optimality theorems for $E_8$ and the Leech lattice were developed by the authors; AI is credited specifically for the twenty-dimensional example. The authors note possible overlap with a prior construction in Li–Pott–Schüler.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author-checked proof; independent review pending

9 Jul 2026Mathematical physics

Vlasov mean-field dynamics — Dobrushin theory formalized

The Lean project formalizes well-posedness, stability, and the mean-field limit for Vlasov dynamics with smooth Lipschitz forces, including a Wasserstein-distance layer and a short-time superposition argument.

Details and sources

AI contribution

Joseph K. Miller directed the formalization and reports that AI agents wrote the Lean proofs. The mathematics follows the existing Dobrushin framework.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Public formal development; project-reported checking, not replayed in this audit

Pinned v2 Lean release

Publication

Research preprint and pinned v2 formalization release

Preprint / manuscript

This is the smooth-force theory, not a solution of singular Coulomb or gravitational mean-field problems. The author reports no sorry proofs and only standard foundational axioms; this audit did not rebuild the development. It is distinct from the index's steady-state Vlasov–Maxwell–Landau record.

Proof documentation

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: AI coding agents; exact models not identified in the inspected paper
Verification: Public formal development; project-reported checking, not replayed in this audit

16 Mar 2026Mathematical physics

Gaussian free field — Osterwalder–Schrader axioms formalized

OSforGFF formalizes the massive Gaussian free field and its Osterwalder–Schrader axioms. The August v3.2 development extends the original four-dimensional result to every dimension at least two.

Details and sources

AI contribution

The project documents human-directed AI formalization by Cherkis, Douglas, Hoback, Mei, and Nissim, using the listed models and several supporting Lean libraries.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Public formal development; project-reported checking, not replayed in this audit

Lean project, model attribution, and August scope Statement and axiom guardrails

Publication

Public formalization project

Preprint / manuscript

Paper and revisions

The current active proof graph is reported to have no sorry proofs or custom axioms; guardrails freeze the headline theorem statements and axiom footprints, and CI includes a second-kernel check. Earlier external inputs were proved or avoided. This audit did not replay CI. Free-field verification does not solve interacting quantum field theory or the Yang–Mills mass-gap problem.

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Opus 4.6, Claude Fable 5, Gemini 3 Pro, and GPT-5.2 Codex
Verification: Public formal development; project-reported checking, not replayed in this audit

28 Aug 2026Mathematical physics

Critical points of signed point-charge potentials restricted to a line

After merging coincident projected source classes and removing cancellations, $m$ effective sources yield at most $2m-1$ line critical points for every finite positive inverse-power exponent. The bound is attained by positive-charge examples.

Details and sources

AI contribution

The main idea emerged in exploratory model conversations; the author reconstructed, checked, and revised the mathematical arguments.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This resolves the corrected signed line statement of Gabrielov–Novikov–Shapiro and the positive-charge line specialization of Edelsbrunner–Fillmore–Oliveira. It does not settle the unrestricted spatial Maxwell problem. Complete cancellation is treated separately.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro
Verification: Author-prepared proof; independent review pending

27 Aug 2026Mathematical physics

Exact variational stiffness in quantum geometric nesting models

The stiffness of the broken continuous symmetry equals its Gaussian variational value in quantum geometric nesting models, proving the cited quantum-many-body-bootstrap conjecture.

Details and sources

AI contribution

The authors used the models to streamline the proof and polish the writing; they do not attribute the original discovery wholly to AI.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The result relies on the specified frustration-free QGN setting and its invariant one-particle–one-hole sector. The authors acknowledge a concurrent independent proof by Wu, Li, and Yao.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5 and GPT-5.6 Sol
Verification: Author-prepared proof; independent review pending

12 Aug 2026Geometric variational problems and nonlocal energies

Minimizers in Gamow's liquid drop problem

Problem statement

For which volumes does Gamow's liquid-drop energy admit a minimizer, and when is that minimizer necessarily a round ball?

For fixed volume $V$ , round balls uniquely minimize $\mathcal E(\Omega)=P(\Omega)+D(\Omega)$ exactly when $V\leq V_*$ , while no minimizer exists for $V>V_*$ . The theorem also determines the minimal binding energy.

Details and sources

AI contribution

Chodosh and Gianocca state that ChatGPT 5.6 Pro obtained the results over a series of chats without significant assistance. They checked and reworked the proof, while retaining its fundamental strategy, and wrote the article themselves.

Problem origin

The minimization problem comes from Gamow's 1928 liquid-drop model, and the sharp threshold conjecture appeared in the human mathematical literature before this work.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint with explicit AI-use statement

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the model's long history, the sharp resolution of both existence and shape, and its connections to geometric analysis, PDEs, and mathematical physics.

This is a full characterization for the three-dimensional Gamow functional studied in the paper. It does not resolve every variant of liquid-drop models with different kernels, dimensions, or constraints.

Liquid-drop minimizer conjecture resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro
Verification: Author-checked proof preprint; external review pending
Open for: Conjectured sharp threshold in the liquid-drop literature
Research activity*: 5/5

11 Aug 2026Spin glasses and random matrices

Critical Sherrington–Kirkpatrick overlap distribution

Problem statement

What is the quenched overlap distribution of the Sherrington–Kirkpatrick model at criticality, and does

N^{2/3}\mathbb E\langle R_{1,2}^2\rangle

converge as Talagrand conjectured?

At inverse temperature $\beta=1$ , the Ising and spherical Sherrington–Kirkpatrick overlaps have scale $N^{-1/3}$ and converge to an explicit random measure built from the reflected $\mathsf{Airy}_1$ point process; the resulting second-moment limit proves Talagrand's conjecture.

Details and sources

AI contribution

The authors state that most arguments were generated with GPT-5.6 Pro while they explored consequences of their earlier sphere-to-cube comparison principle. The problem framing and companion ideas came from the human research program.

Problem origin

The limiting critical-overlap question and the second-moment statement were posed in Talagrand's human-developed spin-glass program, including Conjecture 11.7.5.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Research preprint AI-generated candidate proof manuscript

Activity evidence

A documented editorial estimate based on the central role of critical overlap laws in spin-glass theory and the explicit resolution of a Talagrand conjecture.

The theorem identifies the critical overlap law and resolves the cited second-moment conjecture. It does not address the separate low-temperature free-energy variance problem highlighted in the paper.

Talagrand conjecture resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Author-checked proof preprint; external review pending
Research activity*: 5/5

11 Aug 2026Interacting particle systems and anomalous transport

Candidate current-fluctuation laws for the stochastic XNOR process

Problem statement

What are the fully normalized long-time current laws for homogeneous biased, biased domain-wall, and zero-magnetization initial data in the continuous-time stochastic XNOR process?

A candidate proof proposes Gaussian, half-normal, and M-Wright limits with explicit amplitudes for three stochastic-XNOR current regimes. Parameter-free simulations support the conjectures, but the mathematical companion has not been checked line by line.

Details and sources

AI contribution

The models assisted the analytic derivation and writing and generated the simulations, analysis scripts, and figures. The separate mathematical proof draft was generated substantially with AI; the author checked its structure and selected calculations but not every technical argument.

Problem origin

The author selected the stochastic model and observables, while the three fully normalized conjectures and their candidate proof were developed in a human–AI workflow from gaps left by earlier human literature.

Verification

Numerically supported candidate; proof draft not fully checked

Claim audit

The author explicitly says the candidate proof has not received independent line-by-line verification and includes material outside the author's expertise.

Publication

Public physics preprint and separate candidate-proof manuscript

Preprint / manuscript

Activity evidence

A documented editorial estimate based on links to anomalous transport and integrable systems, balanced against the co-generated formulation and incomplete proof verification.

The primary paper deliberately states the three results as conjectures. The companion is not being submitted for peer review in its current form and is indexed as a candidate advance, not a resolved theorem.

AI-generated candidate proof; verification incomplete

Problem origin: Human--AI co-generated
System: ChatGPT 5.5 / GPT-5.6 Sol
Verification: Numerically supported candidate; proof draft not fully checked
Claim audit: Issue documented
Research activity*: 3/5

3 Aug 2026Discrete harmonic analysis and mathematical physics

Discrete unique continuation on a lattice simplex

Problem statement

What support size is forced when a function on the lattice simplex obeys every oriented-simplex relation and is nonzero at the balanced point?

For functions on $\Delta_{nR}^{(n)}$ satisfying the complete oriented-simplex relations, a nonzero balanced coefficient forces $|\operatorname{supp}g|\geq c_nR^{\lceil n/2\rceil}$ . Explicit constructions show that the exponent is optimal.

Details and sources

AI contribution

The author reports human-guided discovery and exploration with GPT-5.6 Sol, then independently verified every argument and wrote the public proof.

Problem origin

The paper formulates and solves the problem in the same project and describes human-guided discovery with GPT-5.6 Sol.

Verification

Independently checked by the author; external review pending

Publication

Public 17-page proof preprint

Preprint / manuscript

Conditional Lean development and trust boundary

Activity evidence

A documented editorial estimate based on the newness of the exact formulation and its connections to Pascal-type uncertainty principles and lattice unique continuation.

This is not presented as a longstanding named conjecture: the paper formulates the simplex problem and solves it in the same work. It is included as a transparent AI-assisted research claim rather than being assigned an inflated time-open estimate.

Newly formulated problem solved sharply

Claimed outcome: Proved
Problem origin: Human--AI co-generated
System: GPT-5.6 Sol
Verification: Independently checked by the author; external review pending
Open for: Newly formulated discrete unique-continuation problem
Research activity*: 2/5

20 Jul 2026Spin glasses and probability

Full replica-symmetry breaking at zero temperature in the SK model

Problem statement

Does the zero-temperature Parisi measure for the zero-field SK model have smooth density and full support on

[0,1)

For the zero-field Sherrington--Kirkpatrick model, the zero-temperature Parisi minimizer is absolutely continuous with smooth density and full support $[0,1)$ . The positive-temperature endpoint also satisfies $q_\beta\to1$ .

Details and sources

AI contribution

The manuscript states that ChatGPT 5.6 generated the proof arguments and prose. Hong-Bin Chen prompted, edited, proofread, verified, and assumes responsibility for the paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked + conditional Lean verification

Publication

Public preprint and Lean 4 repository

Preprint / manuscript

Counterexample preprint and supplement

Activity evidence

A documented editorial estimate based on the central role of the Parisi measure, replica-symmetry breaking, and overlap structure in spin-glass theory.

The Lean development checks the two main new theorems only after assuming seven named analytic inputs covering Parisi theory, PDE and stochastic analysis, convergence, and the origin-support fact. It is therefore a precise conditional certificate, not an assumption-free formalization of the entire paper.

Zero-temperature support question proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6
Verification: Author checked + conditional Lean verification
Open for: Longstanding zero-temperature structure question
Research activity*: 4/5

29 Jul 2026Atomistic representations and geometric machine learning

Fixed-order incompleteness of atom-centered structural descriptors

Problem statement

Are five-body trispectrum descriptors, or any fixed finite order of atom-centered density correlations, universally sufficient to distinguish noncongruent three-dimensional atomic environments?

There are noncongruent three-dimensional atomic environments with identical complete correlations through seven-neighbor order. More generally, for every finite correlation-order and angular-cutoff pair $(\nu_{\max},l_{\max})$ , continuous families of noncongruent environments have identical retained features.

Details and sources

AI contribution

Interactive model conversations located and translated reconstruction results from distant fields, including musical $Z$ -relations, into counterexamples for atom-centered descriptors. The authors checked the constructions manually and with AI-generated code and report separate systematic search experiments in the supplement.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; validation archive pending

Publication

Public preprint with constructions, proofs, and supplementary information

Preprint / manuscript

Activity evidence

A documented editorial estimate based on sustained work on completeness, stability, and expressive power in atomistic machine learning.

This answers the stated trispectrum-sufficiency question negatively and rules out universal completeness for any fixed finite truncation of this descriptor hierarchy. The authors stress that the examples are far from plausible chemical structures, and that the result does not invalidate alternative provably complete representations.

Universal fixed-order completeness disproved

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Claude Opus 4.8 + Fable + Codex 5.5/5.6 Sol
Verification: Author checked; validation archive pending
Open for: Open descriptor-completeness questions since 2020
Research activity*: 3/5

12 Feb 2026Scattering amplitudes

Single-minus gluon tree amplitudes

Problem statement

Do the tree-level amplitudes

A_n(1^-,2^+,\ldots,n^+)

vanish identically, or can they be nonzero in half-collinear kinematics—and if nonzero, what is their all-

n

closed form?

Single-minus tree amplitudes, often presumed to vanish, are shown to be nonzero on half-collinear complex kinematics, with a piecewise-constant closed formula for every multiplicity.

Details and sources

AI contribution

GPT-5.2 Pro conjectured the all- $n$ formula from human-computed low-point cases; a scaffolded internal model produced a proof later checked analytically by the authors.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Analytically checked by authors

Publication

Public preprint submitted for publication

Preprint / manuscript

Activity evidence

Scattering amplitudes are a large international research program, and the degenerate single-minus configuration had been a recurring specialist question for roughly fifteen years.

This is a new theorem in mathematical physics rather than the settlement of a named conjecture, so it is classified as a variant result.

OpenAI research account

Long-standing presumption overturned

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: GPT-5.2 Pro
Verification: Analytically checked by authors
Open for: Question pursued for about 15 years
Research activity*: 4/5

2 Jun 2026Mathematical statistical physics

FullRSB jamming identity $a+b=1$

Problem statement

Can the numerically observed fullRSB critical-exponent relation

a+b=1

at the jamming transition be derived analytically from the scaling equations?

Parisi and Zamponi proved analytically that the full replica-symmetry-breaking jamming exponents satisfy $a+b=1$ , an identity previously observed only numerically to high precision.

Details and sources

AI contribution

The authors state that the proof was obtained through interaction with Claude Sonnet 4.6 and Opus 4.7 and then verified by them.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author verified and journal published

Publication

Journal article and public arXiv manuscript

Preprint / manuscript

Activity evidence

The identity controls scaling relations for gap, force, and overlap exponents in a heavily studied theory of jamming.

The theorem is conditional on the fullRSB scaling framework and takes existence and uniqueness of the relevant profile as given. Within that framework it closes the analytic identity.

Journal publication

Previously numerical identity proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Claude Sonnet 4.6 + Claude Opus 4.7
Verification: Author verified and journal published
Open for: Unproved since the 2014 fullRSB analysis
Research activity*: 4/5

29 Jul 2026Classical electrostatics and critical-point theory

Maxwell conjecture for point-charge equilibria

Problem statement

n

positive point charges have only nondegenerate electrostatic equilibria, can their potential have more than

(n-1)^2

critical points?

Five positive point charges in $\mathbb R^3$ are arranged so that their electrostatic potential has at least $24$ nondegenerate critical points, exceeding Maxwell’s proposed bound $(n-1)^2=16$ . The construction can also be iterated to give $3+2m$ charges with at least $4+20m$ critical points.

Details and sources

AI contribution

The authors state that GPT-5.6 Sol suggested the construction idea. They verified the mathematical details and wrote the proof in their own words.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three-author verification; Mathematica and Maple checks

Publication

Public arXiv manuscript; computer-algebra checks; peer review pending

Preprint / manuscript

Primary 2007 conjecture paper Recent peer-reviewed problem context Author discussion on LinkedIn Author discussion on X

Activity evidence

A classical electrostatics problem traced to Maxwell, with modern work in critical-point theory, fewnomials, and computational geometry.

This refutes the general $(n-1)^2$ upper bound at $n=5$ . It does not determine the exact maximum number of equilibria; the maximum is still unknown even for three unequal charges.

Conjectured quadratic upper bound disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI GPT-5.6 Sol
Verification: Three-author verification; Mathematica and Maple checks
Open for: Upper-bound problem posed in 1969; named conjecture formulated in 2007
Research activity*: 5/5

8 Jul 2026Cosmohedron combinatorics

Factorial asymptotics of the Matryoshka numbers

Problem statement

Do the Matryoshka numbers satisfy

a_n\sim c\,n!n^4

, as conjectured in OEIS A177384, and what is the constant

c

For OEIS A177384, $a_n\sim S n!n^4$ , with the rigorous enclosure $0.00542831750\leq S\leq0.00542831848$ .

Details and sources

AI contribution

After receiving only the problem statement, Danus autonomously proved existence of the limit and derived the explicit certified enclosure. The authors report no mathematical guidance beyond the common final check.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human-expert-checked elementary proof

Publication

Public proof note and Danus case-study paper

Preprint / manuscript

Proof note

Activity evidence

A documented editorial estimate based on the OEIS conjecture and its role in the combinatorics of the cosmohedron.

The interval is rigorous rather than a numerical fit. Since no closed form for $S$ is known, the certified enclosure is the stated form of the constant determination.

Danus case-study paper Readable case study

OEIS asymptotic conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Danus / Claude Opus 4.8 / GPT-5.5 workers
Verification: Human-expert-checked elementary proof
Research activity*: 3/5

13 Jul 2026Interacting particle systems

Type-D ASEP Tracy–Widom marginals

Problem statement

Do the one-species current marginals of type-D ASEP have the predicted Tracy–Widom long-time asymptotics despite the model’s two-species interactions?

An exact current-decoupling identity proves the predicted Tracy–Widom marginal asymptotics in the fixed- $q$ regime and establishes the stated hydrodynamic and weak-asymmetry limits.

Details and sources

AI contribution

Claude wrote the paper outside the abstract and introduction; Aristotle produced the Lean development. The human author manually verified the arguments and curated the formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-reviewed proof; foundational Lean tiers checked

Lean development Documented remaining formalization steps

Publication

Public arXiv paper, Lean repository, and generated proof blueprint

Preprint / manuscript

Research paper Original type-D ASEP conjecture

Activity evidence

A focused conjecture inside active integrable-probability and interacting-particle-system research, with a public formal development and additional unresolved correlation questions.

The repository’s own REMAINING_STEPS file says only its first two foundational tiers are sorry-free. The headline Tracy–Widom and SPDE layers still use intentional placeholders or black-box hypotheses, so this entry is not classified as an end-to-end formal proof. A separate covariance conjecture also remains open.

Formalization blueprint

Marginal conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Opus 4.8 / Claude Fable 5 / Aristotle
Verification: Author-reviewed proof; foundational Lean tiers checked
Open for: 4 years
Research activity*: 3/5

21 Jul 2026Quantum many-body theory

Free energy and spectral edge of fixed-even-order SYK models

Problem statement

Determine the leading asymptotic of the largest eigenvalue of the

N

-Majorana SYK Hamiltonian at each fixed even interaction order

q\geq4

N\to\infty

For each fixed even interaction order $q\geq4$ , the revised manuscript claims convergence of both annealed and quenched normalized pressures to the Schwinger–Dyson pressure at every fixed positive inverse temperature, together with the asymptotic largest eigenvalue and a zero-temperature limit.

Details and sources

AI contribution

The author reports GPT-5.6 assistance with literature search, development of technical arguments, and manuscript preparation, while retaining responsibility for the result.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-controlled arXiv preprint; review pending

Claim audit

A new single-author preprint with no independent specialist review or formal verification located.

Publication

Public arXiv manuscript and TeX source; no independent review, code, or formal proof located

Preprint / manuscript

Activity evidence

SYK spectral statistics connect probability, operator algebras, quantum gravity, and many-body physics and have generated intensive work since the model’s modern revival.

The 1 September v3 adds all-temperature free-energy convergence and rewrites the microscopic argument using cavity expansions, conditional factorization, and a Majorana-bath representation. Its Hamiltonian uses thermodynamic normalization; the old record's square-root scaling and numerical constant are not carried over across normalizations. Independent specialist review remains pending.

Every fixed even interaction order resolved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.6
Verification: Author-controlled arXiv preprint; review pending
Claim audit: Issue documented
Open for: Primary SYK spectral-edge question active for roughly a decade
Research activity*: 5/5

5 Mar 2026Mathematical physics

Cosmic-string radiation integral

Problem statement

Evaluate, for arbitrary loop angle

\alpha

and integer harmonic

N

I(N,\alpha)=\int_{S^2}\frac{[1-(-1)^N\cos(N\pi e_1)][1-(-1)^N\cos(N\pi e_2)]}{(1-e_1^2)(1-e_2^2)}\,d\Omega,

the singular integral determining the cosmic-string radiation power

P_N=32G\mu^2 I(N,\alpha)/(\pi^3N^2)

A Gemini Deep Think tree-search system derived six analytical methods for the singular sphere integral governing the harmonic power spectrum of gravitational radiation from arbitrary cosmic-string loop geometries.

Details and sources

AI contribution

The model proposed symbolic derivations while executable high-precision numerical feedback pruned erroneous branches across roughly 600 candidates.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Analytical derivation + numerical checks

Publication

Detailed arXiv manuscript with prompts, search constraints, code, and derivations

Preprint / manuscript

Research preprint and checker

Activity evidence

A live mathematical-physics calculation with recent partial attempts, rather than a decades-old named conjecture.

Previous work had only asymptotic or odd-harmonic results. The paper gives a unified exact treatment and six independent derivation routes, but it has not been proof-assistant formalized.

Exact analytical solution derived

Claimed outcome: Proved
Problem origin: Human-source problem
System: Gemini Deep Think + tree search
Verification: Analytical derivation + numerical checks
Open for: Under 1 year
Research activity*: 2/5

Number theory

8 Sep 2026Multiplicative arithmetic functions

Fibonacci numbers are not an additive uniqueness set

A positive integer-valued multiplicative function can satisfy $f(F_n+F_m)=f(F_n)+f(F_m)$ for all $n,m\ge1$ without being the identity. The smallest displayed construction exchanges the prime signatures of $557$ and $2417$ .

Details and sources

AI contribution

Poo-Sung Park reports that GPT-6 Astra proposed the prime-signature switching mechanism, designed the exact-integer search, found the first pair, and helped compress the search into a finite modular criterion.

Problem origin

C. A. Spiro posed the Fibonacci additive-uniqueness question in 1992; a 2025 survey still listed it as open.

Verification

Author proof with reproducible exact certificate checker

Publication

Public arXiv proof with embedded Python checker

Preprint / manuscript

The checker validates the finite modular certificate for each displayed prime pair; the paper separately proves why that certificate covers every Fibonacci sum. Independent review is pending.

Spiro's 1992 question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra
Verification: Author proof with reproducible exact certificate checker

1 Sep 2026Number theory

Erdős Problem #1 — distinct subset sums

Epoch reports a disproof of the distinct-subset-sums conjecture: for every positive $\epsilon$ , arbitrarily large dissociated sets of size $n$ fit in $\{1,\ldots,N\}$ with $N\leq\epsilon 2^n$ .

Details and sources

AI contribution

Problem origin

Adamczewski and Bloom identify this numbered, previously open Erdős problem in Appendix B of FrontierMath: Erdős. It was not generated by the model.

Verification

Public Lean/Comparator proof independently replayed; canonical tracker accepted

Publication

Public formal proof and FrontierMath: Erdős report

Preprint / manuscript

Benchmark announcement and experimental scope Public statements and benchmark harness Canonical Erdős Problems record Independent build and statement audit

An additional, enlarged-budget experiment, outside the benchmark's two successful runs. The construction is ineffective: it does not give a usable threshold for each epsilon. The canonical tracker now accepts the Lean disproof. The public audit reports 4,608 Lean lines and two independent successful runs implementing essentially the same lattice argument. Independent builds found zero sorry outside statement stubs, no added axioms, unsafe features, or native_decide, and only Lean's standard foundational axioms. Formal statements for 68 targets do not mean 68 solved problems.

Claimed full resolution; disproof reported

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Public Lean/Comparator proof independently replayed; canonical tracker accepted

1 Sep 2026Number theory

Erdős Problem #126 — prime divisors of pairwise sums

For an $n$ -element set of positive integers, Epoch reports that the number of distinct primes dividing sums of two distinct elements is $\gg\sqrt n$ , strengthening the requested superlogarithmic growth.

Details and sources

AI contribution

Problem origin

Adamczewski and Bloom identify this numbered, previously open Erdős problem in Appendix B of FrontierMath: Erdős. It was not generated by the model.

Verification

Public Lean/Comparator proof independently replayed; canonical tracker accepted

Publication

Public formal proof and FrontierMath: Erdős report

Preprint / manuscript

Benchmark announcement and experimental scope Public statements and benchmark harness Canonical Erdős Problems record Independent build and statement audit

One of the two successes in the 68-target benchmark. Successively stronger intermediate exponents are not counted as separate problem resolutions. The canonical tracker now accepts the Lean proof. The public audit reports 7,866 Lean lines and four independent successful runs; the indexed theorem records the qualitative superlogarithmic conclusion without multiplying the intermediate bounds into separate discoveries. Independent builds found zero sorry outside statement stubs, no added axioms, unsafe features, or native_decide, and only Lean's standard foundational axioms. Formal statements for 68 targets do not mean 68 solved problems.

Claimed full resolution; proof reported

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Public Lean/Comparator proof independently replayed; canonical tracker accepted

3 Sep 2026 updateNumber theory

Erdős Problem #4 — improved maximal-prime-gap lower bound

The new Lean development proves $G(X)\gg (\log X)(\log_2 X)^2\log_4 X/(\log_3 X)^2$ for all sufficiently large $X$ , asymptotically improving the 25 August GPT-5.6 Sol bound by an unbounded factor.

Details and sources

AI contribution

OpenAI attributes the new proof and Lean formalization to GPT-6 Astra; the project also publishes an abridged reasoning trace and exact formal statement.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Public Lean proof independently built and kernel-replayed

New Lean proof and manuscript

Publication

Public formalization project

Preprint / manuscript

Superseded 25 August manuscript

VibeMathed reports a clean build of all 8,707 jobs at commit 03a1190d, a leanchecker replay, and only Lean's three standard foundational axioms. This is an unconditional lower-bound advance, unlike the separate 186 short-gap claim with three explicit assumptions. No named mathematician has yet reviewed the new argument. This remains an advance toward Erdős Problem #4, not Cramér's conjecture or a determination of the true maximal-gap order.

Independent build and statement audit Original problem and community discussion

Proved partial advance; optimal prime-gap growth remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra
Verification: Public Lean proof independently built and kernel-replayed

2 Sep 2026Number theory

Prime gaps at most 186 — conditional formal proof

The OpenAI project derives infinitely many consecutive prime gaps at most 186 from two named exponential-sum estimates and an explicit package of numerical integral bounds, via an admissible 40-tuple.

Details and sources

AI contribution

The formalization manifest identifies GPT-6 Astra operating through Codex, with human-guided edits, automated checks, and instructions to retain unresolved inputs as explicit axioms.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Conditional Lean proof; three project axioms plus standard foundations

Lean proof and explicit assumptions Numerical certificate checker

Claim audit

Conditional formalization: three mathematical inputs remain assumptions; there is no independent human semantic review.

Publication

Public formalization project

Preprint / manuscript

Numerical manuscript

The three explicit assumptions are kloosterman3_bound, kloosterman2_correlation_bound, and physical_integral_bounds. The release reports Comparator and a second-kernel check with those assumptions allowed. Its numerical checker requires a disclosed FLINT modification; this audit did not run it. This is not an unconditional end-to-end Lean proof and does not establish twin primes.

Official release and verification limitations Formalization provenance manifest

Claimed improved bound; three formal inputs remain assumed

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra
Verification: Conditional Lean proof; three project axioms plus standard foundations
Claim audit: Issue documented

Aug 2026Number theory

Bounded prime gaps of 246 — AxiomProver formalization

PrimeGapsLib formally derives infinitely many prime gaps at most 246 from the Bombieri–Vinogradov theorem, including the required numerical certificate. It also formalizes the earlier bound of 600.

Details and sources

AI contribution

AxiomProver generated Lean proofs from a human mathematical blueprint, with the project page documenting the human contributors.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Public Lean proof conditional on Bombieri–Vinogradov

Lean library and exact conditional scope

Publication

Public formalization project

Preprint / manuscript

No public preprint or manuscript located.

The bound of 246 was already known from Polymath's human work. Bombieri–Vinogradov remains an external hypothesis of this formalization. The 600 and 246 statements are one project, not two newly solved open problems. The README's Comparator/Challenge.lean link returned 404 during this audit; the repository and main-result documentation remain available.

Project, blueprint, and contributor attribution

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: AxiomProver (Axiom Math)
Verification: Public Lean proof conditional on Bombieri–Vinogradov

Date unconfirmedNumber theory

Hyperelliptic curves over finite fields — Gauss formalization

Gauss formalizes the Bombieri–Stepanov square-root point-count bound for $y^2=f(x)$ over a finite field of size $q$ : the discrepancy from $q$ is less than $5m\sqrt q$ when $m=\deg f\geq3$ , $q>6m$ , and $f$ is not a square over the algebraic closure.

Details and sources

AI contribution

The repository attributes the statements, proofs, and documentation to Gauss, working from a human-supplied blueprint based on Iwaniec–Kowalski, Chapter 11.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Public formal development; project-reported checking, not replayed in this audit

Final Lean theorem with exact hypotheses

Publication

Public formalization project

Preprint / manuscript

Human proof blueprint

The nonsquare hypothesis is present in the inspected final Lean theorem and must not be dropped. This is known finite-field mathematics, not the Riemann hypothesis for the zeta function or a new theorem for all varieties. Release date was not established from the retrieved primary metadata; backfilled in the 4 September audit. The project was not rebuilt here.

Project and AI attribution

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: Gauss (Math Inc.)
Verification: Public formal development; project-reported checking, not replayed in this audit

4 Sep 2026Number theory

Vinogradov's three-primes theorem — formalization announcement

Anthropic's FLT announcement also reports a Prove2Me formalization of Vinogradov's theorem that every sufficiently large odd integer is a sum of three primes.

Details and sources

AI contribution

The announcement attributes this separate three-day formalization exercise to Claude agents using Prove2Me. It does not identify the exact model version used for this experiment.

Problem origin

The source identifies an existing human mathematical theorem. AI generated the formal proof, not the original problem.

Verification

Primary-source announcement only; separate proof artifact not located

Claim audit

Formalization announced, but a separate public proof artifact and independent check were not located.

Publication

Public formalization project

Preprint / manuscript

No public preprint or manuscript located.

Kept provisional: no separately inspectable theorem file, dependency audit, or kernel replay for this exercise was located. FLT's repository is not evidence for this theorem. Vinogradov's sufficiently-large result is known mathematics and is not the full all-odd-integers weak Goldbach theorem. The announcement endpoint was intermittently unavailable during source rechecking.

Primary announcement describing the separate experiment

Formalization milestone; previously known mathematics

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude through three Max subscriptions; exact model unspecified
Verification: Primary-source announcement only; separate proof artifact not located
Claim audit: Issue documented

4 Sep 2026Number theory · formal verification

Fermat's Last Theorem: complete Lean formalization

For every natural exponent $n\geq3$ and positive natural numbers $a,b,c$ , $a^n+b^n\ne c^n$ . The public Lean development derives Mathlib's FermatLastTheorem statement with only propext, Classical.choice, and Quot.sound.

Details and sources

AI contribution

Claude agents worked through Prove2Me with occasional high-level guidance from Tianyi Peng, adapting the Darmon–Diamond–Taylor proof route and building on Mathlib, Imperial's FLT project, and flt-regular. The release does not identify the model as the public Fable 5.1 model.

Problem origin

Fermat's human-origin theorem was proved by Wiles and Taylor–Wiles in 1995. The previously outstanding task here is complete formal verification, listed as challenge 33 in Wiedijk's 100-theorem benchmark, not a newly open mathematical conjecture.

Verification

Public Lean proof; Buzzard reports independent build and comparator check

Pinned Lean proof and verification instructions Final theorem and axiom guard

Publication

Public code, proof walkthrough, and technical report; announced 4 September 2026

Preprint / manuscript

Technical report and selected model traces

Kevin Buzzard reports compiling the code and running comparator successfully. The repository also reports a second-kernel check using nanoda with four disclosed patches; that is a project-reported check, not an independently reproduced audit here. This audit inspected the artifacts but did not rebuild the large proof. Intermediate named theorems are formalized only in the strengths needed by the proof; this is not a new proof discovery of FLT.

Proof scope and provenance manifest Anthropic release and model attribution Kevin Buzzard's independent check and discussion Original formalization benchmark, challenge 33

Formalization milestone; mathematical theorem proved in 1995

Claimed outcome: Proved
Problem origin: Human-source problem
System: Anthropic internal research model, roughly comparable to Claude Fable 5.1
Verification: Public Lean proof; Buzzard reports independent build and comparator check

1 Sep 2026Number theory

Largest prime factor of $n^2+1$ : improved uniform lower bound

The largest prime factor of $n^2+1$ is at least a constant times $(\log_2 n)^2/\log_4 n$ , where $\log_j$ denotes the iterated logarithm, improving the previous denominator $\log_3 n$ .

Details and sources

AI contribution

GPT autonomously obtained Corollary 1.2 and an initial proof. Hector Pasten developed the stronger theorem through model interactions and wrote the proof; Claude proofread.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

A quantitative advance in a longstanding prime-factor problem, not a proof that infinitely many numbers $n^2+1$ are prime and not a resolution of the abc conjecture.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro; Claude Opus 5
Verification: Author-prepared proof; independent review pending

24 Aug 2026Number theory

Logarithmic Chowla correlations across all shift scales

The revised preprint claims a fixed power-logarithmic saving for maximal logarithmically weighted two-point Liouville correlations outside a single shift set sparse in every initial segment, extending to all positive shifts.

Details and sources

AI contribution

The author credits assistance with mathematical exploration, proof development, checking, and literature organization.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author preprint using additional preprint inputs; specialist review pending

Publication

Public research preprint

Preprint / manuscript

The September 1 version is substantially broader than v1. Its separate all-shifts conditional estimate assumes GRH. Neither statement proves ordinary Cesàro two-point Chowla; the appendix identifies which steps rely on other preprints.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Codex GPT-5.6 Sol
Verification: Author preprint using additional preprint inputs; specialist review pending

17 Aug 2026Divisibility and binomial coefficients

Erdős--Straus binomial-divisibility density question

Problem statement

For fixed

n>1

, do almost all large

m

admit

1\leq k\leq m-n

with

\binom{n+k}{n}\mid\binom{m+k}{k}

For every fixed $n\geq2$ , the natural density of integers $m$ for which some $1\leq k\leq m-n$ satisfies $\binom{n+k}{n}\mid\binom{m+k}{k}$ exists and equals $1$ .

Details and sources

AI contribution

FAR recovered the open-ended question from the literature. A GPT-5.5 xhigh opencode run supplied the construction using Kummer's theorem and congruence conditions; automated judging preceded expert review.

Problem origin

Erdős and Straus posed the density question in their 1977 paper on products of consecutive integers.

Verification

Checked by a paper author or domain expert; public complete proof; independent peer review pending

Publication

Public proof in the reviewed-solutions appendix

Preprint / manuscript

FAR pipeline and released data

Activity evidence

A documented editorial estimate based on answering a long-standing open-ended density question in elementary number theory.

This answers the almost-all density alternative for the precisely bounded range $1\leq k\leq m-n$ . It should not be conflated with other consecutive-product divisibility questions carrying nearby Erdős catalogue labels.

Almost-all alternative proved for every fixed

n\geq2

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 xhigh in the FAR pipeline (OpenAI)
Verification: Checked by a paper author or domain expert; public complete proof; independent peer review pending
Open for: Question of Erdős and Straus (1977)
Research activity*: 3/5

15 Aug 2026Finite fields and primitive polynomials

Primitive quartic polynomial conjecture for sufficiently large fields

Problem statement

For

q\ne13

, does every

\alpha\in\mathbb F_{q^2}\setminus\mathbb F_q

admit a translate

x^2+x+\lambda-\alpha

that is primitive over

\mathbb F_{q^2}

For every sufficiently large odd prime power $q$ and every $\alpha\in\mathbb F_{q^2}\setminus\mathbb F_q$ , some $\lambda\in\mathbb F_q$ makes $x^2+x+\lambda-\alpha$ primitive over $\mathbb F_{q^2}$ .

Details and sources

AI contribution

The authors report that Aristotle assisted with the proof, especially the key structural lemma. A model-generated character-sum argument was not used in the final paper; the authors replaced it with a published Fu--Wan theorem and wrote the manuscript themselves.

Problem origin

The primitive-quartic statement was posed in the human finite-field literature before this AI-assisted asymptotic proof.

Verification

Author-presented proof; independent review pending

Publication

Public asymptotic proof preprint

Preprint / manuscript

AI-assisted asymptotic manuscript

Activity evidence

A documented editorial estimate reflecting a rigorous asymptotic advance whose full human-only resolution appeared contemporaneously.

This proves only the sufficiently-large- $q$ form and gives no explicit threshold. A separate contemporaneous paper by Zhou and Wu proves the full conjecture for all admissible $q$ without an AI contribution, so this record is classified as an AI-assisted variant rather than the full resolution.

Separate human-only full proof

AI-assisted asymptotic proof; full conjecture separately proved without AI

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle (Harmonic)
Verification: Author-presented proof; independent review pending
Open for: Primitive quartic polynomial conjecture
Research activity*: 2/5

13 Aug 2026Riemann zeta zeros and critical-line density

Two-thirds critical-line simplicity bound for Riemann zeta zeros

Problem statement

What unconditional proportion of the nontrivial zeros of

\zeta(s)

can be proved to be simple and on the critical line, and what proportion can be proved distinct?

The original manuscript proves unconditionally that at least $\tfrac{2}{3}$ of the nontrivial zeros of $\zeta(s)$ are simple and lie on the critical line and at least $\tfrac{5}{6}$ are distinct. Lamzouri's independent proof sharpens these to more than $67.25\%$ and $83.62\%$ , and adds two further density estimates.

Details and sources

AI contribution

The paper states that Claude discovered and wrote the mathematical argument after Jarred Sumner posed and guided the problem. Levent Alpöge and Ralph Furman checked the proof and accept responsibility; Eric Easley orchestrated the Lean formalization.

Problem origin

The proportions of zeta zeros known to be simple, distinct, and on the critical line are longstanding human research questions with earlier unconditional records in the analytic-number-theory literature.

Verification

Independent conventional proof plus two public Lean certificate paths

Pinned Lean 4 development (v1.0) Headline formal theorem Formalization audit Pinned AxiomProver certificate for the independent proof

Publication

Public arXiv manuscript with a pinned, auditable Lean 4 artifact

Preprint / manuscript

Research manuscript Independent conventional proof, revised v2

Activity evidence

A documented editorial estimate based on the strength of the new unconditional constants, the longstanding zeta-zero program, the author audit, and the complete public Lean artifact.

This is a major unconditional density advance, not a proof of the Riemann hypothesis or of simplicity for every nontrivial zero. Lamzouri's independent argument is conceptually different and adds bounds of $88.76\%$ for zeros that are simple or on the line and $83.62\%$ for the average of the two proportions. Its AxiomProver certificate treats the analytic inputs as explicit assumptions.

New unconditional density records; the Riemann hypothesis remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude (Anthropic)
Verification: Independent conventional proof plus two public Lean certificate paths
Open for: Longstanding density questions in analytic number theory
Research activity*: 5/5

11 Aug 2026Inverse Galois theory and Belyi maps

Mathieu group $M_{23}$ as a Galois group over $\mathbb Q$

Problem statement

Does there exist a Galois extension of

\mathbb Q

with Galois group equal to the Mathieu sporadic simple group

M_{23}

An explicit degree- $23$ polynomial has splitting field with Galois group $M_{23}$ over $\mathbb Q$ , and the authors construct a regular $M_{23}$ -extension of $\mathbb Q(t)$ , completing the sporadic-group program.

Details and sources

AI contribution

The authors used agents to generate and test code, analyze data, operate mathematical packages, organize computations, and recover from failures. Their contribution statement says humans chose the mathematical strategy, coordinates, and redirections; autonomous attempts did not make meaningful progress.

Problem origin

Realizing every sporadic simple group over the rationals is a classical human research program; $M_{23}$ was the final unresolved sporadic case.

Verification

Magma and PARI/GP certified; expert preprint review pending

Exact Magma and PARI/GP computations

Publication

Public proof preprint, code repository, and contribution account

Preprint / manuscript

Author account of the human–AI collaboration FrontierMath attribution discussion

Activity evidence

A documented editorial estimate based on four decades of attempts and completion of the final sporadic-group case in the inverse Galois program.

Epoch classifies this as solved by humans rather than solved by AI because the core-idea boundary is not unambiguously model-led. It is included here as an AI-assisted human result, with that distinction preserved.

Last sporadic-group case resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Multiple AI agents (models not disclosed)
Verification: Magma and PARI/GP certified; expert preprint review pending
Open for: 42 years
Research activity*: 5/5

7 Aug 2026Divisor covers and arithmetic progressions

Umans–Wang arithmetic-progression divisor-cover conjecture

Problem statement

Do the Umans–Wang arithmetic-progression divisor covers exist at the conjectured parameters, including

(\alpha,\beta)=(1/3,1/3)

The arithmetic-progression version fails throughout an explicit parameter range that includes the proposed $(1/3,1/3)$ setting, even after the exponent relaxation considered by Umans and Wang.

Details and sources

AI contribution

The model discovered the counterexample argument and produced an initial write-up. The author checked the construction, clarified its scope, and prepared the paper.

Problem origin

Umans and Wang proposed the arithmetic-progression divisor-cover conjecture as a route toward their higher-rank Strong Divisor Conjecture.

Verification

Author-checked counterexample preprint; external review pending

Publication

Public counterexample preprint

Preprint / manuscript

AxiomProver Lean certificate

Activity evidence

A documented editorial estimate based on the conjecture's role as a proposed route toward arithmetic-circuit lower bounds.

This refutes the one-dimensional arithmetic-progression conjecture, not the higher-rank Strong Divisor Conjecture. The broader program remains open.

One-dimensional conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra via Codex
Verification: Author-checked counterexample preprint; external review pending
Research activity*: 3/5

6 Aug 2026Rogers--Ramanujan identities and point counts

Modularity of point counts for $X^a=Y^b$ : the $a=3$ layer

Problem statement

For coprime

1<a<b

, is the point-counting series for commuting nilpotent pairs satisfying

X^a=Y^b

the conjectured explicit theta-product

P_{a,b}(q)

For every admissible $b$ , the $a=3$ point-counting series is identified with an explicit theta-product modular function, proving an infinite family of Rogers--Ramanujan identities and giving a geometric origin for Warnaar's products.

Details and sources

AI contribution

AxiomProver generated the public Lean certificate for the new identities. The mathematical proof and manuscript are by Kenny Lau and Ken Ono.

Problem origin

Huang, Jiang, and Oblomkov conjectured the product formula for the point-counting series; the statement is layered by the human-chosen parameter $a$ .

Verification

Lean checked for the new identities, relative to cited literature results

Publication

Public proof preprint and Lean certificate repository

Preprint / manuscript

Current combinatorial Lean development AI manuscripts and early Lean artifacts

Activity evidence

A documented editorial estimate based on the conjecture's links among finite-field point counts, modular forms, representation theory, and Rogers--Ramanujan identities.

This is a genuine partial resolution: the classical $a=2$ layer was known, the paper settles $a=3$ , and the general $a\geq4$ conjecture remains open. The Lean certificate assumes four named results from the existing literature.

Full

a=3

layer proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: AxiomProver
Verification: Lean checked for the new identities, relative to cited literature results
Open for: Conjecture of Huang, Jiang, and Oblomkov
Research activity*: 4/5

4 Aug 2026Arithmetic statistics and algebraic groups

Counting algebraic tori over $\mathbb Q$ by Artin conductor

Problem statement

How quickly can the number

N_n^{\mathrm{tor}}(X)

n

-dimensional algebraic tori over

\mathbb Q

grow when ordered by Artin conductor?

For every $n\geq2$ , the number of $n$ -dimensional algebraic tori over $\mathbb Q$ of Artin conductor at most $X$ is at most $X^{\exp(C(\log n)^2)}$ up to an $n$ -dependent constant.

Details and sources

AI contribution

ChatGPT 5.6 Pro generated initial proofs for central representation-theoretic and counting arguments. Through iterative prompting and critical revision, the author strengthened the bound, independently checked every argument, and wrote the final manuscript.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Independently checked by the author; external review pending

Publication

Public proof preprint

Preprint / manuscript

Upper-bound preprint

Activity evidence

A documented editorial estimate based on the connection to counting number fields, Malle-type conjectures, and the lack of prior arbitrary-dimensional bounds avoiding case-by-case classification.

This is a substantial unconditional upper bound in every dimension, not a proof of the conjectured asymptotic $N_n^{\mathrm{tor}}(X)\sim c_nX(\log X)^{n-1}$ . It is therefore classified as partial progress.

First arbitrary-dimensional upper bound; asymptotic conjecture open

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Pro
Verification: Independently checked by the author; external review pending
Open for: Higher-dimensional counting problem and Malle-type asymptotic conjecture
Research activity*: 4/5

28 Jul 2026p-adic arithmetic geometry

Inclusions among two-dimensional $p$ -bounded crystalline loci

Problem statement

How can the simple inclusions

{\mathcal Z}(\underline r)\subseteq{\mathcal Z}(\underline r')

be classified as

\underline r

ranges over

p

-bounded Hodge types, and can those inclusions be detected on closed points?

Apart from two degenerate cases, simple inclusions between reduced special fibers of two-dimensional $p$ -bounded crystalline loci are classified by three operations on Hodge types. With one exception, inclusion is detected on closed points, equivalently on semisimple mod- $p$ Galois representations.

Details and sources

AI contribution

GPT-5.5 Pro found the first proof after an extended continuation process. The authors restarted and reorganized the argument, while Codex and GPT-5.5 formalized an early version in Lean and exposed errors that were corrected; the final exposition was written by the authors.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Combinatorial core Lean checked; stack-theoretic transfer author checked

Publication

Public author-written preprint, historical AI artifacts, and two Lean repositories

Preprint / manuscript

Classification preprint

Activity evidence

A documented editorial estimate based on active work linking Emerton--Gee stacks, Serre weights, and mod- $p$ Galois representations.

The current Lean development covers the combinatorial statements about semisimple point sets, not the Emerton--Gee stack arguments. The record therefore distinguishes the machine-checked core from the author-checked geometric transfer.

Classification theorem proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.5 Pro + Codex 5.5
Verification: Combinatorial core Lean checked; stack-theoretic transfer author checked
Open for: Classification question arising in ongoing work
Research activity*: 3/5

1 Jun 2026Multiplicative number theory

Divisibility set for a generalized Euler totient

Problem statement

Define

\varphi_k(n)=\sum_{1\leq a\leq n,(a,n)=1}a^k

and

\mathcal D_s=\{k\geq s:\varphi_s(n)\mid\varphi_k(n)\text{ for every }n\}

. Is

\mathcal D_1=\{1,3,15\}

Campbell proves the exact classification $\mathcal D_1=\{1,3,15\}$ conjectured by Büyükaşik and collaborators in 2024.

Details and sources

AI contribution

The author says the proof is based on extensive interactions with GPT-5.5 Pro.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked preprint

Publication

Complete arXiv proof submitted for publication

Preprint / manuscript

Activity evidence

A concrete 2024 conjecture supported by computation, but with a short and specialist literature trail.

This is a young but precise published conjecture, included separately from long-running legacy problems.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author-checked preprint
Open for: 2 years
Research activity*: 2/5

21 May 2026Extremal number theory

Erdős Problem #12 — divisor-avoiding sets

Problem statement

Let

A\subset\mathbb N

be infinite with no distinct

a,b,c\in A

such that

a\mid(b+c)

and

b,c>a

. Can

|A\cap[1,N]|/\sqrt N

have positive lower limit? Must every such

A

fall below

N^{1-c}

infinitely often for some absolute

c>0

AlphaProof Nexus constructs a set of size at least $N/(\log N)^{O(\!\log\log\log N\!)}$ up to $N$ while avoiding the forbidden divisibility pattern. This answers part (i) positively and disproves part (ii); the reciprocal-sum part remains open.

Details and sources

AI contribution

The system autonomously developed a block-and-CRT construction using progression-free sets and produced its Lean proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Lean artifacts

Publication

AP Nexus preprint, natural-language proof, and Lean source

Preprint / manuscript

AP Nexus paper

Activity evidence

A decades-old Erdős problem with several substantial specialist partial results by Schoen, Baier, and Elsholtz–Planitzer.

The parent record has three clauses. The site therefore keeps this as partial even though the first two clauses receive definitive answers.

Two of three parts resolved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: AlphaProof Nexus
Verification: Lean checked
Open for: 56 years
Research activity*: 4/5

4 Feb 2026Bernoulli numbers and class groups

Almost all primes are partially regular

Problem statement

For how large an initial range of even indices

2k

can one prove that almost every odd prime

p

does not divide the numerator of

B_{2k}

, and hence that the corresponding even class-group eigenspaces vanish?

For every $\alpha>1/2$ , a density-one set of odd primes $p$ avoids divisibility by $p$ in all relevant Bernoulli numerators up to $\sqrt p/(\log p)^\alpha$ , yielding corresponding initial class-group vanishing by reflection.

Details and sources

AI contribution

AxiomProver autonomously converted the natural-language argument into a complete Lean development; humans wrote and checked the exposition.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Published and Lean checked

Lean source

Publication

Archiv der Mathematik (2026), with public Lean package

Preprint / manuscript

Activity evidence

The exact density-one bound is new, while irregular primes, Bernoulli numerators, and Vandiver’s conjecture form a long-running international program.

This is a new partial regularity theorem. It does not solve the Kummer–Vandiver conjecture.

Published article

Density-one range proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: AxiomProver
Verification: Published and Lean checked
Open for: New theorem inside a 100+ year program
Research activity*: 4/5

31 Mar 2026Modular forms

Ramanujan’s tau function misses almost all primes

Problem statement

Let

S(X)=\#\{\ell\leq X:\ell\text{ prime and }|\tau(n)|=\ell\text{ for some }n\}

. Does abc imply

S(X)=o(\pi(X))

Assuming the abc conjecture, the primes occurring as absolute values $|\tau(n)|$ have density zero among all primes.

Details and sources

AI contribution

AxiomProver autonomously proved and formalized the main counting engine and conditional theorem using the stated abc hypothesis and a cited prior proposition.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Published and Lean checked

Lean source

Publication

Indagationes Mathematicae article and public Lean source

Preprint / manuscript

Activity evidence

Values omitted by Ramanujan’s tau function and related Diophantine questions form a substantial, long-running modular-forms program.

The theorem is conditional on abc. It neither proves Lehmer’s nonvanishing conjecture nor settles whether $\tau$ takes infinitely many prime values.

Published article

Conditional density theorem

Claimed outcome: Proved
Problem origin: Human-source problem
System: AxiomProver
Verification: Published and Lean checked
Open for: New conditional theorem in a classical program
Research activity*: 4/5

18 Jun 2026Prime factors in intervals

Erdős Problem #451 — blocks avoiding middle-sized prime factors

Problem statement

Let

n_k

be the least integer greater than

2k

for which

\prod_{i=1}^{k}(n_k-i)

has no prime factor in

(k,2k)

. How rapidly must

n_k

grow?

The conjectured superpolynomial growth is established: for all sufficiently large $k$ , $n_k>\exp(\!\log^2k/(20\log\log k)\!)$ . Determining the sharper order of $n_k$ remains open.

Details and sources

AI contribution

GPT-5.5 Pro and Quanyu Tang generated the argument; Tang and Wouter van Doorn then wrote and checked the human paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human-checked arXiv proof

Publication

Public arXiv preprint and official problem-page update

Preprint / manuscript

Official problem record Research discussion

Activity evidence

A 1979 specialist conjecture with classical bounds, a focused modern discussion, and a dedicated 2026 paper.

The growth conjecture is proved, but the original problem also asks for an estimate of $n_k$ , so the entry remains partial.

Superpolynomial growth proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Human-checked arXiv proof
Open for: 47 years
Research activity*: 3/5

10 Jun 2026Multiplicative combinatorics

Erdős Problem #539 — growth of cofactor sets

Problem statement

For

|A|=n

, how small can

Q(A)=\{a/\gcd(a,b):a,b\in A\}

be? Equivalently, estimate

h(n)=\min_{|A|=n}|Q(A)|

ProofCouncil proves $h(n)\leq n^{1/2}\exp(O(\sqrt{\log n}))$ . With the classical lower bound this determines $h(n)=n^{1/2+o(1)}$ , while sharper subpolynomial factors remain open.

Details and sources

AI contribution

A GPT-5.5-Pro-driven author–critic council developed and stress-tested the proof through multiple specialized agents.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Official update + Lean record

Publication

Public agent paper, code, and official problem-page update

Preprint / manuscript

ProofCouncil paper

Activity evidence

A sustained specialist line involving Erdős–Szemerédi, Freiman–Lev, Granville–Roesler, and a new multi-author agent paper.

The exponent is settled, which is the principal asymptotic milestone, but the exact subpolynomial behavior is not.

Official problem record Public agent code

Main exponent determined

Claimed outcome: Proved
Problem origin: Human-source problem
System: ProofCouncil / GPT-5.5 Pro
Verification: Official update + Lean record
Open for: 53 years
Research activity*: 3/5

19 Jun 2026Unit fractions

Erdős Problem #306 — reciprocal semiprimes

Problem statement

a/b\in\mathbb Q_{>0}

and

b

is squarefree, can

a/b

always be written as a finite sum of reciprocals

1/n_i

where the

n_i

are distinct products of two distinct primes?

A public Lean development gives an affirmative proof for every positive rational with squarefree denominator, subject to two explicitly isolated Rosser–Schoenfeld analytic inputs. The official problem page still lists the problem as open.

Details and sources

AI contribution

The repository discloses AI assistance but does not name the model; a separate earlier Claude-assisted result addressed only a partial case.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked modulo two named inputs

Lean repository Archived artifact

Publication

Public repository and archived formal artifact

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

An established unit-fraction problem with historical constructions, a substantive forum thread, and a reproducible formal artifact.

This is deliberately not labeled a confirmed full resolution until the analytic inputs are fully discharged and the official record or a human manuscript accepts the proof.

Paper and Lean repository

Candidate full proof

Problem origin: Human-source problem
System: AI-assisted Lean development
Verification: Lean checked modulo two named inputs
Open for: 46 years
Research activity*: 3/5

14 Jun 2026Distribution of powerful numbers

Erdős Problem #942 — powerful numbers between squares

Problem statement

Let

h(n)

count powerful integers

m\in[n^2,(n+1)^2)

, where

p\mid m

implies

p^2\mid m

. What is the extremal order of

h(n)

Infinitely often, the number of powerful integers in $[n^2,(n+1)^2)$ is at least a constant times $\log n/(\log\log n\,\log\log\log n)$ , improving the earlier exponent- $1/3$ result.

Details and sources

AI contribution

The author reports assistance from several models in developing and formalizing the fixed-parameter construction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Official problem update with linked paper and Lean source

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A classical 1976 question with published 2004 progress and a substantial modern proof-and-formalization thread.

The extremal order of the counting function remains open, so this is a strong lower-bound advance rather than a resolution.

Official problem record Research discussion and artifacts

Lower bound improved

Problem origin: Human-source problem
System: Claude / Codex / Aristotle
Verification: Lean checked
Open for: 50 years
Research activity*: 3/5

25 Feb 2026Additive number theory

Odd composites beyond powers of two and primes

Problem statement

Can the odd integers not representable as

2^k+p

, with

p

prime, be written as an infinite arithmetic progression together with a density-zero exceptional set?

A Lean development formalizes the negative answer proved by Chen in 2023.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Antigravity / Gemini 3.1 Pro
Verification: Lean checked

24 Nov 2025Additive bases

Density-zero additive complements

Problem statement

For every infinite

A\subseteq\mathbb N

, is there a density-zero set

B

such that

A+B

contains all sufficiently large integers?

ChatGPT expanded Lorentz's classical argument and Aristotle produced a checked Lean formalization.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: ChatGPT / Aristotle
Verification: Lean checked

27 May 2026Extremal number theory

Pairwise-coprime extremal sets

Problem statement

Let

N\geq p_k

, where

p_k

is the

k

th prime. If

A\subseteq\{1,\ldots,N\}

contains no

k+1

pairwise-coprime elements, is

|A|

at most the number of integers in

[N]

divisible by one of the first

k

primes?

The known negative answer and its counterexample now have an AI-assisted Lean verification.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / GPT
Verification: Lean checked

4 May 2026Multiplicative number theory

Erdős primitive-set inequality

Problem statement

Among primitive sets of integers greater than one, is the weighted reciprocal sum from Erdős's conjecture maximized by the primes?

The modern affirmative proof now has an AI-assisted Lean formalization.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Primitive-set resolution preprint

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

Problem, history, and proof record Discussion thread Community AI ledger

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Codex
Verification: Lean checked

24 Nov 2025Sidon sets

Sidon sets meeting every arithmetic progression

Problem statement

Must the complement of every infinite Sidon set contain an infinite arithmetic progression?

AlphaProof found the explicit Sidon set $\{(n+1)!+n:n\geq0\}$ meeting every infinite arithmetic progression; the construction was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Formal proof source Earlier statement file (contains sorry)

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

Problem, history, and proof record Discussion thread Community AI ledger

AI-assisted Lean formalization

Problem origin: Human-source problem
System: AlphaProof / Aristotle / GPT
Verification: Lean checked

21 Dec 2025Binomial coefficients

Binomial-coefficient divisibility depth

Problem statement

Let

S(n)

be the largest exponent such that every nontrivial

\binom nk

is divisible by some prime to that exponent. Is

\limsup S(n)=\infty

Seed Prover 1.5 found a new proof of the affirmative result, and a Lean proof artifact is recorded.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Seed Prover 1.5
Verification: Lean checked

27 Dec 2025Elementary number theory

Product-minus-sum representations

Problem statement

Does some fixed

k

let every sufficiently large integer be written as

\prod_{i=1}^k a_i-\sum_{i=1}^k a_i

with all

a_i\geq2

AI systems found and formally checked the short affirmative construction with $k=2$ .

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle / ChatGPT
Verification: Lean checked

15 Mar 2026Covering systems

Divisor covering systems with coprime overlaps

Problem statement

Does there exist

n

and, for every divisor

d\mid n

with

d>1

, a residue class

a_d\pmod d

such that the classes cover every integer and any two intersecting classes have coprime moduli?

Adenwalla's negative solution was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

28 Apr 2026Unit fractions

Greedy Egyptian underapproximations

Problem statement

For almost every real

x

, are its best

n

-term unit-fraction underapproximations eventually produced by the greedy algorithm?

Kovač's theorem that the eventually-greedy set has measure zero was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

31 Jan 2026Additive bases

Sparse additive complements to powers of two

Problem statement

Is there a set

A

with

|A\cap[1,N]|=O(N/\log N)

such that every sufficiently large integer is

2^k+a

for some

a\in A

Ruzsa's affirmative construction, using van Doorn's exposition, was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

4 Apr 2026Additive prime number theory

Unbounded prime-plus-set representations

Problem statement

Let

A\subseteq\mathbb N

satisfy

|A\cap\{1,\ldots,N\}|\gg\log N

for all sufficiently large

N

, and let

f(n)

count representations

n=p+a

. Must

\limsup f(n)=\infty

The stronger Chen–Ding affirmative theorem was formalized in Lean conditional on a named Maynard–Tao input.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked conditional on Maynard–Tao input

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

The Lean file is explicit about its custom Maynard–Tao axiom. It verifies the deduction conditional on that analytic input, not the input itself.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked conditional on Maynard–Tao input

28 Dec 2025Complete sequences

Completeness of mixed-power sequences

Problem statement

For coprime integers

a,b>1

, is every sufficiently large integer a sum of distinct numbers

a^k b^\ell

Birch's affirmative theorem was reconstructed and verified in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

21 Apr 2026Irrationality

Irrational squarefree Möbius series

Problem statement

Is the series

\sum_{n\geq1}\mu(n)^2n/2^n

irrational?

The stronger Chen–Ruzsa infinite-subseries theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

26 May 2026Reciprocal sums

Interior of reciprocal subset-sum triples

Problem statement

Do the three shifted reciprocal sums formed from convergent infinite subsets have a set of values with nonempty interior in

\mathbb R^3

Kovač's affirmative theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

20 Jan 2026Covering systems

Covering consecutive integers by congruences

Problem statement

r

congruence classes cover

2^r

consecutive integers, must those classes cover all integers?

The affirmative theorem of Balister, Bollobás, Morris, Sahasrabudhe, and Tiba was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

18 Apr 2026Covering systems

Sparse covering-congruence obstruction

Problem statement

Let

n_1<n_2<\cdots

and choose classes

a_k\pmod{n_k}

, with

n_k>(1+\epsilon)k\log k

for some

\epsilon>0

and every

k

. Must the number of

m<n_k

uncovered by the first

k

classes fail to be

o(k)

Cambie's powers-of-two construction disproving the proposed obstruction was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

14 Jan 2026Unit fractions

Denominator drops in harmonic intervals

Problem statement

For every starting point

a

, can extending a consecutive reciprocal sum by one term reduce its denominator in lowest terms?

Van Doorn's affirmative construction with a linear bound was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

26 May 2026Unit fractions

Disjoint unit-fraction decompositions

Problem statement

How many pairwise-disjoint subsets of

[N]

can each have reciprocal sum one?

The asymptotic value $(1-o(1))\log N$ was formalized from the Hunter–Sawhney observation and Bloom's theorem.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

21 Dec 2025Ramsey theory for unit fractions

Monochromatic unit-fraction equations

Problem statement

Does every finite coloring of the positive integers contain distinct monochromatic

a,b,c

satisfying

1/a=1/b+1/c

The Brown–Rödl affirmative theorem received a public Lean proof.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Earlier statement file (contains sorry) Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Seed Prover 1.5
Verification: Lean checked

1 Apr 2026Unit fractions

Near-unit harmonic intervals

Problem statement

For the first consecutive harmonic block beginning at

n

whose sum reaches one, is the scaled overshoot characterized by

\liminf n^2\epsilon(n)=0

The Lim–Steinerberger affirmative theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

31 Jan 2026Unit fractions

Vardi-constant extremality

Problem statement

Does every non-Sylvester reciprocal decomposition of one have a smaller doubly-exponential growth constant than the Vardi constant?

Kamio's affirmative extremal theorem was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

10 Dec 2025Additive bases

Sparse additive-basis doubling

Problem statement

For every zero-density additive basis

A

, must

|(A+A)\cap[1,N]|/|A\cap[1,N]|

tend to infinity?

The Ruzsa–Turjányi counterexample was formalized in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked

25 Nov 2025Additive combinatorics

Reciprocal sums of dissociated sets

Problem statement

If every subset sum of a finite set

A\subset\mathbb N

is distinct, must

\sum_{n\in A}1/n<2

ChatGPT supplied a proof explanation of Ryavec's affirmative theorem and Aristotle formalized it in Lean.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: ChatGPT / Aristotle
Verification: Lean checked

29 May 2026Additive number theory

Extremal Frobenius-number asymptotics

Problem statement

For coprime

k

-element sets

A\subseteq\{1,\ldots,n\}

, is the maximum Frobenius number asymptotic to

n^2/(k-1)

Dixmier's affirmative theorem was formalized in an unconditional Lean development using a mixed open-harness and proprietary-model workflow.

Details and sources

AI contribution

AI systems contributed proof search, proof reconstruction, or formal proof engineering; Lean's kernel checked the resulting artifact.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem, history, and proof record Discussion thread Community AI ledger

Publication

Official problem record, discussion, and public formal-proof source

Preprint / manuscript

No public preprint or manuscript located.

This record distinguishes an AI-produced proof or formalization from mathematical priority: the underlying result may have been known before the AI work.

AI-assisted Lean formalization

Problem origin: Human-source problem
System: Gemini 3.1 Pro / Gemini 3.0 Flash / Claude Sonnet 4.6 / Project Numina / Aristotle / Claude Opus 4.8 / ulam.ai CLI harness
Verification: Lean checked

4 May 2026Multiplicative number theory

Erdős Problem #456 — totient preimages and least primes

Problem statement

Let

p_n

be the least prime congruent to

1\pmod n

and

m_n

the least integer with

n\mid\varphi(m_n)

. Is

m_n<p_n

almost always, does

p_n/m_n\to\infty

almost always, and when is

m_n=p

unique?

A 71-page discussion manuscript claims unconditional negative answers to the first two questions and a Dickson-conditional answer to the third.

Details and sources

AI contribution

The author reports using an automated multi-turn GPT-5.5 Pro scaffold to develop and repeatedly audit the argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Community manuscript; expert review requested

Claim audit

The official problem page still marks the problem open, and the author explicitly requests expert review or formalization.

Publication

Public discussion thread and linked manuscript

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A multi-part 1979 Erdős problem with classical input from Linnik-type prime bounds and an active recent discussion.

The first two conclusions are claimed unconditional; the third depends on Dickson's conjecture. This record documents the claim without promoting it to a confirmed resolution.

Official problem record Claim, manuscript, and discussion

Mixed solution claimed

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 Pro audit-and-revise scaffold
Verification: Community manuscript; expert review requested
Claim audit: Issue documented
Open for: 47 years
Research activity*: 3/5

27 Apr 2026Sieve theory and gaps in sifted sets

Erdős Problem #1101 — subexponential good sequences

Problem statement

Does there exist a good pairwise-coprime sequence

u_n

with

\sum 1/u_n<\infty

and polynomial growth? What if one only requires

u_n\leq e^{o(n)}

A community note reports a subexponential good-sequence construction with help from GPT-5.5 Pro. The full polynomial-growth question remains open.

Details and sources

AI contribution

The contributor credits GPT-5.5 Pro with assisting the partial-result argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Community partial result

Publication

Official problem record, discussion, and linked note

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A specialist 1981 Erdős problem linking sieve gaps and structured sequences, with a small but substantive recent discussion.

The official page explicitly classifies the discussion as partial and continues to mark both main questions open.

Official problem and partial-result status Discussion and linked note

Partial construction claimed

Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Community partial result
Open for: 45 years
Research activity*: 2/5

7 May 2026Covering systems

Erdős Problem #7 — failed odd-covering-system formalization

Problem statement

Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than

1

A claimed proof that no distinct covering system can have only odd moduli does not establish the conjecture. Its central Lean axiom states that a product of factors greater than one is less than one.

Details and sources

AI contribution

The author used Lean and an Aristotle audit to check the monotonicity layer, while leaving three purported consequences of the BBMST sieve as axioms. GPT-assisted community review helped identify the fatal mismatch.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Failed axiom and statement-fidelity audit

Submitted manuscript and Lean code Archived release

Claim audit

The encoded sieve product omitted the BBMST initial LP factor $c_0\approx0.098$ . Under the submitted definition every update factor is greater than $1$ , so the axiom `bbmst_sf_lt_one` is impossible. The author acknowledged the mistranslation.

Publication

Public manuscript, Lean repository, archived release, and corrective discussion

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A longstanding prize problem with major sieve-theoretic progress and an unusually detailed public audit of an AI-assisted formal claim.

The underlying Erdős–Selfridge odd covering-system problem remains open. A sorry-free derivation from a false imported axiom is not an end-to-end formal proof.

Official problem record Discussion, objection, and correction

Claim withdrawn after a false axiom was found

Problem origin: Human-source problem
System: Aristotle audit + GPT-assisted discussion
Verification: Failed axiom and statement-fidelity audit
Claim audit: Issue documented
Open for: Still open after the failed claim
Research activity*: 5/5

24 Jun 2026Analytic number theory

Erdős Problem #1061 — divisor-sum solution growth

Problem statement

For

S(x)=\#\{(a,b)\in\mathbb N^2:a+b\leq x,\ \sigma(a)+\sigma(b)=\sigma(a+b)\}

, is

S(x)\sim cx

for some

c>0

A preprint claims that the number $S(x)$ of ordered pairs with $a+b\leq x$ and $\sigma(a)+\sigma(b)=\sigma(a+b)$ grows faster than $x(\log x)^R$ for every fixed $R>0$ , ruling out the proposed linear asymptotic.

Details and sources

AI contribution

The author reports extensive ChatGPT use for technical lemmas, calculations, code, proof development, and auditing while retaining the strategic direction and final responsibility.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public self-contained preprint; independent review pending

Publication

Public arXiv proof and official discussion record

Preprint / manuscript

Claimed proof

Activity evidence

The question appears in Guy’s collection and intersects active work on divisor sums, prime patterns, and additive equations.

The manuscript is a substantive claimed resolution, but the official problem record still treats it as open while the proof is reviewed. This entry therefore does not imply community acceptance.

Variant proof and exact scope

Claimed resolution under expert review

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT
Verification: Public self-contained preprint; independent review pending
Open for: Official record still open
Research activity*: 4/5

27 Jun 2026Probabilistic number theory

Erdős Problem #731 under dyadic regularity

Problem statement

A(n)

is the least positive integer not dividing

\binom{2n}{n}

, is there a reasonable function

f(n)

such that

A(n)\sim f(n)

for almost all

n

For the least positive integer $A(n)$ not dividing $\binom{2n}{n}$ , a preprint proves a density-tight logarithmic scale and rules out an asymptotic equivalent for every dyadically regular deterministic normalization.

Details and sources

AI contribution

The author reports extensive ChatGPT use for proof exploration, technical lemmas, calculations, code, and auditing.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public proof; no independent review or formal certificate located

Claim audit

Scope is deliberately narrower than the literal informal question: the no-asymptotic-equivalent theorem assumes dyadic regularity, and sharper limiting-distribution questions remain open.

Publication

Public arXiv proof and official problem discussion

Preprint / manuscript

Activity evidence

The least-nondivisor problem has a long history involving Kummer carries, missing primes, and almost-all asymptotics.

The word “reasonable” in the original problem is informal. This paper gives and resolves an explicit broad block-smooth interpretation; it does not exclude every possible irregular deterministic scale.

Certificates and independent implementations Verification catalog and runnable checks

Explicit regularity variant claimed

Problem origin: Human-source problem
System: ChatGPT
Verification: Public proof; no independent review or formal certificate located
Claim audit: Issue documented
Open for: Variant of an open-ended Erdős problem
Research activity*: 4/5

12 Jun 2026Experimental number theory

Sun Conjecture 4.6(ii) on trigonometric permanents

Problem statement

For an odd prime

p

, do Sun’s normalized trigonometric permanents satisfy

s_p<0\iff p\equiv5\pmod{12}

and

s'_p<0\iff p\equiv7\pmod8

Exact calculations at $p=29$ refute both proposed sign laws: $s_{29}>0$ despite $29\equiv5\pmod{12}$ , while $s'_{29}<0$ despite $29\not\equiv7\pmod8$ .

Details and sources

AI contribution

The pipeline generated counterexamples and four algorithmically independent exact layers using cyclotomic arithmetic, finite fields, CRT uniqueness, and subset dynamic programming.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Multiple independent exact implementations; not externally refereed

Publication

Public note, certificates, programs, and frozen source

Preprint / manuscript

Original conjecture Result manuscript

Activity evidence

A precise computational number-theory conjecture whose published table stopped at the prime immediately before the first failure.

Part (i), the divisibility assertion for odd composite $n$ , was not refuted and remains open. No replacement sign law is claimed.

Result page and exact scope

Both sign clauses in part (ii) refuted

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Demonstrandum multi-agent pipeline
Verification: Multiple independent exact implementations; not externally refereed
Open for: Unchanged since the 2021 conjecture
Research activity*: 2/5

24 Jul 2026Factorial ratios and divisibility

OEIS A211420 fixed-divisor conjecture

Problem statement

For

a_n=\dfrac{(8n)!\,n!}{(4n)!(3n)!(2n)!}

, and fixed

k\in\{1,2,3\}

and

r\geq1

, does there exist a constant

C(k,r)

such that

\frac{a_n C(k,r)}{\prod_{i=1}^{r}(kn+i)}\in\mathbb Z\qquad\text{for every }n\geq0?

A readable proof and two Lean developments establish the original existence claim with the explicit uniform choice $C(k,r)=\operatorname{lcm}(1,\ldots,8r-1)^r$ for every $k\in\{1,2,3\}$ .

Details and sources

AI contribution

GPT-5.6 Pro produced the ordinary proof under human direction. Codex built the primary Lean development, while Aristotle supplied a separate formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two Lean developments kernel checked

Proof and verification repository Primary Lean development

Publication

Public proof manuscript, primary Lean file, separate second formalization, and verification scripts

Preprint / manuscript

Activity evidence

A recent sequence-specific conjecture with a public OEIS record and a reproducible formal proof packet, but little documented prior literature.

The checked theorem strengthens the OEIS conjecture by giving an explicit constant. The release does not claim specialist peer review, historical priority, or minimality of the constant.

Original OEIS conjecture

Stronger explicit bound proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Pro / Codex / Aristotle
Verification: Two Lean developments kernel checked
Open for: Conjectured on OEIS in 2025
Research activity*: 2/5

17 Jul 2026Additive number theory

Erdős Problem #123 — primitive smooth sums

Problem statement

For pairwise-coprime integers

a,b,c>1

, is every sufficiently large integer a sum of distinct monomials

a^i b^j c^k

that form a divisibility antichain?

For every pairwise-coprime triple $a,b,c>1$ , every sufficiently large integer is a sum of distinct terms $a^i b^j c^k$ such that no selected term divides another.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + community status accepted

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle and accepted Erdős Problems record

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

The literal website wording allowed $a=b=c=1$ and is false there. The accepted Lean theorem proves the intended nondegenerate $a,b,c>1$ formulation used in the source literature and formal-conjecture encoding.

Problem resolved; Lean checked

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked + community status accepted
Research activity*: 3/5

13 Jul 2026Complete sequences

Erdős Problem #254 — complete sequences from phase divergence

Problem statement

A\subseteq\mathbb N

has unbounded dyadic-shell counts and

\sum_{n\in A}\lVert\theta n\rVert=\infty

for every

0<\theta<1

, must

A

be complete?

The proposed Lean theorem proves that dyadic-shell abundance together with divergence of $\sum_{n\in A}\lVert\theta n\rVert$ for every $0<\theta<1$ forces every sufficiently large integer to be a sum of distinct elements of $A$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 2/5

13 Jul 2026Irrationality

Erdős Problem #267 — lacunary Fibonacci reciprocal sums

Problem statement

n_1<n_2<\cdots

and

n_{k+1}/n_k\geq c>1

, must the reciprocal Fibonacci sum

\sum_k 1/F_{n_k}

be irrational?

The proposed Lean proof establishes irrationality of $\sum_k 1/F_{n_k}$ for every infinite sequence with a uniform ratio gap $n_{k+1}/n_k\geq c>1$ , closing the range $1<c<2$ left by earlier criteria.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 3/5

16 Jul 2026Unit fractions

Erdős Problem #320 — distinct unit-fraction subset sums

Problem statement

Estimate the number

S(N)

of distinct reciprocal subset sums

\sum_{n\in A}1/n

with

A\subseteq\{1,\ldots,N\}

If $S(N)$ counts the distinct values of $\sum_{n\in A}1/n$ over $A\subseteq\{1,\ldots,N\}$ , then $\log S(N)$ has the exact order $\frac{N}{\log N}\prod_{j\geq3}\log_jN$ , with the iterated product stopped at a fixed threshold.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + community status accepted

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle and accepted Erdős Problems record

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

The formal artifact is public and the community problem record now reflects the result.

Problem resolved; Lean checked

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked + community status accepted
Research activity*: 3/5

16 Jul 2026Unit fractions

Erdős Problem #321 — reciprocal-dissociated sets

Problem statement

How large can

A\subseteq\{1,\ldots,N\}

be if all sums

\sum_{n\in S}1/n

, for

S\subseteq A

, are distinct?

The maximum size $R(N)$ of a subset of $\{1,\ldots,N\}$ with all reciprocal subset sums distinct is $\Theta\!\left(\frac{N}{\log N}\prod_{j=3}^{k(N)}\log_jN\right)$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + community status accepted

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle and accepted Erdős Problems record

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

The formal artifact is public and the community problem record now reflects the result.

Problem resolved; Lean checked

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked + community status accepted
Research activity*: 3/5

13 Jul 2026Additive bases

Erdős Problem #336 — exact order of asymptotic bases

Problem statement

h(r)

is the maximal finite exact order attainable by a basis of order

r

, what is

\lim_{r\to\infty}h(r)/r^2

The proposed Lean proof identifies the sharp limit $\lim_{r\to\infty}h(r)/r^2=1/3$ for the maximal exact order of an asymptotic basis of variable order at most $r$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 3/5

13 Jul 2026Multiplicative number theory

Erdős Problem #394 — average least starts of divisible products

Problem statement

For the least

t_k(n)

with

n\mid t_k(n)(t_k(n)+1)\cdots(t_k(n)+k-1)

, do the conjectured logarithmic saving and adjacent-length little-

o

estimates hold on average?

The proposed Lean proof answers both average-order questions affirmatively: one may take $c=1/2048$ in the $t_2$ bound, and the adjacent averages satisfy $\sum_{n\leq x}t_{k+1}(n)=o(\!\sum_{n\leq x}t_k(n))$ for fixed $k\geq2$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 2/5

13 Jul 2026Divisor theory

Erdős Problem #450 — divisors in short translated intervals

Problem statement

How large must

y(\varepsilon,n)

be so that every interval

(x,x+y)

contains at most

\varepsilon y

integers having a divisor in

(n,2n)

The proposed Lean proof gives the sharp fixed- $\varepsilon$ order $y=\Theta_\varepsilon(n)$ uniformly in the translate: a linear upper bound and a matching obstruction to $o(n)$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 2/5

13 Jul 2026Sieve theory

Erdős Problem #489 — squared gaps in a sparse sieve

Problem statement

For a sufficiently sparse forbidden-divisor set

A

, must

x^{-1}\sum_{b_i<x}(b_{i+1}-b_i)^2

converge to a finite limit in the sifted set

B=\{b_1<b_2<\cdots\}

The proposed Lean theorem proves that the normalized sum of squared consecutive gaps in the integers divisible by no member of a set $A$ with $|A\cap[1,x]|=o(\sqrt{x})$ converges to a finite limit.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 2/5

13 Jul 2026Multiplicative combinatorics

Erdős Problem #538 — reciprocal mass under prime-product multiplicity

Problem statement

If each integer has at most

r

representations

m=pa

with

p

prime and

a\in A\subseteq[1,N]

, what is the best upper bound for

\sum_{a\in A}1/a

The proposed Lean proof gives the matching order $\Theta_r(\log N/\log\log N)$ for the maximum reciprocal mass of an admissible set.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 2/5

13 Jul 2026Multiplicative combinatorics

Erdős Problem #796 — second-order product-Sidon extremal term

Problem statement

g_3(n)

is the largest size of

A\subseteq[1,n]

with fewer than three representations of every product

a_1a_2

, does its conjectured second-order normalized term converge?

A Lean-checked candidate proves that the normalized second-order residual for $g_3(n)$ converges to the explicit constant $M+\text{variationalLimit}$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 3/5

13 Jul 2026Covering systems

Erdős Problem #1188 — counting minimal covering systems

Problem statement

Estimate the number

F(x)

of minimal distinct covering systems whose moduli all lie in

[1,x]

The proposed Lean proof gives $\log\log F(x)/\log x\to1$ , equivalently $F(x)=\exp(x^{1+o(1)})$ , for the number of minimal distinct covering systems with moduli at most $x$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a proposed solution with a reproducible Lean artifact. It remains classified conservatively until the independent Erdős Problems record accepts the mathematical status change.

Candidate full solution; tracker review pending

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 3/5

13 Jul 2026Extremal divisibility

Erdős Problem #709 — distinct divisible representatives in intervals

Problem statement

How long must an interval be to contain distinct representatives

x_i

, with

a_i\mid x_i

, for every

n

-element set of moduli

A=\{a_1,\ldots,a_n\}

A Lean-checked construction improves the 1959 upper bound to $f(n)\leq14n^{3/7}$ and gives an explicit logarithmic lower bound. Matching bounds or an asymptotic formula remain open.

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; community status pending

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle; Erdős Problems record still open

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

This is a formally checked exponent improvement on a still-open problem, not a full resolution.

New exponent bound; asymptotic remains open

Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked; community status pending
Research activity*: 3/5

14 Jul 2026Primitive sets

Erdős Problem #793 — second term for strongly $2$ -primitive sets

Problem statement

Determine the second-order term in the maximum size of a set

A\subseteq\{1,\ldots,n\}

for which no element divides the product of two other members.

The exact second-order asymptotic is $F(n)=\pi(n)+(27/2+o(1))n^{2/3}/\log^2n$ for the extremal family with $a\nmid bc$ .

Details and sources

AI contribution

Star Fleet ran parallel GPT-5.6 proof agents, used Claude Fable 5 as an automated referee, and then released the accepted Lean project for external reconstruction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + community status accepted

Official problem record Discussion thread Star Fleet report

Publication

Public verification bundle and accepted Erdős Problems record

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

The score is a documented editorial estimate based on the problem record, cited prior work, and public discussion; it is not a difficulty rating.

Przemek Chojecki first posted the solution; Star Fleet independently reconstructed and formalized it. The official problem record now lists the result as proved in Lean.

Problem resolved; Lean checked

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 starships / Claude Fable 5 reviewer
Verification: Lean checked + community status accepted
Research activity*: 3/5

2 Jul 2026Binomial coefficients and divisors

Erdős Problem #387 — binomial divisors avoiding an interval

Problem statement

Must every binomial coefficient

\binom nk

, with

1\leq k\leq n/2

, have a divisor at most

n

that is larger than a fixed positive multiple of

n

Bui, Naprienko, Pratt, and Zaharescu gave an unconditional negative resolution. A separate public Lean project formalizes the unconditional covering input, while a later GPT-5.5 Pro collaboration found a parallel covering route.

Details and sources

AI contribution

The headline paper is human-authored. The indexed AI contribution is a later parallel proof route and proof engineering around the covering lemma, not priority for the mathematical resolution.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Human preprint + partial Lean formalization

Supporting Lean repository

Publication

Complete arXiv proof, supporting manuscript, and Lean repository

Preprint / manuscript

Activity evidence

A fifty-year-old Erdős–Graham problem resolved in a substantial 62-page number-theory paper; the score is an editorial estimate.

The Lean repository proves the covering components with two named standard analytic inputs, but explicitly says the complete combination with the paper’s later sections is outside that repository. The canonical Erdős record lists the problem solved and the formal status unformalized.

Primary single-file Lean proof Second Lean development

Human resolution; AI-assisted route partly formalized

Problem origin: Human-source problem
System: GPT-5.5 Pro / Lean
Verification: Human preprint + partial Lean formalization
Research activity*: 4/5

21 Jul 2026Divisor theory

Erdős Problem #469 — primitive pseudoperfect reciprocals

Problem statement

Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?

The reciprocal sum of the primitive pseudoperfect numbers converges. The public proof also yields density results for pseudoperfect numbers.

Details and sources

AI contribution

The source file credits GPT-5.6 Sol Ultra, Claude Fable 5, and Lewis on the informal proof, and GPT-5.6 Sol Ultra with Lewis on the formal proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked in two public developments

Publication

Public preprint, single-file Lean proof, second Lean development, and official accepted record

Preprint / manuscript

Official problem record Discussion and proof-claim thread

Activity evidence

A specialist divisor-theory problem with a dedicated preprint, two formalizations, and an accepted community status change; the score is an editorial estimate.

The AI provenance comes from the header of the primary Lean source and the linked preprint materials, while the official database supplies the mathematical status.

Problem proved; two Lean developments

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra / Claude Fable 5 / Zachary J. Lewis
Verification: Lean checked in two public developments
Research activity*: 3/5

3 Sep 2026Binomial coefficients and primes

Erdős Problem #684 — revised logarithmic-limsup claim

Problem statement

For the least

k

at which the small-prime part of

\binom nk

exceeds

n^2

, how large can

f(n)

be?

The 3 September v3 manuscript replaces the withdrawn argument and claims $\limsup [f(n)/\log n][\log\log\log n/\log\log n]\geq1/2$ , implying an unbounded logarithmic limsup. A new public Lean development accompanies it.

Details and sources

AI contribution

The claim circulated in AI-math tracking discussions, but no primary disclosure identifying a model or a raw run was located.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Revised proof and author-reported Lean checks; independent reassessment pending

Pinned v3 Lean development and build report Correction and counterexample audit

Claim audit

The author explicitly withdraws the v1–v2 proof, which used an unjustified weighted extension of Timofeev's method. The earlier Lemma 18 counterexample remains part of the audit history; it does not by itself refute the independent v3 argument.

Publication

Rewritten v3 preprint, pinned Lean source, and earlier correction report

Preprint / manuscript

Rewritten v3 proof and withdrawal notice

Activity evidence

The claim received a complete preprint and a subsequent public technical audit; the score is an editorial estimate of documented activity.

The v3 repository reports a clean build and standard-axiom audit, with its prime-number-theorem input supplied by PrimeNumberTheoremAnd. This audit inspected the source documentation but did not replay Lean. The historical failure is preserved, and independent reassessment and official problem-database acceptance were not established.

Official open problem record Discussion thread

New v3 proof claim; earlier argument withdrawn

Claimed outcome: Proved
Problem origin: Human-source problem
System: Model not publicly disclosed
Verification: Revised proof and author-reported Lean checks; independent reassessment pending
Claim audit: Issue documented
Research activity*: 3/5

25 Jul 2026Multiplicative number theory

Erdős Problem #768 — Sylow divisor condition

Problem statement

A(x)

counts integers satisfying the Sylow divisor condition, determine the constant

c

A(x)/x=\exp(-(c+o(1))\sqrt{\log x}\log\log x)

Eric Li proves the exact constant $c=1/(2\sqrt{\log2})$ in the asymptotic density of integers satisfying the Sylow divisor condition.

Details and sources

AI contribution

The author disclosed AI assistance on the proof, paper, and Lean development. A separate formalizer and a third public audit independently rebuilt the theorem.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three clean Lean checks; community tracker pending

Primary Lean repository Independent rebuild and audit

Publication

Public arXiv paper, primary Lean repository, independent formalization, and third-party rebuild

Preprint / manuscript

Activity evidence

A named asymptotic problem with a dedicated paper, multiple independent formal checks, and active public review; the score is an editorial estimate.

The paper and multiple formal checks support a full resolution, while the canonical Erdős repository still showed the problem as open at the July 28 audit cutoff. The conservative status preserves that distinction.

Full resolution claimed; tracker update pending

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT / Aristotle + Eric Li
Verification: Three clean Lean checks; community tracker pending
Research activity*: 4/5

3 Jul 2026Divisor theory

Erdős Problem #884 — divisor-rich consecutive integers

Problem statement

d_1<\cdots<d_t

are the divisors of

n

, is

\sum_{i<j}(d_j-d_i)^{-1}\ll1+\sum_{i<t}(d_{i+1}-d_i)^{-1}

with an absolute implied constant?

Daniel Larsen’s unconditional human disproof is now represented by a public Lean project; the canonical Erdős database marks the problem disproved in Lean.

Details and sources

AI contribution

Claude Fable 5 is credited in the public formalization history. The underlying mathematical disproof is Larsen’s human result, so this entry records formalization rather than discovery priority.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; official record updated

Publication

Public proof manuscript, Lean repository, and official record

Preprint / manuscript

Human proof manuscript

Activity evidence

A specialist problem with an unconditional proof and a later formalization; the score is an editorial estimate.

This is deliberately classified as a formalization milestone. It must not be read as an AI-first discovery claim.

Human disproof formalized in Lean

Problem origin: Human-source problem
System: Claude Fable 5 + human formalizer
Verification: Lean checked; official record updated
Research activity*: 2/5

13 Jul 2026Euler totients and permutation patterns

Erdős Problem #415 — consecutive-totient ordering patterns

Problem statement

What is the growth of the longest universally realized consecutive-totient ordering length, which pattern fails first, and is the natural ordering most frequent?

The formal development proves $F(n)=o(\log\log\log n)$ , gives a finite witness showing the decreasing order is not always first to fail, and refutes both natural-order maximality readings.

Details and sources

AI contribution

Star Fleet independently reconstructed and formalized the conclusions after Przemek Chojecki had solved the problem first; the report gives him full priority.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + exact Euler-sieve witness

Official problem record Community discussion Star Fleet report

Publication

Public formal proof bundle and independent finite checker

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

This entry records the machine-checked reconstruction rather than assigning discovery priority to the AI system.

Three negative answers independently formalized

Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked + exact Euler-sieve witness
Research activity*: 3/5

13 Jul 2026Covering systems

Erdős Problem #1189 — irreducible covering sets

Problem statement

For irreducible covering sets of size

k

, determine their count, the possible largest modulus, maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

The development proves the exact largest modulus $3\cdot2^{k-3}$ for $k\geq5$ , near-linear order for the least largest modulus, reciprocal mass $\Theta(\log k)$ , and an infinite divisor-family construction. Its counting asymptotic is reduced to the published BBMST enumeration theorem.

Details and sources

AI contribution

Star Fleet independently reconstructed and formalized the result after Wouter van Doorn had solved it first; the report assigns him priority.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked with one named literature input

Official problem record Community discussion Star Fleet report

Publication

Public Lean bundle, exact finite census, independent checker, and explicit BBMST interface

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

The counting asymptotic is not kernel-proved from first principles: BBMST’s published enumeration theorem is carried as a named hypothesis. Every surrounding reduction and the other three answer families are checked in Lean.

Four answers formalized; counting input explicit

Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked with one named literature input
Research activity*: 4/5

26 Jul 2026Additive number theory

Erdős Problem #1112 — lacunary sequences and $k$ -fold sumsets

Problem statement

For fixed

1\leq d_1<d_2

and

k\geq3

, does some lacunarity ratio force the existence of a sequence

A

with gaps in

[d_1,d_2]

whose

k

-fold sumset avoids every sufficiently lacunary sequence

B

The ratio $r_k(d_1,d_2)$ exists exactly when $d_2\geq k+1$ . In the positive range, $r=192d_2$ works; when $d_2\leq k$ , a stronger variable-ratio nonexistence statement holds.

Details and sources

AI contribution

Johan Land reports that Claude Fable 5 and Opus 4.8 carried out the core mathematics, with GPT-5.5 and Gemini 3.1 used for advice and adversarial review. Land orchestrated, audited, and accepts responsibility.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean; official tracker status open (Lean)

Complete paper, Lean proof, and audit repository Final Lean theorems Frozen formal statement

Publication

Public human-readable paper, frozen Lean statement, complete proof, certificate data, and two Python harnesses

Preprint / manuscript

Human-readable proof manuscript

Activity evidence

A documented editorial estimate based on the official Erdős record, the novelty search, and the formal-audit activity; not a difficulty rating.

The Lean build and certificate checks have been independently rebuilt, and the statement fidelity has been checked. The official Erdős database still labels the problem open (Lean), because an independent human review of the mathematics has not been completed.

Official Erdős problem record Official status-review discussion

Machine-verified resolution; official status open (Lean)

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5 / Claude Opus 4.8 / GPT-5.5 / Gemini 3.1
Verification: Sorry-free Lean; official tracker status open (Lean)
Open for: Official status: open (Lean)
Research activity*: 3/5

30 Jul 2026Arithmetic combinatorics over function fields

Non-covering congruence systems over $\mathbb F_q[x]$

Problem statement

What is the asymptotic size of the largest minimum modulus degree in a distinct-modulus congruence system over

\mathbb F_q[x]

that does not cover the whole polynomial ring?

The extremal quantity for non-covering congruence systems satisfies $D_q(n)=n/(q-1)+O_q(1)$ . The construction and matching sieve argument correct the direction suggested by an earlier sharper conjecture while determining the leading asymptotic term.

Details and sources

AI contribution

Rongyin Wang credits the model with the construction in Example 2 and the idea of a truncated Chinese-remainder-theorem sieve in Lemma 6. Wang verified the work, added details, and filled gaps.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-verified proof; peer review pending

Publication

Public arXiv manuscript with theorem-specific AI attribution

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the classical covering-systems program and recent function-field analogues.

The theorem determines the leading asymptotic up to a bounded $q$ -dependent term; it does not give an exact formula for every $n$ .

Sharp asymptotic growth established

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Author-verified proof; peer review pending
Open for: Extremal function-field covering-system problem
Research activity*: 3/5

14 Jul 2026Diophantine approximation and symbolic dynamics

Superlinear complexity of the $(3/2)^n$ steering word

Problem statement

What structure can be proved for the nearest-integer coding of

(3/2)^n

, and can it illuminate the unresolved distribution of

(3/2)^n

modulo one?

For the nearest-integer steering word $t_n=2m_{n+1}-3m_n$ associated with $(3/2)^n$ , the subword complexity satisfies $p_T(k)/k\to\infty$ .

Details and sources

AI contribution

Ralf Stephan reports that the strategy, Lean development, and draft were produced with LLM assistance. The public manuscript more narrowly discloses Claude Code assistance for the Lean formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked modulo one cited theorem

Lean 4 development

Claim audit

The formalization is not axiom-free: one deep published theorem is represented as an external axiom. The broader equidistribution problem remains open.

Publication

Public arXiv preprint and Lean 4 repository

Preprint / manuscript

arXiv preprint

Activity evidence

The distribution of fractional parts of $(3/2)^n$ is a long-standing Diophantine problem. This paper isolates a new symbolic-complexity consequence and supplies a public formal artifact.

This is a rigorous structural advance around the classical $(3/2)^n$ problem, not a proof of equidistribution or density modulo one. The Lean development assumes the cited $S$ -arithmetic Subspace Theorem and derives the specialized consequences used in the paper.

Formal partial advance

Problem origin: Human-source problem
System: Claude Fable / Opus (author disclosure)
Verification: Lean checked modulo one cited theorem
Claim audit: Issue documented
Open for: Advance on a classical open direction
Research activity*: 4/5

16 Jul 2026Digit expansions and combinatorics on words

Aperiodicity and subword complexity in the binary expansion of powers of three

Problem statement

How periodic or combinatorially simple can long windows in the binary expansion of

3^m

be as

m

grows?

For every fixed period $p$ , the number of breaks of $p$ -periodicity in the binary expansion of $3^m$ grows on the order of $\log m/\log\log m$ ; the low-order word also has subword complexity at least $n+1$ for each fixed $n$ once $m$ is large.

Details and sources

AI contribution

Ralf Stephan reports LLM involvement in the strategy, Lean work, and paper drafting. The public preprint itself discloses Claude Code assistance for formalization.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked modulo cited deep theorems

Lean 4 development

Claim audit

The Lean artifact deliberately imports cited deep results as axioms, so “Lean checked” describes the new derivation conditional on those published inputs.

Publication

Public arXiv preprint and Lean 4 repository

Preprint / manuscript

arXiv preprint

Activity evidence

The work connects linear forms in logarithms, binary digit structure, and combinatorics on words, with a complete public formal development of the new deductions.

The 30 August v3 clarifies its formalization coverage: Sections 1–4 are checked subject to explicitly imported Baker–Wüstholz and Dimitrov–Howe inputs, while Section 5 generalizations and a stronger condition-dropping remark are not fully formalized. This remains adjacent digit-structure progress, not a resolution of the $(3/2)^n$ equidistribution problem.

Formal partial advance

Problem origin: Origin not yet traced
System: Claude Fable / Opus (author disclosure)
Verification: Lean checked modulo cited deep theorems
Claim audit: Issue documented
Open for: Advance on an open digit-structure direction
Research activity*: 3/5

15 Apr 2026Sunflowers and least common multiples

Erdős Problem #856 — harmonic LCM patterns

Problem statement

Estimate the maximum harmonic weight of a subset of

[N]

containing no

k

elements with a common pairwise least common multiple.

The manuscript proves that $f_k(N)=(\log N)^{\gamma_k+o(1)}$ for an exponent characterized by a weighted sunflower pressure.

Details and sources

AI contribution

Przemek Chojecki’s public research index credits GPT-5.5 Pro with the claimed solution. This record follows the scope and caveats stated in the manuscript and community discussion.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public manuscript; claimed advance

Claim audit

The source page’s “full solution” label is broader than the manuscript’s own scope statement.

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Primary problem record Community discussion Ulam research index

Activity evidence

The official page has 17 comments and ties the problem to contemporary sunflower-capacity bounds. The new pressure formula identifies an exact exponent but does not compute it.

Despite the Ulam page calling this a full solution, the manuscript explicitly leaves open computing $\gamma_k$ or expressing it purely through the ordinary sunflower capacity. The index therefore records a major partial resolution, not a closed problem.

Exact exponent characterized; value open

Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Public manuscript; claimed advance
Claim audit: Issue documented
Open for: Asked by 1970
Research activity*: 4/5

7 Sep 2026Digital number theory

Binary disjunctivity of the Erdős–Borwein constant

Problem statement

For the Erdős–Borwein constant

E=\sum_{n\geq1}\frac{1}{2^n-1},

does the block

11

occur infinitely often in the base-2 expansion of

E

A Lean development proves, conditional on two explicitly isolated published analytic-number-theory inputs, that every finite binary word occurs infinitely often in the binary expansion of the Erdős–Borwein constant. This strictly strengthens the earlier block- $11$ result.

Details and sources

AI contribution

The new disjunctivity argument and Lean development are attributed to GPT-6 Astra under human direction. John M. Campbell's earlier conventional proof of infinitely many occurrences of $11$ was developed through extensive GPT-5.5 Pro interaction.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean proof conditional on two published analytic inputs

Pinned disjunctivity Lean release

Publication

Pinned Lean release plus earlier arXiv proof

Preprint / manuscript

Earlier block-11 research paper

Activity evidence

A recognized 2012 question of Crandall, later repeated by Shallit, with a limited rather than large sustained literature.

The formal conclusion is conditional on an Alford–Granville–Pomerance estimate and a prime-interval supply consequence of the prime number theorem, which are passed into Lean as hypotheses rather than formalized. Specialist review of statement fidelity and the analytic interface remains pending.

Community scope and dependency audit

Open problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra / GPT-5.5 Pro
Verification: Lean proof conditional on two published analytic inputs
Open for: 14 years
Research activity*: 2/5

Jun 2026Additive number theory

Erdős Problem #477 — polynomial tiling complements

Problem statement

Does there exist an integer polynomial

f

of degree at least two and a set

A\subseteq\mathbb{Z}

such that every integer has a unique representation

n=a+f(k),\qquad a\in A,\ k\in\mathbb{Z}?

A June 2026 manuscript claims that the thirteenth powers have a tiling complement in the integers, which would answer the existence question positively.

Details and sources

AI contribution

The pipeline repository says GPT-5.5 Pro generated the proof and that human contributors polished and verified the resulting manuscript.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked manuscript; official record open

Solver–verifier pipeline

Publication

Public proof manuscript and open pipeline

Preprint / manuscript

Activity evidence

A substantial but narrow additive-tiling question recorded by Erdős and Graham in 1980, with limited documented follow-up.

The candidate is existential and constructive, using $f(m)=m^{13}$ . The official Erdős Problems record still listed #477 as open when checked on 25 July 2026.

Lean formalizations and axiom interfaces Finite-quotient verifier Reproducibility record

Candidate construction

Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author-checked manuscript; official record open
Open for: 46 years
Research activity*: 2/5

27 Jul 2026Algebraic number theory

Explicit presentation of the $2$ -adic absolute Galois group

Problem statement

Give an explicit profinite presentation of the absolute Galois group

\operatorname{Gal}(\overline{\mathbb{Q}}_2/\mathbb{Q}_2)

Roe and Turturean give an explicit marked profinite presentation of $G_{\mathbb Q_2}$ with four generators, two word relations, and a pro- $2$ condition on the wild generators, completing the missing dyadic case of the classical explicit local theory.

Details and sources

AI contribution

Turturean’s long-horizon ChatGPT Pro harness found a candidate and informal proof. Roe and Turturean then independently developed Lean 4 formalizations with Fable, Opus, GPT, and Codex assistance while revising the manuscript against the formal interfaces.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked modulo 7–9 named literature interfaces

Publication

Public full manuscript, two Lean developments, finite-quotient verifier, and reproducibility record; external peer review pending

Preprint / manuscript

Full preprint PDF

Activity evidence

The odd-prime presentations date to Jannsen–Wingberg’s early-1980s work. Epoch’s contributor survey rated the missing dyadic case a publishable solid result in a specialized but active area.

The manuscript contains a complete proof but has not yet been externally peer reviewed. Roe’s formalization reduces the theorem to nine named interfaces to the classical literature; Turturean’s records seven external inputs after adaptation. The independent finite-quotient verifier matches all 5,402 test groups but is corroboration rather than a proof by itself.

Interactive paper Development history FrontierMath problem and earlier progress update Lean Zulip discussion Author announcement Epoch AI announcement

Explicit presentation proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.5 Pro (GPT-5.6 A/B) / Claude Fable 5 / Claude Opus 4.8
Verification: Lean checked modulo 7–9 named literature interfaces
Open for: Missing explicit dyadic case since the early 1980s
Research activity*: 4/5

5 Mar 2026Diophantine equations

Two small Diophantine equations

Problem statement

Prove that each of the nine listed small Diophantine equations has infinitely many integer solutions. GPT-5.4 Pro resolved

z^2+y^2z-z+x^3+2=0

and

z^2+y^2z+x^3+x+1=0

by finding three distinct solutions with

|x|>10^{50}

and then giving parametric families.

GPT-5.4 Pro found direct substitutions giving infinitely many integer solutions for two equations in a nine-equation FrontierMath portfolio. The authors adapted one substitution to a third equation; six remain open.

Details and sources

AI contribution

The model produced two explicit parametric families satisfying the large-solution requirement.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Problem authors checked

Publication

Epoch AI write-up and public six-page solution note

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Epoch reports 2–4 mathematicians familiar with and seriously attempting the portfolio, and rates a full solution as moderately interesting.

The parent challenge is not solved. This record is deliberately classified as partial because the benchmark requires all nine equations to be resolved.

Problem and status Two AI-found families

Two of nine equations solved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.4 Pro
Verification: Problem authors checked
Open for: Date not fixed
Research activity*: 2/5

13 Apr 2026Number theory

Erdős Problem #1196

Problem statement

A\subseteq[x,\infty)

is primitive—no member divides another—must

\sum_{a\in A}1/(a\log a)\leq1+o(1)

x\to\infty

A von Mangoldt Markov-chain method proves the primitive-set bound and settles several related conjectures, including Erdős Problems #1217 and #164.

Details and sources

AI contribution

Liam Price launched the autonomous query; a human team including Terence Tao then checked, developed, and generalized the method.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Multi-author human proof

Publication

Detailed arXiv preprint and public expert exposition

Preprint / manuscript

Original ChatGPT conversation Tao’s exposition Official problem record

Activity evidence

A 1966 conjecture with several published partial bounds and a dedicated multi-author resolution.

This is a particularly transparent case: the initial model conversation, the human mathematical development, and the resulting preprint can be compared directly. No formal proof assistant artifact was located.

Resolved and extended

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Multi-author human proof
Open for: 60 years
Research activity*: 4/5

6 Jan 2026Number theory

Erdős Problem #728

Problem statement

For fixed

C>0

and sufficiently small

\epsilon>0

, are there infinitely many

a,b,n

with

a,b\geq\epsilon n

a!b!\mid n!(a+b-n)!

, and

a+b>n+C\log n

A logarithmic-gap theorem for a factorial divisibility problem resolves the agreed formulation and also gives solutions to Problems #729 and #401.

Details and sources

AI contribution

GPT-5.2 supplied the informal argument and Harmonic’s Aristotle produced a kernel-checked Lean proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + human writeup

Official problem and discussion Revision history Formalization post-mortem

Publication

Lean source and a detailed arXiv translation are public

Preprint / manuscript

Human-readable proof

Activity evidence

One original source, extensive later discussion, and a dedicated human writeup.

The first generated proof addressed an ambiguous weaker reading. A second autonomous run handled the formulation accepted by the forum, after which the result gained community consensus.

Public writeup chat

Fully resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.2 Pro + Aristotle
Verification: Lean checked + human writeup
Open for: 51 years
Research activity*: 3/5

25 Nov 2025Extremal number theory

Erdős Problem #56 — missing-hypothesis audit

Problem statement

For

N\geq p_k

, if

A\subseteq\{1,\ldots,N\}

contains no

k+1

pairwise relatively prime elements, is

A

no larger than the set of multiples of the first

k

primes?

Aristotle found a Lean-checked counterexample to the first formal statement at $N=k=2$ . The counterexample exposed a missing hypothesis, $N\geq p_k$ , rather than disproving the intended Erdős problem.

Details and sources

AI contribution

Aristotle independently falsified the faulty encoded statement. After the missing condition was identified, ChatGPT explained the literature proof and Aristotle formalized the corrected theorem.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; target statement was faulty

Claim audit

Missing hypothesis: the initial formal statement omitted $N\geq p_k$ , making it trivially false.

Publication

Official Erdős discussion, corrected problem statement, and repaired Lean formalization

Preprint / manuscript

No public preprint or manuscript located.

This is not a failure of Lean’s kernel. Lean correctly certified a counterexample to the proposition it was given; the failure was the mismatch between that proposition and the intended mathematical claim.

Formal statement repaired

Problem origin: Human-source problem
System: Aristotle + ChatGPT
Verification: Lean checked; target statement was faulty
Claim audit: Issue documented

28 Nov 2025Diophantine approximation

Erdős Problem #480 — variable-mismatch audit

Problem statement

For every sequence

x_1,x_2,\ldots\in[0,1]

, must

\inf_{n\geq1}\liminf_{m\to\infty}n\lvert x_{m+n}-x_m\rvert\leq5^{-1/2}\,?

Aristotle automatically proved an encoded statement that was not the intended theorem: the Lean hypothesis said $m\ne0$ where it should have said $n\ne0$ . The proof also exploited Lean’s totalized division convention, where $1/0=0$ .

Details and sources

AI contribution

The prover completed the supplied Lean goal, revealing that the goal itself admitted an unintended route.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; wrong variable in target

Claim audit

Low-level specification bug: $m\ne0$ replaced the intended condition $n\ne0$ , enabling a proof of the wrong proposition.

Publication

Official Erdős discussion, public Lean proof, and filed correction issue

Preprint / manuscript

No public preprint or manuscript located.

The mathematical theorem was already known from Chung and Graham. This case is retained because it cleanly separates kernel correctness from correctness of the formal specification.

Official problem and audit discussion Formalization post-mortem

Misformalization detected

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked; wrong variable in target
Claim audit: Issue documented

27 Nov 2025Multiplicative number theory

Erdős Problem #488 — corrected-target audit

Problem statement

For a finite set

A

, let

B

be the positive integers divisible by some

a\in A

. For every

m>n\geq\max(A)

, must

\frac{\lvert B\cap[1,m]\rvert}{m}<2\frac{\lvert B\cap[1,n]\rvert}{n}\,?

Aristotle found and Lean-checked a finite counterexample to the statement then encoded in Formal Conjectures. Source comparison showed that this was likely a misstated “non-divisibility” version; the corrected “divisibility” problem remains open.

Details and sources

AI contribution

Given only the formal statement, Aristotle produced the counterexample $n=13$ , $m=200$ , and $A=\{2,3,5,7,11,13\}$ .

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; source wording corrected

Claim audit

Source mismatch: the encoded version followed wording now treated as a likely typo in one Erdős source.

Publication

Official problem history, community discussion, and formal counterexample

Preprint / manuscript

No public preprint or manuscript located.

The certificate is valid for the encoded proposition. It does not settle the corrected version now displayed by the Erdős database.

Current problem record Counterexample and correction discussion Problem revision history Formalization post-mortem

Counterexample to obsolete wording

Problem origin: Human-source problem
System: Aristotle
Verification: Lean checked; source wording corrected
Claim audit: Issue documented

23 Apr 2026Covering systems

Erdős Problem #202

Problem statement

Let

n_1<\cdots<n_r\leq N

and choose pairwise-disjoint residue classes

a_i\pmod{n_i}

. How large can

r

be as a function of

N

The conjectured asymptotic statement for a covering-system problem was proved by GPT-5.4 Pro under human prompting and checking.

Details and sources

AI contribution

Boon Suan Ho prompted the model, checked the output, and presented the result to the Erdős Problems community.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Community checked + formal artifact reported

Publication

Public problem discussion and community record

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

7 cited source records and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The entry records the accepted problem-site status. It does not treat the absence of a conventional journal article as evidence against the proof.

Problem record Discussion thread Community AI ledger

Conjectured asymptotic proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Community checked + formal artifact reported
Open for: 65 years
Research activity*: 4/5

4 Feb 2026Complete sequences

Erdős Problem #347

Problem statement

Is there an integer sequence with

a_{n+1}/a_n\to2

whose finite subset sums have density 1 even after deleting any finite number of terms?

A problem of Erdős and Graham on complete sequences was resolved through a multi-person, multi-system collaboration.

Details and sources

AI contribution

AI systems proposed and formalized components while human collaborators reconciled the statement, proof structure, and historical sources.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Public problem record and formal proof artifacts

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 15 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The result is a full solution, but it is not an autonomous single-model discovery; the provenance is inherently collaborative.

Fully resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle + Claude + Codex + GPT
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

7 Mar 2026Number theory

Erdős Problem #650

Problem statement

Given an arbitrary

m

-element set

A\subseteq[1,N]

, how many integers can every interval of length

2N

contain so that each is divisible by a distinct element of

A

? In particular, is the extremal quantity

O(\sqrt m)

The exact value of the matching function f(m) was determined as min(m, ⌈2√m⌉), resolving an Erdős problem on matching integers to distinct multiples.

Details and sources

AI contribution

ChatGPT proposed the proof strategy, AlphaEvolve supported numerical optimization, and Aristotle produced the formal certificate.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + human paper

Claim audit

The first GPT lower-bound proof had a genuine gap: after deleting the midpoint, it still assumed that the next multiple lies in the positive half. Standard human and model checks missed this. Aristotle nevertheless completed the formalization by finding and encoding a repair independently; the final theorem remains valid.

Publication

Public arXiv preprint

Preprint / manuscript

Problem record Discussion and proof-gap timeline Community AI ledger

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The final exposition is human written and states an optimal theorem stronger and cleaner than the original problem formulation. The Lean proof is valid, but its route is not identical to the initially circulated informal proof.

Optimal formula proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro + Aristotle + AlphaEvolve
Verification: Lean checked + human paper
Claim audit: Issue documented
Open for: 31 years
Research activity*: 1/5

5 Jun 2026Divisor theory

Erdős Problem #696

Problem statement

Compare the longest prime chain

p_{i+1}\equiv1\pmod{p_i}

inside the divisors of

n

with the analogous longest chain of arbitrary divisors. Does their length ratio tend to infinity for almost every

n

A forty-seven-year-old divisor problem was resolved in a collaborative workflow ending in a kernel-checked Lean proof.

Details and sources

AI contribution

Several models contributed proof ideas and proof engineering while human collaborators coordinated statement fidelity.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Independent tracker entry

Publication

Public problem record and formal development

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This is a full mathematical resolution, but the evidence supports a distributed collaboration rather than a single autonomous run.

Aletheia research report Dedicated resolution preprint

Fully resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle + Claude Code + Claude Opus 4.7 + GPT-5.5 Pro
Verification: Lean checked
Open for: 47 years
Research activity*: 1/5

29 Jan 2026Irrationality

Erdős Problem #1051

Problem statement

a_1<a_2<\cdots

and

\liminf a_n^{1/2^n}>1

, must

\sum_{n\geq1}1/(a_na_{n+1})

be irrational?

An irrationality problem of Erdős and Graham was solved by Aletheia and certified in Lean.

Details and sources

AI contribution

The generator–verifier–reviser agent produced the argument and a formal system checked the final theorem.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Public problem record and autonomous-math evaluation

Preprint / manuscript

Activity evidence

2 cited source records and 6 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The result is one of the clearest autonomous Aletheia research cases, although the public record is a paper-level interaction report rather than a raw chat export.

Formalization portal Lean repository

Problem proved and formalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aletheia
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

Sep 2025Analytic number theory

Strong Prime Number Theorem

The strong Prime Number Theorem and substantial supporting complex-analysis infrastructure were formalized in Lean.

Details and sources

AI contribution

Gauss generated most statements and proofs from a human-prepared blueprint, with targeted scaffolding and review of key lemmas.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked

Publication

Public blueprint, documentation, and Lean repository

Preprint / manuscript

Formalization blueprint

This did not discover the Prime Number Theorem. It is classified separately because the contribution is large-scale verification of established mathematics.

Known theorem autoformalized

Problem origin: Origin not yet traced
System: Gauss
Verification: Lean checked

100

6 Apr 2026Number Theory, Divisors

Erdős Problem #26

Problem statement

Let

A\subset\mathbb{N}

be infinite. Must there exist some

k\geq 1

such that almost all integers have a divisor of the form

a+k

for some

a\in A

Davenport and Erdős had already given a negative answer to the literal problem in 1951. A DeepMind prover agent later found and Lean-verified a stronger counterexample to a variant.

Details and sources

AI contribution

DeepMind prover agent is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Stronger-variant proof preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 6 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This is an AI contribution to a stronger variant, not the first historical resolution of the literal problem.

Problem and proof record Discussion thread Community AI ledger

Stronger variant disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: DeepMind prover agent
Verification: Lean checked
Open for: Known result; 2026 variant
Research activity*: 1/5

101

25 Apr 2026Number Theory

Erdős Problem #38

Problem statement

Does there exist

B\subset\mathbb{N}

which is not an additive basis, but is such that for every set

A\subseteq\mathbb{N}

of Schnirelmann density

\alpha

and every

N

there exists

b\in B

such that

\lvert (A\cup (A+b))\cap \{1,\ldots,N\}\rvert\geq (\alpha+f(\alpha))N

where

f(\alpha)>0

for

0<\alpha <1

? The Schnirelmann density is defined by

d_s(A) = \inf_{N\geq 1}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}.

The problem was proved after remaining open for 70 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 70 years
Research activity*: 1/5

102

30 Mar 2026Number Theory, Base Representations

Erdős Problem #125

Problem statement

Let

A

be the integers using only digits 0 and 1 in base 3, and

B

the integers using only digits 0 and 1 in base 4. Does

A+B

have positive lower density?

The problem was disproved after remaining open for 30 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

DeepMind prover agent is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

2 cited source records and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: DeepMind prover agent
Verification: Lean checked
Open for: 30 years
Research activity*: 2/5

103

10 Jan 2026Number Theory

Erdős Problem #205

Problem statement

Must every sufficiently large

n

equal

2^k+m

with

\Omega(m)<\log\log m

? Can the bound be reduced to

\epsilon\log\log m

, or to a still slower-growing function?

The problem was disproved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, GPT-5.2 Thinking is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 21 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Aristotle, GPT-5.2 Thinking
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

104

14 Apr 2026Irrationality

Erdős Problem #258

Problem statement

a_1,a_2,\ldots

are positive integers with

a_n\to\infty

, must

\sum_n \tau(n)/(a_1\cdots a_n)

be irrational?

The problem was proved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

2 cited source records and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

105

3 May 2026Number Theory, Unit Fractions

Erdős Problem #283

Problem statement

Let

p:\mathbb Z\to\mathbb Z

be polynomial with positive leading coefficient and no fixed divisor. Can every sufficiently large

m

be written as

p(n_1)+\cdots+p(n_k)

for

1\leq n_1<\cdots<n_k

with

1/n_1+\cdots+1/n_k=1

The problem was proved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 2 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 1/5

106

24 Apr 2026Number Theory, Additive Basis

Erdős Problem #330

Problem statement

Does there exist a minimal additive basis of positive density such that deleting any one of its members prevents a positive-density set of integers from being represented?

The problem was proved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 14 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

107

25 Dec 2025Number Theory, Additive Basis

Erdős Problem #333

Problem statement

For every density-zero set

A\subseteq\mathbb N

, does there exist

B

with

A\subseteq B+B

and

|B\cap[1,N]|=o(N^{1/2})

A 1977 theorem of Erdős and Newman already implied the negative answer. The 2025 AI event rediscovered and formalized that conclusion.

Details and sources

AI contribution

Claude Opus 4.5, GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Erdős case-study preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 14 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The AI contribution is rediscovery and formalization, not mathematical priority.

Problem and proof record Discussion thread Community AI ledger

Known theorem reconstructed

Problem origin: Human-source problem
System: Claude Opus 4.5, GPT-5.2 Pro
Verification: Lean checked
Open for: Known by 1977
Research activity*: 2/5

108

3 May 2026Number Theory, Complete Sequences

Erdős Problem #351

Problem statement

For

p(x)\in\mathbb Q[x]

with positive leading coefficient, is the set

\{p(n)+1/n:n\in\mathbb N\}

strongly complete—does every cofinite subcollection have finite subset sums containing all sufficiently large integers?

The problem was proved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 9 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 1/5

109

26 Mar 2026Number Theory

Erdős Problem #369

Problem statement

For every

\epsilon>0

and

k\geq2

, do all sufficiently large intervals

[1,n]

contain

k

consecutive integers that are all

n^\epsilon

-smooth?

The literal database statement is trivial; the AI work addressed an intended stronger variant related to results already present in the literature.

Details and sources

AI contribution

GPT is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This record is classified as a variant because the literal and intended formulations differ.

Intended variant treated

Problem origin: Human-source problem
System: GPT
Verification: Lean checked
Open for: Variant record
Research activity*: 1/5

110

31 Mar 2026Number Theory

Erdős Problem #380

Problem statement

Call

[u,v]

bad when the greatest prime factor of

\prod_{u\leq m\leq v}m

occurs with exponent greater than 1. Is the count of integers up to

x

lying in a bad interval asymptotic to the count of

n\leq x

with

P(n)^2\mid n

The problem was proved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 5 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Erdős Problems site confirmed
Open for: 46 years
Research activity*: 1/5

111

10 Jan 2026Number Theory, Binomial Coefficients

Erdős Problem #397

Problem statement

Are there only finitely many identities

\prod_i {2m_i\choose m_i}=\prod_j {2n_j\choose n_j}

when all the

m_i

and

n_j

are distinct?

The negative answer was rediscovered and formalized in 2026; an essentially identical problem had appeared in a 2012 China TST.

Details and sources

AI contribution

Aristotle, GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Erdős case-study preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 30 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The Lean artifact is a new verification, but the mathematical result was not new in 2026.

Problem and proof record Discussion thread Community AI ledger

Known result rediscovered

Problem origin: Human-source problem
System: Aristotle, GPT-5.2 Pro
Verification: Lean checked
Open for: Known by 2012
Research activity*: 3/5

112

11 Jan 2026Number Theory, Factorials

Erdős Problem #401

Problem statement

Can one find a function

f(r)\to\infty

such that infinitely many

n

admit

a_1+a_2>n+f(r)\log n

while

a_1!a_2!

divides

n!2^n3^n\cdots p_r^n

The problem was proved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Resolution write-up

Activity evidence

1 cited source record and 18 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle, GPT-5.2 Pro
Verification: Lean checked
Open for: 46 years
Research activity*: 2/5

113

2 Mar 2026Number Theory

Erdős Problem #457

Problem statement

Is there

\epsilon>0

such that infinitely many

n

have every prime

p\leq(2+\epsilon)\log n

dividing

\prod_{1\leq i\leq\log n}(n+i)

The problem was proved after remaining open for 47 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Later formalization preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

2 cited source records and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle, GPT-5.2 Pro
Verification: Lean checked
Open for: 47 years
Research activity*: 2/5

114

21 Jan 2026Number Theory, Group Theory

Erdős Problem #543

Problem statement

For a finite abelian group

G

of order

N

, let

f(N)

be the smallest size of a random subset that generates every element as a subset sum with probability at least 1/2. Is

f(N)\leq\log_2N+o(\log\log N)

The problem was disproved after remaining open for 53 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

2 cited source records and 28 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.2 Pro
Verification: Erdős Problems site confirmed
Open for: 53 years
Research activity*: 4/5

115

8 May 2026Number Theory

Erdős Problem #690

Problem statement

For fixed

k

, is the density

d_k(p)

of integers whose

k

th-smallest prime factor is

p

unimodal as

p

ranges over the primes?

Cambie’s 2025 result had already refuted universal unimodality. The Multiscalar Fields work was a later AI contribution to the resolved problem.

Details and sources

AI contribution

Multiscalar Fields System is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 2 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This record does not attribute the first resolution to the 2026 AI event.

Later AI contribution

Problem origin: Human-source problem
System: Multiscalar Fields System
Verification: Erdős Problems site confirmed
Open for: Resolved before AI event
Research activity*: 1/5

116

1 May 2026Number Theory

Erdős Problem #694

Problem statement

f_{\max}(n)

and

f_{\min}(n)

are the largest and smallest solutions of

\phi(m)=n

, how large can

f_{\max}(n)/f_{\min}(n)

be for

n\leq x

The problem was resolved after remaining open for 47 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 2 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Lean checked
Open for: 47 years
Research activity*: 1/5

117

10 Jan 2026Number Theory, Factorials

Erdős Problem #729

Problem statement

For each constant

C>0

, are there infinitely many

a,b,n

with

a+b>n+C\log n

such that the denominator of

n!/(a!b!)

has only bounded prime factors?

The problem was proved after remaining open for 51 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Resolution write-up

Activity evidence

1 cited source record and 30 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle, GPT-5.2 Pro
Verification: Lean checked
Open for: 51 years
Research activity*: 3/5

118

20 Nov 2025Number Theory

Erdős Problem #848

Problem statement

Is the largest subset

A\subseteq[1,N]

for which

ab+1

is never squarefree attained by the residue class

7\pmod{25}

GPT-5 reduced the question to a finite computation. The official record classifies it as decidable, not as a completed full resolution.

Details and sources

AI contribution

GPT-5 is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Primary GPT-5 research report

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 18 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

A finite decision procedure is meaningful progress, but the remaining finite check has not been reported complete.

Problem and proof record Discussion thread Community AI ledger

Resolved up to a finite check

Problem origin: Human-source problem
System: GPT-5
Verification: Erdős Problems site confirmed
Open for: 33 years
Research activity*: 2/5

119

5 Feb 2026Number Theory

Erdős Problem #851

Problem statement

For every

\epsilon>0

, is there a bounded

r

such that integers of the form

2^k+n

, with

n

having at most

r

prime factors, have density at least

1-\epsilon

The problem was proved after remaining open for 41 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.2 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.2 Pro
Verification: Erdős Problems site confirmed
Open for: 41 years
Research activity*: 1/5

120

15 Apr 2026Number Theory, Primitive Sets

Erdős Problem #858

Problem statement

How large can

\frac1{\log N}\sum_{n\in A}1/n

be when

A\subseteq[1,N]

contains no relation

at=b

whose multiplier

t

has least prime factor greater than

a

The problem was resolved after remaining open for 56 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Erdős Problems site confirmed
Open for: 56 years
Research activity*: 1/5

121

5 Jan 2026Number Theory, Additive Basis

Erdős Problem #871

Problem statement

A

is an additive basis of order 2 and its representation count tends to infinity, can

A

be partitioned into two disjoint additive bases of order 2?

The problem was disproved after remaining open for 38 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Claude Opus 4.5, Gemini 3 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 13 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Claude Opus 4.5, Gemini 3 Pro
Verification: Lean checked
Open for: 38 years
Research activity*: 2/5

122

25 Apr 2026Number Theory, Squares

Erdős Problem #888

Problem statement

How large can

A\subseteq[1,n]

be if every ordered quadruple

a\leq b\leq c\leq d

A

with

abcd

a square must satisfy

ad=bc

The problem was resolved after remaining open for 28 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 20 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle, GPT-5.5 Pro
Verification: Erdős Problems site confirmed
Open for: 28 years
Research activity*: 2/5

123

26 Apr 2026Number Theory

Erdős Problem #896

Problem statement

For

A,B\subseteq[1,N]

, how many integers can have exactly one factorization

m=ab

with

a\in A

and

b\in B

The problem was resolved after remaining open for 54 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 1 problem-page discussion comment were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Erdős Problems site confirmed
Open for: 54 years
Research activity*: 1/5

124

26 Dec 2025Number Theory

Erdős Problem #897

Problem statement

If an additive function satisfies

\limsup_{p,k}f(p^k)/\log p^k=\infty

, must

\limsup_n(f(n+1)-f(n))/\log n=\infty

? Or even

\limsup_n f(n+1)/f(n)=\infty

Wirsing published the counterexample in 1981. The 2025 AI work rediscovered the construction and produced a Lean verification.

Details and sources

AI contribution

Archivara, Aristotle is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 28 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

The formal proof is new; the underlying counterexample is not.

Known counterexample formalized

Problem origin: Human-source problem
System: Archivara, Aristotle
Verification: Lean checked
Open for: Known by 1981
Research activity*: 3/5

125

21 Jun 2026Number Theory, Ramsey Theory

Erdős Problem #948

Problem statement

Can one choose a function

f

and a number of colours

k

so that every

k

-colouring of the integers contains a slowly growing sequence whose finite subset sums omit at least one colour?

The problem was resolved after remaining open for 49 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Aristotle, GPT-5.5 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

1 cited source record and 12 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Aristotle, GPT-5.5 Pro
Verification: Lean checked
Open for: 49 years
Research activity*: 2/5

126

25 Apr 2026Number Theory, Primes

Erdős Problem #1138

Problem statement

Let

x/2<y<x

C>1

, and let

d

be the largest prime gap below

x

. Must

\pi(y+Cd)-\pi(y)\sim Cd/\log y

The problem was disproved after remaining open for 27 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.5 Pro, GPT-5.5 Thinking is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Author-hosted obstruction manuscript

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

1 cited source record and 0 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 Pro, GPT-5.5 Thinking
Verification: Lean checked
Open for: 27 years
Research activity*: 1/5

127

9 Apr 2026Number Theory, Primes

Erdős Problem #1141

Problem statement

Are there infinitely many

n

such that

n-k^2

is prime for every

k

coprime to

n

with

k^2<n

The problem was disproved after remaining open for 27 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

OpenAI internal model is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 5 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI internal model
Verification: Lean checked
Open for: 27 years
Research activity*: 1/5

128

16 Mar 2026Number Theory

Erdős Problem #1148

Problem statement

Can every sufficiently large integer

n

be represented as

x^2+y^2-z^2

with each of

x^2,y^2,z^2

at most

n

The problem was proved after remaining open for 27 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

Gemini 3 Pro, Gemini 3.1 Pro, GPT-5.2 Pro, GPT-5.2 Thinking, GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Problem and proof record Discussion thread Community AI ledger

Activity evidence

1 cited source record and 4 problem-page discussion comments were located. The score is a conservative proxy for documented research attention.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Gemini 3 Pro, Gemini 3.1 Pro, GPT-5.2 Pro, GPT-5.2 Thinking, GPT-5.4 Pro
Verification: Lean checked
Open for: 27 years
Research activity*: 1/5

129

1 Apr 2026Number Theory, Primes

Erdős Problem #1202

Problem statement

Given

\epsilon,\eta>0

, does some

k

force the following? For any primes

p_1<\cdots<p_k<n^{1-\epsilon}

and any choice of

(p_j-1)/2

residue classes modulo each

p_j

, fewer than

\epsilon n

integers

m\leq n

avoid all the chosen classes.

The problem was resolved after remaining open for 46 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Problem and proof record Discussion thread Community AI ledger

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Specialist sieve question with limited prior dedicated literature.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Erdős Problems site confirmed
Open for: 46 years
Research activity*: 2/5

130

16 Apr 2026Number Theory, Divisors, Primitive Sets

Erdős Problem #1217

Problem statement

Let

A=\{a_1<a_2<\cdots\}

have positive lower logarithmic density. Must it contain a divisibility chain

a_{n_i}\mid a_{n_{i+1}}

such that

\limsup_{x\to\infty}\frac{\#\{i:a_{n_i}<x\}}{\log\log x}\geq\limsup_{x\to\infty}\frac{1}{\log\log x}\sum_{a_n<x}\frac{1}{a_n\log a_n}\,?

The problem was proved after remaining open for 60 years. The community trackers classify the result as a full resolution.

Details and sources

AI contribution

GPT-5.4 Pro is credited on the public resolution record.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Erdős Problems site confirmed

Primitive-set resolution preprint

Publication

Erdős Problems record and community AI ledger

Preprint / manuscript

Activity evidence

Sustained primitive-set literature with multiple substantial advances.

This entry follows the full-resolution classification in the public trackers. The displayed statement is taken from the problem record; the primary page gives the proof links and literature notes.

Problem and proof record Discussion thread Community AI ledger

Problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.4 Pro
Verification: Erdős Problems site confirmed
Open for: 60 years
Research activity*: 4/5

Probability & statistics

7 Sep 2026High-dimensional statistics

Growing-degree AMP equivalence for the Bernoulli rank-one model

For the Gaussian planted-submatrix model with Bernoulli signal, polynomial estimators of degree $D(n)=o(n^{1/60})$ have the exact limiting Bayes-AMP mean-square error when $D(n)\to\infty$ .

Details and sources

AI contribution

Zhangsong Li supplied the auxiliary-Gaussian-channel strategy; GPT-6 Astra identified the conditional-cumulant bound and generated most formal arguments through interactive refinement.

Problem origin

The growing-degree AMP-versus-low-degree question was discussed in prior human research on computational-statistical gaps.

Verification

Author-checked preprint; external review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

Pinned Lean proof and trust audit

This resolves the Bernoulli rank-one case in the stated degree range, not the general growing-degree equivalence question for arbitrary priors or models.

Bernoulli rank-one case resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra
Verification: Author-checked preprint; external review pending

31 Aug 2026High-dimensional central limit theory

Cubic-root Gaussian approximation under unrestricted covariance

Under coordinatewise subexponential tails and marginal variance bounded below, rectangle Gaussian approximation has an $n^{-1/3}$ rate up to logarithms for bounded envelope and polynomial dimension, contradicting the earlier $n^{-1/4}$ near-optimality claim.

Details and sources

AI contribution

The authors say ChatGPT 5.6 Pro generated the initial proof attempt; they corrected and rewrote it. A separate Lean project formalizes the resulting all-dimensions theorem.

Problem origin

Chernozhukov and coauthors conjectured near-optimality of the quarter-power rate in their 2023 human-authored work.

Verification

Pinned Lean proof and trust audit; specialist review pending

Claim audit

This audit checked the repository's theorem map and trust report but did not independently rebuild the development. The natural-language correspondence still requires specialist review.

Publication

Public arXiv proof and pinned Lean 4 development

Preprint / manuscript

Pinned Lean proof, guide, and audit

The paper explicitly formulates the previously informal conjecture in the fixed-envelope, polynomial-dimensional regime. The Lean repository reports no sorry, admit, project axiom, or native_decide in its principal declarations.

Quarter-power near-optimality conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Pro
Verification: Pinned Lean proof and trust audit; specialist review pending
Claim audit: Issue documented

3 Sep 2026Percolation theory

Absence of critical Bernoulli bond percolation on $\mathbb Z^d$

For nearest-neighbor Bernoulli bond percolation on $\mathbb Z^d$ , the probability that the origin belongs to an infinite open cluster vanishes at $p_c$ for every $d\ge 2$ . The Lean development proves Kozma–Nitzan Conjecture 3 and re-proves the classical inputs it uses.

Details and sources

AI contribution

Justin Leder selected and directed the project. The repository says Claude autonomously produced the mathematical argument, exposition, and all Lean code; no human wrote or edited the formal sources.

Problem origin

Vanishing of the percolation probability at the critical point was classical and remained open in dimensions $3$ through $10$ ; Kozma and Nitzan isolated a sufficient gluing conjecture in 2024.

Verification

Lean and independent-kernel replay recorded; statement review pending

Claim audit

No independent human mathematician has yet refereed the argument or the statement–mathematics correspondence. This audit did not rerun the roughly 87,000-line project, whose recorded clean build used a 128-core host.

Publication

Pinned public Lean project with 15-page proof guide and audit log

Preprint / manuscript

No public preprint or manuscript located.

The formal theorem is $\theta(p_c)=0$ , not a separately formalized continuity theorem for the whole function $p\mapsto\theta(p)$ . It says nothing new about site percolation, slabs, other lattices, or Kozma–Nitzan Conjectures 1, 2, and 4.

Kozma–Nitzan reduction and gluing conjectures Community scope and formal-artifact audit

Claimed full resolution; Lean checked

Claimed outcome: Proved
Problem origin: Human-source problem
System: Anthropic Claude models (exact versions undisclosed)
Verification: Lean and independent-kernel replay recorded; statement review pending
Claim audit: Issue documented

8 Sep 2026Symbolic dynamics and hidden Markov processes

Sharp finite Markov order in intrinsic sofic dimension

A stationary finite-alphabet law with finite real Hankel dimension $n$ and finite Markov order has order at most $n(n-1)/2$ , and for every $n\ge2$ a rational nonnegative stationary presentation attains the bound.

Details and sources

AI contribution

The repository says all original mathematical contributions, the manuscript, and the paper-specific Lean development were generated by AI. Eugene Gilburg packaged and published the artifact without claiming subject expertise.

Problem origin

The manuscript attributes the finite-order bound question to work of Béal, Jugé, Mairesse, and Perrin in 2026.

Verification

Pinned Lean proof with claim map; specialist review pending

Pinned manuscript and Lean proof

Claim audit

The publisher explicitly disclaims professional review. Lean checks the encoded theorem, but mathematical statement fidelity and priority have not been independently assessed by a specialist.

Publication

Public manuscript, sources, and paper-specific Lean project

Preprint / manuscript

No public preprint or manuscript located.

The theorem assumes both finite real Hankel dimension and finite Markov order; it supplies the optimal universal bound within that scope rather than deciding whether an arbitrary sofic process has finite Markov order.

Community scope and artifact audit

Sharp bound claimed and Lean checked

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra (runtime provenance not retained)
Verification: Pinned Lean proof with claim map; specialist review pending
Claim audit: Issue documented

5 Sep 2026Probability & statistics

Ibragimov--Iosifescu $\varphi$ -mixing central-limit conjecture

A strictly stationary, centered, square-integrable $\varphi$ -mixing process has diverging partial-sum variance but a normalized subsequence converging to zero in probability, contradicting the claimed central limit theorem and the stronger invariance-principle conjecture.

Details and sources

AI contribution

Epoch's autonomous run generated the counterexample and approximately 13,000-line Lean development without human steering; Tom Adamczewski published the artifacts.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Lean/Comparator checked; statement semantics not independently audited

Lean proof and audit artifacts

Claim audit

The benchmark statement was AI-autoformalized. No independent probabilist has audited the formal definitions or the long construction.

Publication

Public Lean disproof repository

Preprint / manuscript

No public preprint or manuscript located.

The public audit reports zero sorry outside statement stubs, no added axioms or unsafe features, a pinned environment, and only the three standard foundational axioms. Mathematical acceptance remains provisional pending semantic and specialist review.

Problem-history source Independent repository audit

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-6 Astra (pre-release)
Verification: Lean/Comparator checked; statement semantics not independently audited
Claim audit: Issue documented

27 Aug 2026Probability & statistics

Lyons–White rate-monotonicity conjecture

Symmetric continuous-time walks on dihedral groups have rate-monotone distance from uniform at every positive even exponent. The paper also constructs failures at all other finite real exponents at least one and extends the positive result to inversion extensions.

Details and sources

AI contribution

Colin Defant and Ken Ono developed the proofs and their Lean formalization in dialogue with AxiomProver.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Lean certificate with imported literature assumptions; author proof

Lean development and formal challenge

Publication

Public research preprint

Preprint / manuscript

Source package and ancillary formalization

The formalization explicitly assumes standard analysis and group-theory inputs. Its README states the converse for $p>1$ ; the manuscript additionally treats $p=1$ . The repository's reported Comparator check was not rerun here.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: AxiomProver (Axiom Math)
Verification: Lean certificate with imported literature assumptions; author proof

26 Aug 2026Probability & statistics

Alon–Peres birthday paradox for fixed-degree non-backtracking walks

For every fixed degree $d\geq3$ and error tolerance, a non-backtracking walk on any $n$ -vertex $d$ -regular graph self-intersects with high probability after $C(d,\epsilon)\sqrt n$ steps.

Details and sources

AI contribution

GPT found the cubic case and central proof ideas. Benjamin Dozier generalized them to all fixed degrees and wrote the paper; Codex assisted Lean formalization.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author proof; Lean ancillary files reported, not replayed here

Publication

Public research preprint

Preprint / manuscript

The constant depends on the fixed degree. This is not a degree-uniform resolution for arbitrary growing degrees. The degree-two version is false.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra and Codex
Verification: Author proof; Lean ancillary files reported, not replayed here

18 Aug 2026Probability and Khintchine inequalities

Jakimiuk's sharp Gaussian-stability coefficient for Rademacher sums

Problem statement

\mu_p-1

the optimal linear Gaussian-stability coefficient for every real

p\geq3

The proposed coefficient is not valid throughout $2<p<4$ , but it is valid and sharp for every $p\geq4$ ; equality and stability are characterized in the paper.

Details and sources

AI contribution

The authors report that initial proofs of the sharp stability inequality were developed with the model. They checked and revised those arguments and independently verified the extensions.

Problem origin

Jakimiuk conjectured the optimal linear coefficient $\mu_p-1$ in the human probability literature.

Verification

Authors independently verified the complete proofs; independent review pending

Publication

Public proof and counterexample preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on identifying the exact phase transition and replacing a false range with a sharp theorem.

The original range is false, so the $p\geq4$ theorem is recorded together with the disproof rather than presented as an unqualified proof of the original conjecture.

Original

p\geq3

range disproved; corrected

p\geq4

theorem proved

Claimed outcome: ProvedDisproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol (OpenAI)
Verification: Authors independently verified the complete proofs; independent review pending
Open for: Jakimiuk Conjecture 1
Research activity*: 4/5

18 Aug 2026Probability and Khintchine inequalities

Finite-dimensional $L_p/L_4$ Khintchine flat-point conjecture

Problem statement

For

p\geq5

, is the finite-dimensional

L_p/L_4

ratio of a Rademacher sum maximized at the flat coefficient vector?

For every $p\geq5$ , the finite-dimensional $L_p/L_4$ ratio for Rademacher sums is maximized by the flat coefficient vector; the paper proves a stronger strict-monotonicity statement.

Details and sources

AI contribution

The model helped develop the initial finite-dimensional extremal argument. The authors checked, revised, and independently verified the proof.

Problem origin

Barański, Murawski, Nayar, and Oleszkiewicz posed the flat-point maximization conjecture before the present work.

Verification

Authors independently verified the complete proof; independent review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on closing a named finite-dimensional extremal problem with a stronger monotonicity theorem.

The record is scoped to the $p\geq5$ flat-point conjecture proved in the manuscript and does not claim a corresponding statement outside that range.

Barański--Murawski--Nayar--Oleszkiewicz conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol (OpenAI)
Verification: Authors independently verified the complete proof; independent review pending
Open for: Barański--Murawski--Nayar--Oleszkiewicz flat-point conjecture
Research activity*: 3/5

18 Aug 2026Probability and Khintchine inequalities

Dimension-free quadratic stability at the third Rademacher moment

Problem statement

Does a dimension-free quadratic stability inequality hold for the third absolute moment of Rademacher sums?

The conjectured dimension-free quadratic stability inequality at $p=3$ holds with the explicit constant $1/100$ .

Details and sources

AI contribution

The model assisted the paper's initial proof development. The authors checked and revised the argument and independently verified the final theorem.

Problem origin

Jakimiuk posed the dimension-free third-moment quadratic-stability conjecture in the human probability literature.

Verification

Authors independently verified the complete proof; independent review pending

Publication

Public proof preprint

Preprint / manuscript

Revised research manuscript

Activity evidence

A documented editorial estimate based on closing a dimension-free stability conjecture with an explicit quantitative bound.

The theorem proves the requested dimension-free quadratic behavior with an explicit, not claimed optimal, constant.

Jakimiuk Conjecture 2 proved with an explicit constant

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol (OpenAI)
Verification: Authors independently verified the complete proof; independent review pending
Open for: Jakimiuk Conjecture 2
Research activity*: 3/5

17 Aug 2026High-dimensional probability and quantized embeddings

Subgaussianity of sign-quantized Gaussian linear maps

Problem statement

Does

Y=\operatorname{sgn}(Wx)

have subgaussian norm bounded independently of dimension when

W

and

x

are Gaussian?

The one-bit vector $Y=\operatorname{sgn}(Wx)$ has dimension-free subgaussian norm in the stated Gaussian setting, via a more general bounded-differences inequality depending on $\sqrt{\kappa(\Sigma)}$ .

Details and sources

AI contribution

Gemini suggested covariance splitting and Gaussian smoothing for the sign case and extended it to bounded coordinate maps. The authors later found that the argument closely resembles Barber--Kolar Lemma 4.5; they supplied the row-partitioning step and revised the attribution.

Problem origin

Simone Bombari posed the sign-quantized subgaussianity question to the authors before their AI-assisted exploration.

Verification

Author-checked proof with public conversations; independent review pending

Claim audit

Version 2 corrects the missing prior-art attribution: Gemini did not cite Barber and Kolar's closely related 2018 covariance-splitting proof.

Publication

Public revised proof preprint with corrected attribution

Preprint / manuscript

Activity evidence

A documented editorial estimate reflecting a useful positive answer and generalization, reduced because the AI-suggested mechanism closely tracks prior literature.

The positive answer and bounded-differences formulation are retained, but the core AI-suggested proof strategy is explicitly a rediscovery of close prior work rather than a wholly new argument.

Translated conversation timeline Original Gemini conversation

Bombari question answered; AI proof route corrected to prior-art rediscovery

Claimed outcome: Proved
Problem origin: Human-source problem
System: Gemini 3.5 Flash (Google)
Verification: Author-checked proof with public conversations; independent review pending
Claim audit: Issue documented
Open for: Question posed by Simone Bombari
Research activity*: 2/5

16 Aug 2026Causal discovery and minimax statistics

Sharp minimax sample complexity for bivariate LiNGAM

Problem statement

What is the sharp sample complexity for determining the causal direction in the bivariate linear non-Gaussian acyclic model?

Under the paper's sub-Gaussian, scale, and non-Gaussianity conditions, the local minimax sample complexity for recovering the causal direction is $N_2^*\asymp \log(1/\delta)/(d_\beta^2+\beta^2\nu^2)$ , with matching upper and lower bounds.

Details and sources

AI contribution

The author states that GPT-5.6 Sol generated the proof in roughly two hours after receiving a human-written prompt. The author checked the mathematics and revised and polished the manuscript.

Problem origin

The manuscript identifies the sharp sample complexity of causal-direction recovery as a pre-existing gap in the LiNGAM and causal-discovery literature.

Verification

Author checked; public proof preprint; independent review pending

Publication

Public preprint with complete upper and lower bounds

Preprint / manuscript

Dimension-free consistency preprint

Activity evidence

A documented editorial estimate based on closing a clearly identified minimax gap with matching bounds, tempered by the restricted bivariate setting and pending independent review.

The result closes the sharply scoped bivariate local-minimax question under the stated assumptions. It does not settle general multivariate LiNGAM sample complexity, which the paper leaves for future work.

Sharp bivariate minimax law proved; multivariate theory remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol via Codex Ultra (OpenAI)
Verification: Author checked; public proof preprint; independent review pending
Open for: A precise gap in the causal-discovery literature
Research activity*: 4/5

7 Aug 2026High-dimensional statistics and signal detection

Polynomial-time exact recovery at the square-Gaussian MIMO threshold

Problem statement

Can an explicit polynomial-time detector attain the first-order

2\log N

maximum-likelihood exact-recovery threshold in the square Gaussian binary-MIMO model?

Rounded LMMSE followed by steepest single-bit descent is claimed to recover uniformly at signal-to-noise ratio $\rho\ge2\log N$ in $O(N^3)$ operations, matching the first-order maximum-likelihood threshold.

Details and sources

AI contribution

The manuscript states that the two models proved the result and drafted the initial paper. The author posed the problem, directed proof simplification, checked the mathematics, edited the manuscript, and takes responsibility.

Problem origin

The author posed the algorithmic threshold question against the established maximum-likelihood and convex-relaxation literature.

Verification

Author-checked manuscript; no independent review or formalization located

Claim audit

The only located manuscript is author-hosted rather than an archival preprint, and no independent specialist review was located.

Publication

Public author-hosted manuscript and discussion post

Preprint / manuscript

Author-hosted manuscript

Activity evidence

A documented editorial estimate based on the mature MIMO-detection literature and the still-unreviewed status of this candidate.

The theorem concerns a specific square real Gaussian binary-MIMO model and a unit-cost exact-real arithmetic model. It claims a first-order threshold, not the complete lower-order transition window.

Author discussion post

Author-hosted candidate theorem

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 / Claude Fable 5
Verification: Author-checked manuscript; no independent review or formalization located
Claim audit: Issue documented
Research activity*: 3/5

17 Jul 2026High-dimensional nonparametric statistics

Dimension-free consistency of empirical spatial depth

Problem statement

Can uniform

L^1

consistency of empirical spatial distribution and depth estimators be proved with a rate independent of the ambient dimension?

The empirical spatial distribution estimator in $\mathbb R^d$ and its plug-in spatial-depth estimator are uniformly $L^1$ -consistent at a rate depending only on the sample size, not on the ambient dimension or tuning parameters.

Details and sources

AI contribution

The authors state that the result originated in a conversation with ChatGPT 5.4 Pro during experiments on mathematical reasoning; they supplied the public mathematical proof.

Problem origin

The authors say the result originated in a mathematical-reasoning conversation with ChatGPT 5.4 Pro rather than in a previously named open problem.

Verification

Author checked; independent review pending

Publication

Public author-written preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on ongoing work in high-dimensional depth functions and distribution-free nonparametric statistics.

The dimension-invariance is the distinguishing advance. This is recorded as a theorem-level research contribution, not as the resolution of a named longstanding problem.

Dimension-invariant consistency theorem proved

Claimed outcome: Proved
Problem origin: Human--AI co-generated
System: ChatGPT 5.4 Pro
Verification: Author checked; independent review pending
Open for: Dimension-robust statistical estimation question
Research activity*: 2/5

30 Jan 2026Random polynomials and probability

Erdős Problem #524 — random Littlewood-polynomial maxima

Problem statement

For i.i.d. Rademacher signs

a_k(t)=(-1)^{\epsilon_k(t)}

, determine the almost-sure order of

M_n(t)=\max_{x\in[-1,1]}\left|\sum_{k=1}^n a_k(t)x^k\right|.

An AI-assisted write-up obtained the sharp almost-sure limsup law and the correct stretched-exponential scale for the lower envelope. It did not determine the exact lower-envelope constant; independent human work subsequently proved the stronger exact result.

Details and sources

AI contribution

After Mehtaab Sawhney outlined the key Gaussian-process reduction, GPT-5.2 completed most of the argument with smaller inputs from Gemini and Grok.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Specialist checked; stronger paper followed

Publication

Public Erdős Problems discussion, community AI ledger, and a later stronger arXiv proof

Preprint / manuscript

Later stronger human proof

Activity evidence

A Salem–Zygmund problem with classical partial bounds, a detailed 2026 community discussion, and a subsequent sharp human proof.

This remains a partial AI contribution, not an AI resolution of the full problem. Sawhney confirmed the stated partial argument, while joint work with Brayden Letwin later determined the exact lower-envelope constant.

Primary problem record AI contribution and expert discussion Community AI ledger

Correct partial result

Problem origin: Human-source problem
System: GPT-5.2 / Gemini / Grok
Verification: Specialist checked; stronger paper followed
Open for: 72 years
Research activity*: 3/5

12 May 2026Entropy and heat flow

Gaussian completely monotone conjecture

Problem statement

Must every time derivative of entropy along the heat flow obey the alternating-sign inequalities predicted by the Gaussian completely monotone conjecture?

Gu and Sellke exhibit an explicit probability measure on the real line whose fifth entropy derivative along heat flow has the forbidden sign. The example also refutes the associated McKean and Toscani conjectures.

Details and sources

AI contribution

The paper states that GPT-5.5 Pro found the explicit counterexample; the authors supplied and checked the proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked preprint

Publication

Complete public arXiv proof

Preprint / manuscript

Activity evidence

The conjecture sits inside a long, internationally active program on entropy, Fisher information, and Gaussian optimality; one consequence had remained open since 1966.

The direct target dates to Cheng and Geng’s 2015 conjecture. Its failure also settles a consequence of McKean’s 1966 Gaussian-optimality proposal.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author-checked preprint
Open for: 11 years
Research activity*: 4/5

18 May 2026Information theory

Log-convexity of Fisher information along heat flow

Problem statement

For every smooth positive density

f

\mathbb{R}^d

, must the Fisher information

t\mapsto I(f*\gamma_t)

be log-convex along the heat flow?

A smooth positive Gaussian-decaying density on the plane gives a hexagonal counterexample. Tensorization disproves the Cheng–Geng conjecture in every dimension at least two.

Details and sources

AI contribution

The authors report that GPT-5.5 Pro found the explicit two-dimensional counterexample.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Proof with explicit numerics

Publication

Public 28-page arXiv preprint

Preprint / manuscript

Activity evidence

A decade-old specialist conjecture embedded in an active information-theory literature, with less broad attention than the parent Gaussian-optimality program.

The 31 August v3 retains the hexagonal counterexample and adds a study of dimension-dependent sharp constants, including the one-dimensional value, dimension monotonicity, and an asymptotic dichotomy. The construction complements the separate one-dimensional entropy counterexample. No new independent formal verification is claimed.

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Proof with explicit numerics
Open for: 11 years
Research activity*: 3/5

10 Jun 2026Discrete probability and extremal combinatorics

Weighted Bernoulli sums above their mean

Problem statement

For nonnegative weights

w_i

with

\sum_i w_i=1

and independent

v_i\sim\mathrm{Bernoulli}(p)

, for which

p

does

\Pr[\sum_i w_i v_i\geq p]\geq p

hold for every choice of weights?

ProofCouncil correctly established several infinite families and counterexample ranges. Referees judged the mathematics genuinely novel and requested only minor revisions, while the full classification remains tied to difficult Manickam–Miklós–Singhi-type questions.

Details and sources

AI contribution

The multi-model harness discovered a new pairing argument for the family $p=2/k$ with $k\geq6$ and correctly handled several other parameter ranges.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Double-blind expert review; minor revisions

Publication

First Proof Second Batch report, complete submission, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

The question connects to the long-running Manickam–Miklós–Singhi conjecture and modern work on weighted Bernoulli sums.

This is recorded as partial rather than resolved because the broad 'for which values of p?' classification is not complete. The referee report nevertheless identifies genuinely new correct mathematics.

Novel partial families proved

Claimed outcome: ProvedDisproved
Problem origin: Human-source problem
System: ProofCouncil (primarily GPT-5.5 Pro; auxiliary Gemini and Claude)
Verification: Double-blind expert review; minor revisions
Open for: Research classification remains open
Research activity*: 4/5

27 Jul 2026Small-deviation probability inequalities

Feige’s $1/e$ conjecture

Problem statement

Must

\mathbb P(S<\mathbb ES+1)\geq1/e

hold for every sum of independent nonnegative random variables with

\mathbb E X_i\leq1

For independent nonnegative $X_i$ with $\mathbb E X_i\leq1$ , the authors prove $\mathbb P(S<\mathbb ES+1)\geq(n/(n+1))^n\geq e^{-1}$ . Their broader theorem is sharp for every slack $\delta\geq1$ .

Details and sources

AI contribution

GPT-5.6 Pro found a proof combining exact Dirichlet calibration with Grünbaum-type centroid inequalities. GPT-5.6 Sol independently produced a second proof the same day, and Codex helped build the Lean development.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked end to end; independent preprint

Complete Lean development

Publication

Two public arXiv manuscripts and a complete public Lean repository

Preprint / manuscript

Primary proof manuscript Independent proof manuscript

Activity evidence

Feige posed the conjecture in 2004; it has a sustained literature in probabilistic inequalities and approximation algorithms.

The standard unit-slack conjecture and the authors’ $\delta\geq1$ extension are settled. The proposed sharp formula for $0<\delta<1$ is not proved. The Lean repository reports no sorry, admit, or custom axioms.

Author announcement and provenance

Unit-slack conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.6 Pro / GPT-5.6 Sol / Codex
Verification: Lean checked end to end; independent preprint
Open for: 22 years
Research activity*: 4/5

25 Jul 2026Asymptotic convex geometry

Sharp thin-shell inequality for log-concave measures

Problem statement

What is the optimal universal constant in the thin-shell variance bound for isotropic log-concave probability measures?

For every centered isotropic log-concave $X\in\mathbb R^n$ , the paper proves $\operatorname{Var}(|X|^2)\leq8n$ . Equality is attained by centered exponential coordinates, and the regular simplex is extremal among uniform measures on convex bodies.

Details and sources

AI contribution

The model found the proof from prompts about log-concave moment measures and a second-trace bootstrap. Chen and Klartag verified, revised, and rewrote the argument.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Two-author proof with public model transcript

Publication

Public arXiv manuscript and ancillary ChatGPT conversation

Preprint / manuscript

Activity evidence

Thin-shell estimates are a major theme in asymptotic convex geometry and connect directly to KLS and high-dimensional sampling.

The theorem determines the sharp constant for the stated isotropic thin-shell variance inequality. It has not yet been peer reviewed or formally checked.

Model transcript

Optimal universal constant determined

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT-5.6 Pro
Verification: Two-author proof with public model transcript
Research activity*: 5/5

3 Jul 2026Reflected Brownian motion and stationary measures

Signed BAR conjecture

Problem statement

Does the finite signed basic adjoint relation determine the invariant signed measure uniquely, and how far beyond the Harrison–Reiman class can uniqueness extend?

Finite signed basic adjoint-relation data are unique for stable Harrison–Reiman systems with a nonsingular $M$ -matrix reflection matrix. The paper also gives an infinite-dimensional obstruction in the larger completely- $\mathcal S$ class.

Details and sources

AI contribution

The authors state that the proof was discovered with ChatGPT-5.5 Pro and then independently checked and developed by them.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two-author verification; public proof

Publication

Public arXiv manuscript and TeX source; peer review pending

Preprint / manuscript

Activity evidence

A decades-old structural question in reflected diffusions, queueing networks, and stationary-measure theory.

The positive theorem is precisely scoped to the Harrison–Reiman nonsingular $M$ -matrix class. The same universal statement is false in a natural completely- $\mathcal S$ extension.

Conjecture resolved in the Harrison–Reiman class

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.5 Pro
Verification: Two-author verification; public proof
Open for: Dai–Dieker question open for more than 35 years
Research activity*: 4/5

13 Jul 2026Random polynomials

Erdős Problem #521 — real roots of random Littlewood polynomials

Problem statement

For iid random signs, does the number

R_n

of real roots of

\sum_{k=0}^n\varepsilon_kz^k

satisfy

R_n/\log n\to2/\pi

almost surely?

Almost surely, the full-line real-root count $R_n/\log n$ does not converge to $2/\pi$ ; fluctuations from roots outside $[-1,1]$ destroy the proposed strong law.

Details and sources

AI contribution

Star Fleet independently reconstructed a Lean proof after Brayden Letwin, Terence Tao, and Vjekoslav Kovač had obtained the result first; the release credits them explicitly.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked; prior human result credited

Official problem record Community discussion Star Fleet report

Publication

Public source audit and complete Lean verification bundle

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

Documented editorial estimate based on the primary problem record, cited literature, and public review activity; not a difficulty rating.

The in-expectation asymptotic remains true, and a separate almost-sure law inside $[-1,1]$ is unaffected. This record tracks formalization, not mathematical priority.

Known negative answer independently formalized

Problem origin: Human-source problem
System: GPT-5.6 Star Fleet / Claude Fable 5 referee
Verification: Lean checked; prior human result credited
Research activity*: 4/5

20 Jul 2026Gaussian moments and polynomial maps

Gaussian Moments Conjecture in dimensions $n\geq3$

Problem statement

If all positive moments of a complex polynomial in independent standard Gaussian variables vanish, must every polynomially weighted moment eventually vanish as well?

Explicit complex polynomials $P,Q$ in three independent real Gaussian variables satisfy $\mathbb E(P^m)=0$ and $\mathbb E(QP^m)=m!\neq0$ for every $m\geq1$ . The same witnesses disprove the conjecture in every higher dimension.

Details and sources

AI contribution

ChatGPT autonomously produced the first four-variable construction after the initial prompt. Given that example, Claude found the smaller three-variable construction and supplied independent algebraic checks.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked explicit moment identities

Publication

Public arXiv manuscript and readable source; peer review pending

Preprint / manuscript

Contemporaneous expert discussion Original Gaussian Moments Conjecture preprint

Activity evidence

The conjecture connects Gaussian moment problems with the Image and Jacobian conjectures and has an active algebraic-probability literature.

The global conjecture is false, and every dimension at least three is settled negatively. Dimension one is known to be true; dimension two remains open.

Disproved in every dimension

n\geq3

;

n=2

remains

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT-5.6 Sol Pro / Claude Fable 5
Verification: Author-checked explicit moment identities
Open for: 9 years
Research activity*: 4/5

30 Jul 2026Gaussian moments, algebraic tori, and invariant theory

Gaussian Moments Conjecture in dimension two

Problem statement

Does the Gaussian Moments Conjecture hold for complex polynomials in two independent real Gaussian variables?

A public author manuscript claims a factorially weighted constant-term theorem on algebraic tori and uses an explicit embedding of the two-variable Gaussian functional to prove the remaining $n=2$ Gaussian Moments Conjecture and its orthogonal-nullcone classification.

Details and sources

AI contribution

Christopher D. Long states that the results were generated interactively with ChatGPT-5.6 Sol, which also assisted drafting and editing, while Claude Fable 5 supplied additional auditing and comments.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-posted manuscript; no independent verification yet

Claim audit

The author’s own disclaimer says the manuscript has not yet been human verified or formalized; independent mathematical review is required.

Publication

Public GitHub manuscript and LaTeX source; no arXiv record located

Preprint / manuscript

Public manuscript PDF Second public proof manuscript

Activity evidence

A documented editorial estimate based on the conjecture’s links to moment problems, invariant theory, and the Image and Jacobian conjectures.

The repository explicitly says the result has not yet been human verified or formalized. A second arXiv manuscript now presents a proof, but its independence from the earlier public artifact is undocumented and it has not supplied independent specialist verification. This entry records the claims rather than endorsing either proof.

Statement, source files, and verification disclaimer Author announcement and discussion

Claimed proof; independent verification pending

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.6 Sol / Claude Fable 5
Verification: Author-posted manuscript; no independent verification yet
Claim audit: Issue documented
Open for: 9 years
Research activity*: 4/5

29 Jul 2026Particle-swarm optimization and random recurrences

Particle-swarm stagnation law is non-Gaussian

Problem statement

Can the limiting position distribution of the canonical one-dimensional particle-swarm optimizer during stagnation be Gaussian?

Throughout the open mean-square stability region for the one-dimensional stagnant particle-swarm recurrence, no invariant—and hence no limiting—position marginal can be Gaussian. Exact moment identities through order eight eliminate every candidate parameter pair.

Details and sources

AI contribution

ChatGPT assisted symbolic-computation scripting, manuscript organization, and language revision. The authors supplied the mathematical ideas and independently reviewed every formula and certificate with exact executable tests.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three-author proof and independent exact modular certificates

Exact verification code and logs

Publication

Public revised arXiv manuscript, exact polynomial artifacts, code, and logs

Preprint / manuscript

Research paper, revised 29 July

Activity evidence

A documented editorial estimate based on the longstanding theoretical PSO literature and the newly formalized community problem list.

This solves the specifically stated Bell-Curve Problem 18 under the paper’s fixed-attractor and equal-uniform-acceleration assumptions, not every stochastic particle-swarm model.

Open Problem 18 answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI ChatGPT (model not disclosed)
Verification: Three-author proof and independent exact modular certificates
Open for: Problem 18 in the 2026 Particle Swarm problem list
Research activity*: 3/5

20 Apr 2026Random polynomials

Erdős Problem #522 — zeros of random Littlewood polynomials

Problem statement

P_n(z)=\sum_{k=0}^n\varepsilon_kz^k

has independent uniform signs, does the number

R_n

of roots in

|z|\leq1

satisfy

R_n/(n/2)\to1

almost surely?

The manuscript proves the strong law $R_n/(n/2)\to1$ almost surely and gives the quantitative estimate $R_n=n/2+O_\omega(n^{149/150})$ .

Details and sources

AI contribution

Przemek Chojecki’s public research index credits GPT-5.5 Pro with the claimed solution. This record follows the scope and caveats stated in the manuscript and community discussion.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Public manuscript; full expert review not located

Publication

Public proof manuscript and an open Erdős Problems discussion record

Preprint / manuscript

Primary problem record Community discussion Ulam research index

Activity evidence

The official problem page has 24 discussion comments and records prior in-probability progress, making the claimed strong-law upgrade a well-defined and actively discussed target.

The Erdős Problems page still labels the problem open and records this as a claimed solution. The result strengthens the previously known convergence in probability to almost-sure convergence.

Claimed full resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Public manuscript; full expert review not located
Open for: Asked by 1961
Research activity*: 4/5

15 May 2026Probability on groups

Return probability for a lamplighter walk

Problem statement

For the switch–walk–switch lamplighter walk on

\mathbb{Z}_2\wr T_d

, prove the sharp asymptotic

p_{2n}(e,e)=\rho_d^{2n}\exp\!\left[-\bigl(\pi^2(\log(d-1))^2+o(1)\bigr)\frac{n}{\log^2 n}\right],\qquad \rho_d=\frac{2\sqrt{d-1}}{d}.

QED derived the sharp return-probability asymptotic for the switch–walk–switch walk on the lamplighter group over a regular tree. The contributing probability expert verified the proof.

Details and sources

AI contribution

A fully automatic decomposition, proof, and verification run used Codex with GPT-5.5 Pro and received no mathematical input beyond the statement.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Domain expert verified

Proof and expert record

Publication

Dedicated arXiv paper, public proof record, and expert comments

Preprint / manuscript

Research paper QED system paper

Activity evidence

An expert-contributed active-research question with a small specialist audience; its proof was judged a publishable, substantive probability result.

The expert assessed this as a solid specialist contribution comparable to work in the Electronic Journal of Probability or Proceedings of the AMS.

Open problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: QED / GPT-5.5 Pro
Verification: Domain expert verified
Open for: Under 1 year
Research activity*: 2/5

15 May 2026Probability on groups

Total variation for a lamplighter walk

Problem statement

For the switch–walk–switch walk on

\mathbb{Z}_2\wr\mathbb{Z}

, with starts

x=(\mathbf{0},0)

and

y=(\mathbf{0},2)

, prove

\lVert P_t^x-P_t^y\rVert_{\mathrm{TV}}\asymp t^{-1/2}.

QED proved the sharp order t^{-1/2} for the total-variation distance between two switch–walk–switch laws started two sites apart on the lamplighter group over the integers.

Details and sources

AI contribution

The multi-agent system autonomously changed proof plans over several rounds before producing the expert-accepted argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Domain expert verified

Proof and expert record

Publication

Public problem, final proof, workflow, and expert comments

Preprint / manuscript

QED system paper

Activity evidence

A newly contributed active-research problem with real technical depth, but no evidence of a large pre-existing research program.

The contributor described this as a technically nontrivial PhD-level probability problem.

Open problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: QED / GPT-5.5
Verification: Domain expert verified
Open for: Under 1 year
Research activity*: 2/5

3 Sep 2025Probability theory

Quantitative two-chaos fourth-moment theorem

Problem statement

Can the qualitative fourth-moment theorem for sums of two Wiener–Itô integrals of different parity be strengthened to an explicit total-variation rate depending only on the fourth cumulant, and can an analogous theorem be established on Poisson space?

GPT-5 helped turn a qualitative fourth-moment theorem into an explicit total-variation convergence bound and extend the analysis to Poisson chaos, including a counterexample showing the added conditions are essentially necessary.

Details and sources

AI contribution

The model supplied the main proof structure and calculations after iterative correction and targeted hints from the authors.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Expert authors checked

Publication

Full arXiv paper with proofs and screenshots of both sessions

Preprint / manuscript

Research paper and interaction annex

Activity evidence

The authors say the precise quantitative question had simply not been addressed before; it was not a longstanding focus of active competition.

The authors explicitly describe this as incremental and heavily human-guided: GPT-5 made a serious error in the Gaussian proof and missed a key positivity fact in the Poisson case until directed to it.

Quantitative extension proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5
Verification: Expert authors checked
Open for: Previously unstudied
Research activity*: 1/5

21 Jul 2026Probability and statistics

Strong log-concavity of Chernoff’s density

Problem statement

Is the density f of argmaxₜ{W(t)−t²}, for two-sided Brownian motion W, strongly log-concave?

A concise proof establishes that the density of the Chernoff random variable is strongly log-concave, resolving the 2014 conjecture of Balabdaoui and Wellner.

Details and sources

AI contribution

The authors state that the proof was generated by GPT-5.6 Sol through several interactions. A first attempt contained a fatal algebraic error; a second produced a correct proof, and a later run found the streamlined strategy. The authors carefully checked the argument and revised the exposition.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked; preprint public

Publication

Public arXiv note; journal review pending

Preprint / manuscript

Activity evidence

A documented 2014 conjecture in an active specialist area, with focused rather than field-wide attention.

This is a compact specialist result in shape-constrained probability. The paper's unusually detailed disclosure records a failed first attempt before the checked proof, a useful example of why the index distinguishes public argument and human checking from raw model output or a proof-assistant certificate.

Conjecture proved in preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author checked; preprint public
Open for: 12 years
Research activity*: 3/5

13 Jul 2026Statistics theory

Benjamini–Hochberg under correlated Gaussian tests

Problem statement

Does the Benjamini–Hochberg procedure always control false-discovery rate at its nominal level for correlated two-sided Gaussian p-values?

A correlated Gaussian factor model gives false-discovery rate above the nominal level, with a rigorous interval-arithmetic certificate valid for all sufficiently large numbers of hypotheses.

Details and sources

AI contribution

Edgar Dobriban reports that GPT-5.6 Pro obtained the proof, which he then checked carefully.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked + interval certificate

Publication

Public arXiv preprint; journal review pending

Preprint / manuscript

Preprint and certificate

Activity evidence

A widely believed question about a foundational multiple-testing procedure, with sustained work on dependence conditions.

The counterexample is narrower than a failure of the Benjamini–Hochberg procedure in every dependence model: it concerns correlated two-sided Gaussian tests.

Twenty-year conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Pro
Verification: Author checked + interval certificate
Open for: 20 years
Research activity*: 4/5

Quantum information & computing

5 Sep 2026Nonlocality and Bell inequalities

$I_{3322}$ Bell inequality requires infinite dimensions

No finite-dimensional quantum strategy attains the maximal violation of $I_{3322}$ , while an infinite-dimensional strategy does attain it.

Details and sources

AI contribution

Andrea Coladangelo reports that GPT 5.5 Pro produced an approximate proof containing the main ideas. He substantially revised it for correctness and completeness, with further help from GPT-5.6 Sol.

Problem origin

Pál and Vértesi conjectured the infinite-dimensional-attainment phenomenon for $I_{3322}$ in 2010.

Verification

Author-revised 78-page preprint; external review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

This concerns attainment of the exact $I_{3322}$ quantum value, not a finite experimental implementation or a general statement about every Bell inequality.

Pál--Vértesi conjecture claimed proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT 5.5 Pro + GPT-5.6 Sol
Verification: Author-revised 78-page preprint; external review pending

2 Sep 2026Quantum information & computing

Depth-one distinctness for pseudorandom unitaries

A tensor product of independent single-qubit Clifford gates is negligibly distinct for polynomially many queries, refuting the authors' conjecture that negligibly distinct ensembles must be entangling.

Details and sources

AI contribution

The paper states that the model proposed Proposition III.5 as a counterexample. Other ChatGPT versions assisted proof strategy and checking; the authors independently verified all proofs.

Problem origin

The target was a human working conjecture of the paper's authors; it was not generated by the model being credited with the counterexample.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The result also permits the logarithmic-depth global design layer in a PFC construction to be replaced by one layer of local designs. It is a new internal conjecture resolution, not a long-standing named problem.

Claimed full resolution by counterexample

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Author-checked proof; independent review pending

2 Sep 2026Quantum information & computing

Quantum-oracle separation of $\mathsf{QMA}(2)$ from $\mathsf{QMA}$

A unitary oracle separates $\mathsf{QMA}(2)$ from $\mathsf{QMA}$ ; the argument also proves that every fixed-error disentangler below total error one needs exponentially many input qubits.

Details and sources

AI contribution

The authors state that the model generated the main proof idea from the She--Yuen framework with minimal additional guidance; they verified, simplified, and developed the final argument.

Problem origin

The oracle-separation question and Watrous's no-disentanglers conjecture predate the model-generated proof idea.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

The full oracle separation and the non-oracle no-disentanglers consequence are distinct claims from one proof and are indexed together. Independent peer review remains pending.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Author-checked proof; independent review pending

3 Sep 2026Quantum information & computing

Exact classical simulation of quantum communication and Bell correlations

The preprint proves that no finite classical message alphabet can exactly simulate quantum communication in dimension at least four, nor all Bell correlations of two entangled systems in those dimensions. It also gives a finite 357-bit protocol for qutrit communication and hence qutrit correlations.

Details and sources

AI contribution

After the authors reduced the problem to state exclusion, model dialogues suggested core ideas for the four-dimensional impossibility proof and an early qutrit protocol. The authors independently completed, generalized, and checked the arguments.

Problem origin

The paper reviews decades of work and states that finiteness of the exact classical simulation cost above the qubit case had remained unknown.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This resolves the finite-versus-infinite question in every dimension, not the exact minimum qutrit cost: 357 bits is an upper bound. Unlimited shared randomness is allowed in the impossibility result. No public formalization, code, or full model transcript was located.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 and GPT-5.6 Sol
Verification: Author-checked proof; independent review pending

20 Aug 2026Quantum shadow tomography and graph coloring

Fractional-coloring conjecture for triply efficient Pauli shadow tomography

Problem statement

For every state

\varrho

and threshold

\epsilon

, is the fractional chromatic number of the anticommutation graph induced by

B_{\epsilon}(\varrho)

bounded by

O(\epsilon^{-2})

A family of states and Pauli-observable sets satisfies $\chi_{\mathrm f}(B_{\epsilon_m})=\Omega(\epsilon_m^{-2.07598})$ , so $\chi_{\mathrm f}(B_{\epsilon_m})\epsilon_m^2$ is unbounded and no proposed universal constant can exist.

Details and sources

AI contribution

The authors state that GPT Sol 5.6 derived the two main theorems. They verified the proofs and placed the counterexamples in the shadow-tomography literature.

Problem origin

Conjecture 13 was published by King, Gosset, Kothari, and Babbush in their 2025 paper on triply efficient shadow tomography.

Verification

Authors verified the complete public proofs; independent review pending

Publication

Public counterexample preprint

Preprint / manuscript

Original Conjecture 13 preprint Published original conjecture and DOI record

Activity evidence

A documented editorial estimate based on refuting a published quantum-information conjecture tied to the efficiency of Pauli shadow tomography.

This disproves the conjectured fractional-chromatic bound and therefore this proposed route to a universal triply efficient Pauli-shadow protocol. It does not rule out every possible triply efficient shadow-tomography algorithm.

King--Gosset--Kothari--Babbush Conjecture 13 disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT Sol 5.6 (OpenAI)
Verification: Authors verified the complete public proofs; independent review pending
Open for: King--Gosset--Kothari--Babbush Conjecture 13 (2025)
Research activity*: 4/5

18 Aug 2026Quantum cryptography and monogamy games

Simultaneous Goldreich--Levin reduction for unclonable encryption

Problem statement

Can simultaneous search security be generically upgraded to unclonable indistinguishability using a common-mask Goldreich--Levin reduction?

A common-mask simultaneous Goldreich--Levin reduction upgrades any search-secure unclonable-encryption game to indistinguishability security, proving the simplest one-bit BB84-state scheme secure and resolving the associated monogamy-game conjecture.

Details and sources

AI contribution

The authors report that GPT-5.6 Ultra first returned a correct direct BB84 security proof and then, when asked for another route, discovered the simultaneous Goldreich--Levin reduction. The authors developed the conceptual explanation and manuscript.

Problem origin

The general common-mask search-to-decision reduction and the security of the simplest BB84-state scheme were explicitly open in the unclonable-cryptography literature.

Verification

Author-checked complete proof plus an independent concurrent preprint; peer review pending

Publication

Public 67-page proof preprint

Preprint / manuscript

Research manuscript Independent information-theoretic proof

Activity evidence

A documented editorial estimate based on a general reduction that resolves the simplest BB84 construction and applies across unclonable-encryption and monogamy games.

The theorem settles the general reduction and the stated BB84 and coset-state corollaries. An independent 19 August preprint proves a compatible but quantitatively weaker information-theoretic implication and explicitly does not provide an efficient extractor. The main paper's stronger explicit reduction remains the basis of this record.

General search-to-decision reduction and BB84 conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Ultra (OpenAI)
Verification: Author-checked complete proof plus an independent concurrent preprint; peer review pending
Open for: Central search-to-decision question in unclonable cryptography
Research activity*: 5/5

5 Aug 2026Quantum nonlocal games

Maximally entangled states and pseudo-telepathy

Problem statement

Does every bipartite nonlocal game with a perfect entangled strategy also admit one using a maximally entangled state?

An explicit $4\times3$ -question, six-output bipartite game has a perfect entangled strategy but no perfect strategy using any finite-dimensional maximally entangled state.

Details and sources

AI contribution

The author invented the game family and hierarchy modification. Codex searched for a concrete matrix, implemented and ran the semidefinite hierarchy, extracted an exact rational certificate, and formalized the no-go theorem in Lean.

Problem origin

Completeness of maximally entangled states for bipartite pseudo-telepathy is a longstanding human open problem in quantum nonlocality.

Verification

Lean-checked exact rational infeasibility certificate

Lean certificate and exact checker

Publication

Public proof preprint and reproducible Lean certificate

Preprint / manuscript

Construction data and exact verifiers

Activity evidence

A documented editorial estimate based on the problem's longstanding status and its foundational role in quantum nonlocality.

The formal theorem covers all finite-dimensional maximally entangled strategies with arbitrary six-outcome POVMs. It does not claim the analogous result for arbitrary infinite-dimensional tracial von Neumann algebras.

Completeness question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: OpenAI Codex
Verification: Lean-checked exact rational infeasibility certificate
Research activity*: 5/5

6 Aug 2026Quantum error correction and entanglement

Five new absolutely maximally entangled-state parameter cells

Problem statement

Do absolutely maximally entangled states exist for

(n,q)=(12,5),(18,11),(18,13),(17,11)

, and

(17,13)

Exact Hermitian self-dual MDS codes establish $\operatorname{AME}(12,5)$ , $\operatorname{AME}(18,11)$ , and $\operatorname{AME}(18,13)$ ; one-party projection also establishes $\operatorname{AME}(17,11)$ and $\operatorname{AME}(17,13)$ .

Details and sources

AI contribution

Under the authors' direction, the models designed and implemented the searches and verifiers, proposed the three candidate matrices, and assisted with the manuscript and bibliography. The authors report that the first cell was found within three hours and the other four during manuscript preparation.

Problem origin

Existence of absolutely maximally entangled states is a parameter-by-parameter classification problem maintained in the Huber--Wyderka and quantum-code tables.

Verification

Author-audited proof plus independent exact-arithmetic certificates

Publication

Public preprint, construction data, search material, and stand-alone verifiers

Preprint / manuscript

AME status table used by the authors

Activity evidence

A documented editorial estimate based on the maintained parameter tables and the connections to quantum MDS codes, secret sharing, perfect tensors, and quantum error correction.

The public verifier package rechecks Hermitian self-duality, every required square minor, and graph-state cut ranks in exact arithmetic. This settles existence for the five listed cells, not the remaining open cells or local-equivalence classification.

Five existence cells settled

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5 / ChatGPT 5.6 Sol
Verification: Author-audited proof plus independent exact-arithmetic certificates
Open for: Open cells in the AME parameter tables
Research activity*: 4/5

4 Aug 2026Quantum cryptography and complexity

Perfectly complete many-round key agreement in the QROM

Problem statement

Can an arbitrary-round, perfectly complete QCCC key-agreement protocol be secure in the quantum random-oracle model?

Every finite-round, perfectly complete quantum-computation/classical-communication key-agreement protocol in the Boolean-output QROM admits a classical-query eavesdropper using $O((q_A+q_B)^5)$ oracle queries.

Details and sources

AI contribution

The authors report that GPT-5.6 Sol Ultra discovered the proof in a one-shot conversation and drafted a preliminary paper. They independently verified every statement, simplified the argument, and wrote the final manuscript.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Independently checked by the authors; peer review pending

Publication

Public proof preprint

Preprint / manuscript

Impossibility proof preprint

Activity evidence

A documented editorial estimate based on the centrality of black-box key-agreement separations, multiple recent restricted results, and the removal of an unproved conjecture.

The theorem removes both the previous two-round restriction and reliance on the Polynomial Compatibility Conjecture. It assumes perfect completeness, classical communication, independent initial private states, and no preshared entanglement; imperfect completeness remains open.

Arbitrary-round security question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra
Verification: Independently checked by the authors; peer review pending
Open for: Natural arbitrary-round extension of prior QROM impossibility results
Research activity*: 5/5

1 Aug 2026Quantum information and nonlocal games

Quantum parallel repetition for all two-player games

Problem statement

For every finite two-player entangled game

G

with

\omega^*(G)<1

, must

\omega^*(G^{\otimes n})

decay exponentially in

n

Every finite two-player, one-round entangled game with value below one has exponentially decaying value under parallel repetition. The published bound is uniform in the repetition count, with game-dependent decay through the soundness gap and answer alphabets.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for quantum parallel repetition Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on the question’s explicit appearance by 2004, its relation to Raz’s classical theorem, and sustained work on entangled nonlocal games.

The theorem settles the qualitative conjecture for arbitrary finite entangled games. The quantitative exponent is not claimed to be optimal. A public sorry-free solution module accompanies the publisher manuscript, with independent review still pending.

Qualitative conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: 22 years
Research activity*: 5/5

27 Apr 2026Absolutely maximally entangled states

Existence of an $\operatorname{AME}(11,5)$ graph state

Problem statement

Does an absolutely maximally entangled state of eleven local systems of dimension five exist?

An explicit circulant graph over $\mathbb Z/5\mathbb Z$ defines an absolutely maximally entangled state of eleven five-level subsystems, establishing the $\operatorname{AME}(11,5)$ existence subcase.

Details and sources

AI contribution

The merged Formal Conjectures source explicitly attributes the existence proof to a DeepMind prover agent. The contribution is exposed as a concrete construction with a complete Lean proof.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked in upstream Formal Conjectures

Pinned Lean theorem and proof Upstream Formal Conjectures merge

Publication

Merged Formal Conjectures theorem and proof; no associated problem-specific preprint located

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A documented editorial estimate based on the continuing classification program for AME states and their links to quantum error-correcting codes.

This settles one parameter pair inside Open Quantum Problem 35, not the broader classification of which $(n,d)$ admit absolutely maximally entangled states.

Open Quantum Problem 35 source record

Open Quantum Problem 35 subcase Lean checked

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: DeepMind prover agent
Verification: Lean checked in upstream Formal Conjectures
Open for: Open Quantum Problem 35 parameter subcase
Research activity*: 3/5

3 Aug 2026Absolutely maximally entangled states

Explicit circulant construction of an $\operatorname{AME}(11,4)$ state

Problem statement

Can

\operatorname{AME}(11,4)

be realized by an explicit circulant graph-state construction?

The circulant graph over $\mathrm{GF}(4)$ with first row $(0,0,0,1,\omega,\omega,\omega,\omega,1,0,0)$ defines an absolutely maximally entangled state of eleven four-level subsystems.

Details and sources

AI contribution

The contributors describe an experimental computational pipeline that found the construction, but the public record does not identify a specific model. They supplied a complete external Lean proof that was reviewed and merged upstream.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked; upstream review and merge

Pinned complete Lean proof Upstream Formal Conjectures merge

Publication

Merged Formal Conjectures construction, public review discussion, and pinned external proof

Preprint / manuscript

No public preprint or manuscript located.

Activity evidence

A documented editorial estimate based on current work classifying and constructing AME states and associated quantum codes.

The merged source states that existence was already implied by a known quantum code. This is therefore indexed as a new explicit, machine-checked construction rather than the first resolution of the existence subcase.

Review and provenance discussion Open Quantum Problem 35 source record

Explicit construction Lean checked

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Experimental search pipeline (model not documented)
Verification: Lean checked; upstream review and merge
Open for: Explicit-construction variant; existence previously known
Research activity*: 3/5

21 May 2026Quantum optics and graph amplitudes

Four-particle monochromatic quantum graphs for all $D\geq4$

Problem statement

For four particles and local dimension

D\geq4

, can a complete edge-coloured, complex-weighted graph have unit amplitude for every monochromatic inherited colouring and zero amplitude for every nonmonochromatic colouring?

AlphaProof Nexus proves that no complex-weighted monochromatic quantum graph exists with $N=4$ particles in any local dimension $D\geq4$ . The formal development also derives the corresponding real-, integer-, and trinary-weight nonexistence results.

Details and sources

AI contribution

The proof-search system generated the algebraic nonexistence argument and a complete Lean proof for this parameter family.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked

Lean proof and problem audit

Publication

AP Nexus preprint and public Lean proof

Preprint / manuscript

AP Nexus paper Origin paper

Activity evidence

The family belongs to an active graph-theoretic program for constructing high-dimensional entangled photonic states.

This is distinct from the already indexed diagonal family $N=D$ : it fixes $N=4$ and resolves every $D\geq4$ . The full two-parameter classification remains open.

Second infinite parameter family ruled out

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: AlphaProof Nexus
Verification: Lean checked
Open for: Studied since 2017–2018
Research activity*: 3/5

18 May 2026Distributed photonic quantum computing

Non-local photonic SWAP, CNOT, Toffoli, and Fredkin gates

Problem statement

Can essential multiphoton gates be implemented non-locally between spatially separated photons without sending the photons to one location, pre-sharing entanglement, or performing Bell-state measurements?

AI-Mandel and PyTheus produced concrete qubit and qudit gate blueprints in which spatially separated photons need neither pre-shared entanglement nor Bell-state measurements. The constructions instead use path identity and quantum erasure, and include a new teleportation-like mechanism.

Details and sources

AI contribution

AI-Mandel proposed the underlying research idea and translated it into PyTheus searches. PyTheus found the concrete gates; the human authors interpreted and generalized the resulting mechanisms.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Peer-reviewed analytical construction

Publication

Physical Review Research letter with public preprint

Preprint / manuscript

Public preprint AI-Mandel provenance paper

Activity evidence

Non-local gates and distributed quantum information processing are active research areas with direct relevance to quantum networks.

The paper verifies the gate action analytically and gives experimental blueprints. It does not report a laboratory realization, so the record concerns the construction problem rather than hardware validation.

Peer-reviewed paper PyTheus source code

Constructive gate architectures found

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: AI-Mandel multi-agent LLM + PyTheus
Verification: Peer-reviewed analytical construction
Research activity*: 4/5

19 Feb 2026Quantum experiment program synthesis

Meta-solutions for quantum-state experiment families

Problem statement

Can one automatically discover a human-readable program that generates correct quantum-optical experiments for every size in a state family, including families for which no construction rule was previously known?

A transformer trained on synthetic state–experiment pairs generated readable Python programs that construct correct experimental setups for six target quantum-state families. The successful programs extrapolate beyond the training sizes and include previously unknown generalizations.

Details and sources

AI contribution

The model synthesized programs representing construction rules for entire infinite families, rather than optimizing one experimental setup at a time.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Peer reviewed; code and checkpoints released

Publication

Nature Machine Intelligence article with open artifacts

Preprint / manuscript

Public preprint

Activity evidence

Interpretable automated experiment design connects quantum optics, program synthesis, and scientific machine learning.

The authors tested twenty target classes: six were solved perfectly, while several other outputs matched only the initial sizes. This entry records the successful families and does not treat the remaining targets as solved.

Peer-reviewed article Code repository

Six of twenty target families solved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: Meta-design sequence-to-sequence transformer
Verification: Peer reviewed; code and checkpoints released
Research activity*: 4/5

31 Jul 2020Photonic quantum gates

High-dimensional multipartite quantum transformations

Problem statement

How can one manipulate and control the general case of

n

-photon,

d

-dimensional quantum states with experimentally meaningful photonic transformations?

MELVIN uncovered the seed architectures for arbitrary high-dimensional multiphotonic transformations. The resulting scheme encodes a transformation in an ancillary state and uses a new high-dimensional quantum non-demolition measurement to mediate the operation.

Details and sources

AI contribution

MELVIN searched an estimated $10^{30}$ – $10^{40}$ optical setups and exposed the core pattern; the researchers then interpreted and generalized it into the published construction.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Peer-reviewed analytical proposal

Publication

Physical Review Letters paper and public preprint

Preprint / manuscript

Journal record MELVIN implementation

Activity evidence

High-dimensional photonic gates are a central resource problem for quantum communication and information processing.

The construction addresses the previously open general $n$ -photon, $d$ -dimensional transformation setting. Several proposed instances are feasible with contemporary optics, but the paper is a theoretical blueprint rather than a universal laboratory demonstration.

General constructive blueprint proposed

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: MELVIN
Verification: Peer-reviewed analytical proposal
Research activity*: 4/5

24 Jul 2023Quantum retrodiction and communication

High-dimensional Mean King’s Problem

Problem statement

Can the Mean King’s quantum retrodiction puzzle be implemented beyond qubits with a scalable photonic scheme that retains a clear advantage over the best classical strategy?

PyTheus found graph-based optical setups for the Mean King’s Problem in dimensions $D=3,5,7$ . The researchers extracted a general prime-dimensional scheme whose reported success probabilities beat the classical $1/D$ benchmark by more than a factor of two.

Details and sources

AI contribution

The search system produced specific interpretable experiment graphs; the human researchers recognized and generalized their common construction.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Peer-reviewed proposal with numerical analysis

Publication

Optica Quantum article and public preprint

Preprint / manuscript

Peer-reviewed article PyTheus source code

Activity evidence

The Mean King’s Problem is a long-running quantum-information puzzle; the high-dimensional experimental realization is a narrower specialist program.

This solves the experimental-design problem for a useful probabilistic scheme in prime dimensions; it is not a deterministic realization, and the reported setups had not yet been built in the laboratory.

Prime-dimensional experimental scheme found

Problem origin: Origin not yet traced
System: PyTheus
Verification: Peer-reviewed proposal with numerical analysis
Research activity*: 3/5

13 Oct 2022Logic AI and quantum experiment design

SAT synthesis of photonic state-preparation experiments

Problem statement

Can the search for an interpretable photonic experiment that prepares a prescribed quantum state be formulated and solved exactly, rather than relying only on continuous optimization and local minima?

Klaus maps preparation of a target photonic quantum state to Boolean satisfiability, finds interpretable optical graphs, and iteratively removes unnecessary resources. Combining its exact logic search with numerical optimization improved the state-preparation designs studied in the paper.

Details and sources

AI contribution

The system converts physical constraints into SAT clauses, returns exact satisfying designs, and supplies unsatisfiability evidence within the encoded resource class.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Peer-reviewed exact method

Publication

Quantum journal article and public preprint

Preprint / manuscript

Public preprint

Activity evidence

Exact synthesis of photonic experiments is an active bridge between quantum optics, Boolean satisfiability, and automated scientific design.

This is an exact algorithmic resolution of encoded state-preparation instances, not a proof that every quantum state has a feasible experiment under arbitrary laboratory constraints.

Peer-reviewed article

Exact synthesis framework demonstrated

Problem origin: Origin not yet traced
System: Klaus logic AI
Verification: Peer-reviewed exact method
Research activity*: 3/5

21 May 2026Quantum optics and graph amplitudes

Monochromatic quantum graphs in the diagonal family

Problem statement

Can a complete edge-coloured, complex-weighted graph be chosen so that perfect-matching amplitudes equal one for every monochromatic inherited vertex colouring and vanish for every nonmonochromatic colouring? The proved nonexistence family has even

N=D\geq4

AlphaProof Nexus proves nonexistence in the diagonal family $N=D$ for every even $N\geq4$ , alongside additional finite parameter cases. The broader two-parameter problem remains open.

Details and sources

AI contribution

The system generated algebraic nonexistence arguments for several parameter families and formalized them in Lean.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked

Lean artifacts

Publication

AP Nexus preprint and public Lean folder

Preprint / manuscript

AP Nexus paper Origin paper

Activity evidence

A specialist quantum-optics problem with several papers and public problem discussions over roughly eight years.

This settles an infinite diagonal family, not the complete $N,D$ classification sought by the parent quantum-optics problem.

Infinite parameter family ruled out

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: AlphaProof Nexus
Verification: Lean checked
Open for: Studied since 2017–2018
Research activity*: 3/5

27 Jul 2026Quantum information and entropy inequalities

Sharp quantum conditional-entropy continuity bound

Problem statement

What is the optimal uniform continuity bound for quantum conditional entropy in terms of trace distance and the dimension of the conditioned system alone?

For trace distance $\delta$ and $d=\dim A$ , the paper proves the optimal bound $h_2(\delta)+\delta\log(d^2-1)$ up to $\delta=1-d^{-2}$ , followed by the sharp cap $2\log d$ ; the bound is tight when $\dim B\geq d$ .

Details and sources

AI contribution

The model supplied the decisive proof idea by adapting a sharp classical conditional-entropy argument; the five authors checked and developed the final proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Five-author arXiv proof; peer review pending

Publication

Public arXiv manuscript and readable HTML source

Preprint / manuscript

Primary exact verification script

Activity evidence

Sharp entropy-continuity inequalities are foundational tools in quantum information and have an extensive active literature.

This closes the dimension-only continuity problem stated in the manuscript. No proof-assistant formalization or external peer-review report was located.

Sharp dimension-only bound proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.6 Sol
Verification: Five-author arXiv proof; peer review pending
Research activity*: 4/5

28 Jul 2026Classical and quantum information theory

Bipartite bound information exists

Problem statement

Does bipartite classical bound information exist: correlations with positive secrecy cost but zero distillable secret-key rate?

An explicit source with two binary variables and one trit satisfies $S(X;Y\|Z)=0<I(X;Y\downarrow Z)\leq I_{\rm form}(X;Y\mid Z)$ : secret correlation costs secrecy to create although no secret key can be extracted.

Details and sources

AI contribution

The construction emerged from extended exploratory dialogue in which the authors directed the model to test conjectures, search for counterexamples, and check calculations.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Authors reconstructed proof line by line; exact ancillary checks

Publication

Public arXiv paper with five exact verification scripts

Preprint / manuscript

Research paper and ancillary verifiers

Activity evidence

Gisin and Wolf posed the problem in 2001 as the classical analogue of bound entanglement; it became a foundational open question in information-theoretic cryptography.

The standard-basis measurements of the historic Horodecki qutrit candidates are proved not to be examples. Two other historical candidate distributions remain unresolved. The new proof is not yet peer reviewed.

Twenty-five-year existence question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Authors reconstructed proof line by line; exact ancillary checks
Open for: 25 years
Research activity*: 5/5

30 Jul 2026Quantum many-body systems and thermal entanglement

Long-range quantum Gibbs-state separability threshold

Problem statement

Do long-range quantum spin systems exhibit a system-size-independent, constant-temperature transition above which every Gibbs state is separable, and what is the sharp dependence on interaction degree and locality?

For long-range Pauli Hamiltonians with interaction degree $s$ and locality $k$ , the exact order of the separability transition is $\beta_{\rm sep}=\Theta(1/(sk))$ , independent of system size. This resolves the constant-temperature death-of-entanglement question posed by Rouzé, França, and Alhambra.

Details and sources

AI contribution

The authors disclose that ChatGPT Pro contributed significantly to technical work in two appendices and helped improve the $k$ -dependence in the main theorem. They state that the central main-text proof ideas were human-generated and that every AI-assisted proof was checked and refined by the authors.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three-author, 97-page proof; author-checked AI assistance

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Research paper and AI disclosure

Activity evidence

A documented editorial estimate based on the recent STOC open question and the active literature on thermal entanglement and many-body Gibbs states.

The theorem covers long-range Pauli Hamiltonians under the paper’s interaction-degree assumptions. It is one of two distinct prior open questions resolved in the same manuscript.

Constant-temperature death of entanglement proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT Pro (through GPT-5.5)
Verification: Three-author, 97-page proof; author-checked AI assistance
Open for: 1 years
Research activity*: 4/5

30 Jul 2026Quantum many-body systems and high-temperature phases

Higher-dimensional quantum Gibbs correlation decay

Problem statement

Does high-temperature quantum correlation decay extend to higher-dimensional geometrically local systems throughout the zero-free regime without imposing logarithmic separation between observables?

For geometrically local, bounded-range quantum systems in any dimension, disjoint observables have exponentially decaying correlations throughout the zero-free high-temperature regime, without the earlier logarithmic-separation condition.

Details and sources

AI contribution

The authors disclose significant ChatGPT Pro contributions to technical work in two appendices and to theorem refinement. They attribute the central main-text ideas to the human authors and report checking and rewriting all AI-assisted arguments.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three-author, 97-page proof; author-checked AI assistance

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Research paper and AI disclosure

Activity evidence

A documented editorial estimate based on the STOC 2020 conjecture and substantial work on quantum Gibbs-state clustering and algorithms.

This resolves the higher-dimensional form of the Harrow–Mehraban–Soleimanifar correlation-decay conjecture within the paper’s zero-free high-temperature regime; it is not a claim about all temperatures.

Higher-dimensional correlation-decay conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT Pro (through GPT-5.5)
Verification: Three-author, 97-page proof; author-checked AI assistance
Open for: 6 years
Research activity*: 4/5

30 Jul 2026Quantum measurements, channels, and steering

Unital-qubit incompatibility-breaking criterion

Problem statement

Which unital qubit channels make every collection of output measurements jointly measurable, and do the projective-measurement and general-POVM thresholds coincide?

An exact necessary-and-sufficient incompatibility-breaking criterion is proved for every unital qubit channel. The projective-measurement and general-POVM boundaries coincide, yielding the exact POVM-steering boundary for two-qubit states with maximally mixed marginals.

Details and sources

AI contribution

The author reports using Codex and ChatGPT for exploratory numerics, proof refinement, and readability. GPT-5.5 specifically suggested the generalization of a key lemma from a norm-induced support function to arbitrary even support functions.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked analytic proof

Publication

Public arXiv manuscript; peer review pending

Preprint / manuscript

Research paper and AI disclosure

Activity evidence

A documented editorial estimate based on sustained work on measurement incompatibility, steering, and qubit-channel geometry.

The exact result is scoped to unital qubit channels. The paper gives only sufficient conditions in the nonunital setting, and higher-dimensional analogues remain open.

Exact criterion proved for every unital qubit channel

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: OpenAI Codex / ChatGPT-5.5
Verification: Author-checked analytic proof
Open for: General unital-qubit boundary previously unresolved
Research activity*: 4/5

29 Jul 2026Bosonic channels and continuous-variable quantum information

Non-Gaussian communication beyond the thermal threshold

Problem statement

Is the standard thermal-state lower bound optimal among Gaussian inputs to a bosonic thermal attenuator, and can non-Gaussian inputs transmit quantum information in a strictly larger noise region?

The thermal-state lower bound is proved optimal among all single-mode Gaussian inputs, then explicit non-Gaussian rank-two inputs are shown to beat it. For one thermal photon, positive quantum capacity is certified down to transmissivity $\eta=0.7841$ , narrowing the gap to the zero-capacity region $\eta\leq0.75$ .

Details and sources

AI contribution

The authors disclose that earlier human attempts and ChatGPT 5.5 failed to find the construction, while GPT-5.6 Sol produced the non-Gaussian counterexample families. The authors developed and checked the rigorous proof and accept responsibility for the result.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Five-author analytic proof with certified finite-dimensional witnesses

Publication

Public arXiv manuscript and certification details; peer review pending

Preprint / manuscript

Research paper, witnesses, and AI disclosure

Activity evidence

A documented editorial estimate based on the problem’s origin in the 1999 Holevo–Werner bound and the extensive literature on bosonic-channel capacities.

This answers two fine-grained open questions—Gaussian squeezing cannot improve the old bound, while non-Gaussian inputs can—and refutes the Gaussian-optimizer conjecture. The exact thermal-attenuator quantum capacity remains unknown.

Gaussian optimum determined and beaten; exact capacity remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Five-author analytic proof with certified finite-dimensional witnesses
Open for: 27 years
Research activity*: 5/5

29 Jul 2026Quantum property testing and spectrum estimation

Optimality of the Keyl–Werner spectrum estimator

Problem statement

Is the Keyl–Werner weak-Schur-sampling estimator asymptotically optimal for learning the spectrum of an unknown

d

-dimensional quantum state?

A new estimator uses $n=O\!\left(d^2(\log\log d/\log d)^2/\varepsilon^4\right)$ copies, giving the first asymptotic improvement over the Keyl–Werner estimator in the relevant regime and refuting a conjectured $\Omega(d^2/\log d)$ lower bound.

Details and sources

AI contribution

The authors state that AI tools were used throughout to discuss ideas and prove low-level results, while the central ideas were human-generated. No provider, model names, or raw transcripts were disclosed.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Accepted at FOCS 2026; four-author proof

Publication

FOCS 2026 accepted paper and public arXiv manuscript

Preprint / manuscript

Accepted research paper and AI disclosure

Activity evidence

A documented editorial estimate based on a question dating to 2001, later lower-bound conjectures, and a large literature on quantum spectrum estimation.

The result disproves optimality of the 2001 estimator and an influential lower-bound conjecture. It does not determine the final optimal copy complexity of quantum spectrum estimation.

FOCS 2026 accepted-paper list

Twenty-five-year optimality question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: AI tools (models not disclosed)
Verification: Accepted at FOCS 2026; four-author proof
Open for: 25 years
Research activity*: 5/5

30 Jul 2026Quantum optimal transport

Fixed-cost quantum transport metrics

Problem statement

Can a single fixed cost operator turn coupling-based quantum optimal transport into a metric, or square-root metric, on the full quantum state space?

For every dimension $d\geq3$ , no fixed cost operator on standard quantum couplings makes either the optimal cost or its square root a metric on all states; commuting states already witness failure. A separate no-go theorem rules out fixed-cost channel-induced metrics in $d\geq2$ .

Details and sources

AI contribution

Minbo Gao, Zhengfeng Ji, and Tianshi Yu state that LLMs were used throughout to brainstorm ideas and explore proof strategies. They report reviewing and validating every final claim and proof.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Three-author proof; peer review pending

Publication

Public arXiv manuscript with general AI-use disclosure

Preprint / manuscript

Prior qutrit counterexample

Activity evidence

A documented editorial estimate based on the recent quantum optimal-transport literature, the Friedland et al. conjectures, and the scope of the uniform no-go theorem.

Miller had already found counterexamples to two SWAP-cost conjectures. This paper proves the stronger uniform impossibility for every fixed standard-coupling cost and a channel-induced no-go result.

Standard-coupling open question answered negatively

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Large language models (models not disclosed)
Verification: Three-author proof; peer review pending
Open for: Open fixed-cost metric question in quantum transport
Research activity*: 4/5

30 Jul 2026Quantum cryptography and monogamy games

Pauli encodings for unclonable encryption

Problem statement

Can efficient one-bit quantum encryption prevent an adversary from producing two systems that both reveal the plaintext after the key is disclosed?

The paper proves a universal $1/2+1/(2\sqrt K)$ lower bound for $K$ -Pauli encodings, rules out $X/Z$ -only strings and pairwise-marginal arguments, solves the third SDP level for the anticommuting protocol at about $0.5556$ , and proves security against bounded-local-dimension adversaries.

Details and sources

AI contribution

The authors report extended model discussions supporting several technical sections. ChatGPT drafted Sections 5.4, 5.5, and Appendix A before Pierre Botteron, Sébastien Designolle, and Omar Fawzi checked, refined, and assumed responsibility for them.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Three-author proof with semidefinite-program evidence

Publication

Public arXiv manuscript; peer review and artifact release pending

Preprint / manuscript

Ananth–Sahai paper Ragavan paper

Activity evidence

A documented editorial estimate based on rapid recent work in unclonable encryption, monogamy games, and semidefinite hierarchies.

This is a multi-part advance rather than a complete answer to the unclonable-bit question. The exact anticommuting-family conjecture and unrestricted strong security remain open.

No-go bounds and bounded-adversary security proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.5 / Gemini 3.5
Verification: Three-author proof with semidefinite-program evidence
Open for: Active unclonable-bit and monogamy-of-entanglement questions
Research activity*: 5/5

23 Jul 2026Quantum cryptography

Unconditional efficient one-bit unclonable encryption

Problem statement

Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?

Two simultaneous papers give efficient, plain-model, one-time information-theoretically secure unclonable encryption for one classical bit with exponentially small advantage. One construction uses random Pauli eigenstates and attains the optimal exponent up to constants.

Details and sources

AI contribution

In the Ananth–Sahai work, Codex generated the construction and main proof ideas inside a Moonshot-style harness; in Ragavan’s independent work, GPT-5.6 Sol Ultra found the proof and drafted a preliminary paper. All authors checked and accept responsibility.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Two independent author-verified proofs; one Lean checked

Ragavan Lean development

Publication

Two independent public arXiv manuscripts and TeX sources; peer review pending

Preprint / manuscript

Activity evidence

Unclonable encryption is an active quantum-cryptography program; recent schemes retained either polynomial security loss or inefficient operations, and two independent groups posted solutions on the same day.

The shared unconditional claim is one-time encryption of a single classical bit with efficient operations and exponential unclonability. Ragavan’s July 27 v2 Lean development checks the Pauli/Haar correctness, security, and existence statements; efficiency and the conditional many-time corollary remain outside the formal scope.

Goal achieved independently twice

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra / Codex
Verification: Two independent author-verified proofs; one Lean checked
Open for: 6 years
Research activity*: 5/5

27 Jul 2026Entanglement theory

Two-copy distillability of Werner states

Problem statement

Are Werner states in arbitrary dimension ever two-copy distillable when they are not one-copy distillable?

Fu, Gao, and Park prove that a Werner state in any dimension is two-copy distillable if and only if it is already one-copy distillable.

Details and sources

AI contribution

The authors report using ChatGPT for exploratory mathematical discussions; some proof-development ideas arose from GPT-5.5 and GPT-5.6 Sol, then were independently examined, reformulated, and verified by the authors.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Authors independently verified the AI-assisted proof

Publication

Public revised arXiv manuscript with an explicit AI-use statement; peer review pending

Preprint / manuscript

Research paper, revised with AI statement Independent AI-assisted manuscript

Activity evidence

The question is a central finite-copy case inside the decades-old NPT bound-entanglement problem and has attracted sustained quantum-information and operator-algebra work.

The result settles the two-copy Werner-state question only. Higher-copy distillability and the existence of NPT bound entanglement remain open.

Two-copy question claimed resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 / GPT-5.6 Sol
Verification: Authors independently verified the AI-assisted proof
Open for: 26 years
Research activity*: 5/5

29 Jun 2026Quantum optimization

FGG conjecture for QAOA on the ring of disagrees

Problem statement

For an even cycle of size

N

and depth

p

satisfying

2p+2\leq N

, is the optimal QAOA approximation ratio for MaxCut exactly

\frac{2p+1}{2p+2}\,?

Claude Fable 5 found a dynamical-symmetry and quantum-signal-processing argument proving the exact optimal depth-p QAOA ratio on an even cycle. The complete proof was checked by the Lean 4 kernel.

Details and sources

AI contribution

After humans formalized the definitions and isolated the open gap, the model supplied the decisive construction and completed the formal proof through an interactive Lean loop.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked end to end

Publication

Public arXiv paper and complete Lean development

Preprint / manuscript

Paper and formal-verification record

Activity evidence

The conjecture dates to the original 2014 QAOA analysis. It had proofs only at small depth and numerical confirmation through larger p inside a heavily studied quantum-optimization program.

Humans audited that the fixed Lean statement faithfully represents the mathematical conjecture. The proof itself compiles using the standard classical axioms in Mathlib.

Conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Fable 5
Verification: Lean checked end to end
Open for: 12 years
Research activity*: 4/5

Theoretical computer science

6 Sep 2026Metric social choice

Randomized metric social-choice distortion at most $2.1441$

An explicit random-size stable-lottery rule has metric distortion at most $2.1441$ , improving the previous general upper bound $2.5$ ; the best universal lower bound remains about $2.1126$ .

Details and sources

AI contribution

Nisarg Shah supplied directions and proof/search strategies. GPT-5.6 Sol derived the proofs; the author verified and simplified them, with GPT-5.6 Sol and Claude Opus 5 helping the exposition.

Problem origin

Determining the optimal randomized distortion from ordinal preferences is an established human research problem.

Verification

Author-checked proof with exact rational certificate

Publication

Revised public arXiv preprint with exact certificate map

Preprint / manuscript

Revised preprint

The exact polynomial nonnegativity checks certify the stated rule's bound. They do not close the remaining gap to the universal lower bound.

Upper bound improved; exact optimum remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Max + Claude Opus 5
Verification: Author-checked proof with exact rational certificate

29 Aug 2026Proportionally fair clustering

$\beta$ -plurality for general metric spaces

When every agent location is an allowed center, a $2$ -Droop-core clustering always exists, the factor $2$ is tight, and centers may be selected only at agent locations; this resolves the general metric-space $\beta$ -plurality problem.

Details and sources

AI contribution

Cookson, Deltl, and Oh report that ChatGPT 5.6 Sol generated the main result through a series of interactions. They verified the proof and rewrote it for clarity.

Problem origin

Aronov and coauthors posed the $\beta$ -plurality question in 2021.

Verification

Three-author checked preprint; external review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

Pinned manuscript and Lean proof

The theorem assumes the feasible center set contains the agent locations. The paper leaves polynomial-time computation and broader center-set generalizations open.

General metric-space problem resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol
Verification: Three-author checked preprint; external review pending

8 Sep 2026Matrix semigroups and automata

Mortality threshold for finite real matrix monoids

If the entire word-product monoid generated by real $n\times n$ matrices is finite and contains zero, then it has a zero word of length at most $2^{n-1}+(2^{n-1}-1)n(n+1)/2$ ; the same bound holds over $\mathbb Q$ .

Details and sources

AI contribution

The repository attributes every original mathematical contribution, the manuscript, and the focused Lean library to AI. Eugene Gilburg packaged and published the output without claiming subject expertise.

Problem origin

The mortality-threshold problem and earlier dimension-only bounds come from Almeida–Steinberg and subsequent matrix-semigroup work.

Verification

Pinned Lean proof with claim map; specialist review pending

Claim audit

The publisher explicitly disclaims professional review; priority and statement fidelity remain unreviewed outside the artifact's own verification package.

Publication

Public manuscript, sources, and paper-specific Lean project

Preprint / manuscript

No public preprint or manuscript located.

This improves dimension-only bounds for the promised finite-product-monoid case. It does not solve the general matrix-mortality decision problem or apply when the generated monoid is infinite.

Almeida–Steinberg source problem Community scope and artifact audit

Exponential upper bound under finite-monoid promise

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-6 Astra (runtime provenance not retained)
Verification: Pinned Lean proof with claim map; specialist review pending
Claim audit: Issue documented

2 Sep 2026Theoretical computer science

Anari's girth-dependent Bethe-permanent conjecture

If a nonnegative matrix has bipartite support graph of even girth at least $g$ , then $\operatorname{per}(A)\leq2^{2n/g}\operatorname{Bethe}(A)$ , with the factor sharp when $g$ divides $2n$ .

Details and sources

AI contribution

The authors had a weaker strategy; Sol Pro developed it to a near-optimal bound and Sol Ultra interactions led to the sharp proof. The authors retain responsibility for the mathematics.

Problem origin

The primary source traces the target to a previously stated human mathematical problem or conjecture.

Verification

Author-checked proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

This confirms Anari's conjectured optimal girth refinement. It is a conventional author-checked preprint; no formalization or independent referee report was located.

Claimed full proved resolution

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro and GPT-5.6 Sol Ultra
Verification: Author-checked proof; independent review pending

3 Sep 2026Theoretical computer science

Exact half-rate point of the binary Delsarte linear program

The manuscript determines $R_D(1/2-1/\pi)=1/2$ for the asymptotic binary Delsarte linear program and the matching Krawtchouk sign-uncertainty asymptotics.

Details and sources

AI contribution

The author credits the model with proofs for Theorem 3.4 and especially the multi-qubit projection construction, alongside literature assistance and drafting.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

Pinned four-term formalization

This determines a linear-programming bound at one rate. It does not determine the true optimal rate-distance tradeoff of binary codes. It builds on OpenAI's earlier packing/coding chapters rather than replacing those records.

Proved partial advance; general problem remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-prepared proof; independent review pending

31 Aug 2026Theoretical computer science

Four-term binary polynomial multiplication needs nine AND gates

The unrestricted Boolean XOR-AND multiplicative complexity of multiplying two four-term binary polynomials is exactly nine, allowing nonlinear intermediate values and their reuse.

Details and sources

AI contribution

GPT assisted research, proof development, checking, and drafting; Opus supplied model review. Gregory Morse reviewed the final work.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Public Lean certificate; repository reports standard axioms only; not replayed in this audit

Publication

Public research preprint

Preprint / manuscript

The source distinguishes the completed four-term theorem from the five-term thirteen-gate claim, whose full Lean lower-bound proof remains in progress. Model review is not independent human peer review.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol and Claude Opus 5
Verification: Public Lean certificate; repository reports standard axioms only; not replayed in this audit

17 Aug 2026Algebraic combinatorics and computational complexity

Many-one $\mathsf{GapP}$ -completeness of binary symmetric-group characters

Problem statement

Is binary symmetric-group character evaluation

\mathsf{GapP}

-complete under many-one reductions, rather than only under Turing reductions?

Computing $\chi^{\lambda}(\mu)$ from binary-encoded partitions is $\mathsf{GapP}$ -complete under many-one reductions, even when $\lambda$ has at most two parts.

Details and sources

AI contribution

FAR recovered and status-checked the conjecture from the literature. A single GPT-5.5 xhigh opencode run produced the proof artifact; three independent model judges screened it before author or domain-expert review.

Problem origin

Ikenmeyer, Pak, and Panova conjectured many-one $\mathsf{GapP}$ -completeness after proving the corresponding Turing-reduction result.

Verification

Checked by a paper author or domain expert; public complete proof; independent peer review pending

Publication

Public proof in the reviewed-solutions appendix

Preprint / manuscript

FAR pipeline and released data Original conjecture paper

Activity evidence

A documented editorial estimate based on resolving a named complexity conjecture and strengthening the known completeness reduction.

The FAR authors report no precedent after literature search and classify this among three results checked by an author or domain expert. The proof is not formally verified or yet independently peer reviewed.

Ikenmeyer--Pak--Panova Conjecture 5.3.2 proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 xhigh in the FAR pipeline (OpenAI)
Verification: Checked by a paper author or domain expert; public complete proof; independent peer review pending
Open for: Ikenmeyer--Pak--Panova Conjecture 5.3.2
Research activity*: 4/5

13 Aug 2026Network information theory and broadcast channels

Markovity conjecture for the two-receiver broadcast channel

Problem statement

Must an optimizer of the relevant Marton dual functional admit auxiliaries satisfying the conditional Markov structure

U-X-V

Two strictly positive ternary-input broadcast channels admit explicit non-Markov constructions whose objective values exceed every construction satisfying the conjectured Markov structure.

Details and sources

AI contribution

The authors state that both counterexamples were obtained with GPT-5.6 Sol. Chandra Nair suggested using AI models to search his conjectures and, together with Yi Liu, helped check the counterexamples.

Problem origin

Gohari, Liu, and Nair formulated the Markovity Conjecture at ISIT 2025, before the AI-assisted counterexample search.

Verification

Author proof, expert checking, and 192-bit interval certificates

TeX source and numerical certificate details

Publication

Public proof preprint with numerical certificate details

Preprint / manuscript

Scripts, certificates, and Arb outputs

Activity evidence

A documented editorial estimate based on its direct role in evaluating Marton's inner bound and the rigorous computational separation.

The certificates refute the Markovity Conjecture as stated after a theoretical cardinality and rectangularity reduction. They do not refute Marton's inner bound or resolve the separate Additivity Conjecture.

Markovity conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author proof, expert checking, and 192-bit interval certificates
Open for: Conjectured at ISIT 2025
Research activity*: 4/5

12 Aug 2026Grothendieck inequalities and approximation algorithms

New lower and upper bounds for the Grothendieck constant

Problem statement

What is the exact real Grothendieck constant

K_G

, and how tightly can it be bounded by explicit rounding schemes and matching limitations?

A human–AI collaboration proves $6\pi/11\leq K_G\leq \pi/(2\log(1+\sqrt2))-10^{-4}$ , determining the previously unknown tenths digit of $K_G$ to be $7$ and positively answering whether increasing rounding-scheme dimension can improve the bound.

Details and sources

AI contribution

The authors report a long-running collaborative research system: OpenAI models served as reasoning models and Anthropic models as coding agents, with extensive human interaction, problem selection, verification, and responsibility.

Problem origin

Determining the real Grothendieck constant is a classical human problem originating in Grothendieck's inequality and decades of work in analysis and theoretical computer science.

Verification

Author-checked proofs plus public interval-arithmetic certificates

Publication

Public proof preprint with scripts, certificates, and Arb outputs

Preprint / manuscript

Companion AI-methodology paper

Activity evidence

A documented editorial estimate based on the constant's central role in Banach-space geometry, approximation algorithms, and quantum information, and on the scale of the new two-sided improvement.

This is a substantial two-sided quantitative advance, not a determination of the exact value of $K_G$ . The computational inequalities are backed by public rigorous-arithmetic artifacts rather than a proof-assistant formalization.

Major bounds improved; exact constant remains open

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro / GPT-5.6 Sol / Claude Opus / Claude Fable 5
Verification: Author-checked proofs plus public interval-arithmetic certificates
Open for: Exact value unknown since Grothendieck's 1953 inequality
Research activity*: 5/5

7 Aug 2026Planar graph algorithms and distance oracles

Tight bound for facial distance patterns in planar graphs

Problem statement

Can the number of facial distance patterns in an unweighted planar graph be bounded by

O(k^2)

The number of distinct consecutive-distance patterns to $k$ vertices on a designated face is $O(k^2)$ , matching the known lower bound and improving the earlier $O(k^3)$ upper bound.

Details and sources

AI contribution

The authors supplied the problem and state of the art in one prompt. The model recognized a cyclic-sum and weak-separation structure and produced the proof; the authors checked, cleaned, and presented it.

Problem origin

The $O(k^2)$ bound was posed as Open Problem 5.6 in the 2022 ISAAC literature on planar distance patterns.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the matching lower-bound gap and the result's consequences for several planar-graph data structures and algorithms.

The theorem also improves known planar metric-compression, distance-oracle, distributed-diameter, and centralized-diameter bounds. Those corollaries are not counted as separate solved problems.

2022 conjecture resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked proof preprint; external review pending
Open for: 4 years
Research activity*: 4/5

7 Aug 2026Spin systems and approximate counting

Pairwise spectral influence and Glauber mixing time

Problem statement

Does bounded pairwise spectral influence imply polynomial, specifically quadratic, mixing for Glauber dynamics?

A pairwise spectral-influence bound with gap $\delta$ implies an $O_\delta(n^2)$ Glauber-dynamics mixing time, improving the prior $O(n^{1+1/\delta})$ dependence and answering the cited open question.

Details and sources

AI contribution

The authors state that GPT-5.6 Sol Ultra found the main ideas of every proof in the paper; they simplified, checked, and wrote the arguments. This second open-problem consequence was made explicit in the paper's 7 August revision.

Problem origin

Leake and Oveis Gharan posed the mixing-time question for pairwise spectral influence in the prior literature.

Verification

Author-checked revised preprint; external review pending

Publication

Version 2 of a public proof preprint

Preprint / manuscript

Prior paper containing the open question

Activity evidence

A documented editorial estimate based on the broad spectral-independence and spin-system mixing literature.

This is a distinct open question resolved inside the same preprint that proves Welsh's planar-colouring approximation conjecture; it is indexed separately by problem, not counted as a second paper.

Open mixing-time question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra
Verification: Author-checked revised preprint; external review pending
Research activity*: 4/5

5 Aug 2026Online algorithms and lower bounds

Online vertex cover under edge arrivals

Problem statement

Can an online vertex-cover algorithm under edge arrivals achieve competitive ratio strictly below two?

No randomized integral or fractional algorithm beats competitive ratio two against an oblivious adversary, even on bipartite graphs; the standard two-competitive algorithm is therefore optimal.

Details and sources

AI contribution

After the authors identified the recent blueprint framework, the model formulated the reduction and assisted with the initial proof draft. The authors verified and rewrote it.

Problem origin

The optimal competitive ratio under edge arrivals is a human online-algorithms question.

Verification

Author-checked lower-bound preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the foundational role of online vertex cover and the tightness across integral, fractional, and bipartite settings.

The lower bound covers randomized and fractional algorithms in the specified edge-arrival model; it should not be transferred to other online-arrival models.

Optimal competitive ratio determined

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-checked lower-bound preprint; external review pending
Research activity*: 4/5

5 Aug 2026Constraint satisfaction and approximation algorithms

Boolean Max- $k$ -CSP approximation guarantee

Problem statement

How well can arbitrary Boolean Max-

k

-CSP be approximated in polynomial time?

A polynomial-time $k/2^k$ -approximation is proved, improving the previous $0.626612\,k/2^k$ guarantee and narrowing the gap to the conditional hardness threshold.

Details and sources

AI contribution

The conceptual algorithm and proof architecture are attributed to the authors. The model assisted with lengthy symbolic and case computations for a technical theorem.

Problem origin

Approximating arbitrary Boolean Max- $k$ -CSP is a human algorithmic research program with established upper and hardness bounds.

Verification

Author proof with AI-assisted computations; external review pending

Publication

Public research preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the centrality of CSP approximation and the remaining gap to Unique-Games-based hardness.

This is a best-known research advance, not a complete resolution of the approximation threshold. It is classified as partial and the AI role is computational support rather than discovery of the main idea.

Best-known guarantee improved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author proof with AI-assisted computations; external review pending
Research activity*: 5/5

5 Aug 2026Neural-network training dynamics

MAIS-O60 single-neuron Fourier alignment

Problem statement

Does gradient training force a single ReLU neuron on modular addition to align with one Fourier frequency?

A single ReLU neuron trained on modular addition need not align with one Fourier frequency; failure occurs on an open set of initial conditions and persists under several strengthened dynamics variants.

Details and sources

AI contribution

The human author constructed and wrote the main counterexample with conversational assistance. GPT-5.6 Sol developed and drafted the strengthened appendix covering every Clarke trajectory, smooth dead-zone approximations, and fixed-step gradient descent.

Problem origin

MAIS-O60 was generated by Claude Fable 5 under human direction as part of an AI-generated open-problem agenda.

Verification

Author-checked counterexample preprint; external review pending

Publication

Public proof preprint with model-authored appendix provenance

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the MAIS research agenda and the narrower scope relative to established human conjectures.

This entry is explicitly flagged as an AI-generated conjecture. The main negative example is human-developed; the model materially strengthened it.

Open problem answered negatively

Claimed outcome: Disproved
Problem origin: AI-generated problem
System: GPT-5.6 Sol
Verification: Author-checked counterexample preprint; external review pending
Research activity*: 3/5

6 Aug 2026Approximate counting on planar graphs

Welsh's planar $q$ -colouring approximation conjecture

Problem statement

For each fixed

q\geq4

, is approximate counting of proper

q

-colourings on planar graphs NP-hard?

For every fixed $q\geq4$ , approximately counting proper $q$ -colourings of planar graphs is NP-hard; equivalently, an FPRAS would imply $\mathrm{NP}=\mathrm{RP}$ . The theorem also closes the corresponding open planar Tutte-polynomial cases.

Details and sources

AI contribution

The authors state that the main ideas of every proof in the paper were found by GPT-5.6 Sol Ultra; they then simplified, streamlined, and wrote the proofs and take responsibility for correctness.

Problem origin

Welsh stated the planar approximate-counting hardness conjecture as Conjecture 8.7.5 in 1993.

Verification

Author-checked proof preprint; external review pending

Publication

Public sixteen-page proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the conjecture's age, its relation to the chromatic and Tutte polynomials, and the extensive literature on approximate counting and spin-system phase transitions.

The same paper also gives a small-activity FPRAS for the hard-core partition function on planar graphs and a small-external-field classification for two-spin systems. The unresolved quantitative phase boundary is not counted as solved.

Conjectured hardness proved for every

q\geq4

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra
Verification: Author-checked proof preprint; external review pending
Open for: 33 years
Research activity*: 4/5

1 Aug 2026Boolean function complexity

Quadratic bound of block sensitivity by spectral sensitivity

Problem statement

Must every total Boolean function satisfy

\operatorname{bs}(f)=O(\!\lambda(f)^2)

, where

\lambda(f)

is spectral sensitivity?

A family of total Boolean functions satisfies $\operatorname{bs}(f_n)=\Omega(\lambda(f_n)^{2.127})$ , disproving the question whether block sensitivity is always $O(\!\lambda(f)^2)$ . The paper also gives an exactly checkable 30-variable counterexample.

Details and sources

AI contribution

The author reports that GPT-5.6 Sol suggested several key proof ideas, especially gated tournaments; he then minimized and optimized the construction, wrote the paper without generative AI, and checked the formal statement. Aristotle produced the Lean formalization.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked + author-verified formal correspondence

Counterexample and proof preprint

Publication

Public proof preprint, complete Lean repository, and embedded Comparator challenge

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the central role of sensitivity measures in query complexity and the question's connection to Huang's sensitivity theorem.

The formal repository checks the main $2.127$ -exponent construction. The stronger exponent near $2.20$ is explicitly only numerical evidence and is not counted as proved; the separate 30-variable witness can be checked exactly on a laptop.

Proposed quadratic bound disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol + Aristotle
Verification: Lean checked + author-verified formal correspondence
Open for: Question of Aaronson, Ben-David, Kothari, Rao, and Tal
Research activity*: 5/5

1 Aug 2026Algebraic complexity theory

Superlinear circuit lower bounds for the permanent

Problem statement

Can one prove a superlinear-in-the-input-size lower bound for unrestricted arithmetic circuits computing the permanent exactly?

Exact division-free arithmetic circuits for the $n\times n$ permanent require $\Omega(n^2\log\log n)$ gates. Arithmetic formulas require $\Omega(n^4/\log n)$ variable occurrences, including valid formulas with division.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for the permanent lower bounds Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on the permanent’s VNP-completeness and the centrality and longstanding difficulty of unrestricted arithmetic lower bounds.

These are unconditional lower bounds in unrestricted arithmetic models, but they do not prove that the permanent lacks polynomial-size circuits and therefore do not establish $\mathrm{VP}\ne\mathrm{VNP}$ .

Unrestricted lower bounds improved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: Central arithmetic-complexity lower-bound program
Research activity*: 5/5

1 Aug 2026Lattice problems and computational complexity

Polynomial-factor hardness of Euclidean closest vector

Problem statement

Is Euclidean closest vector NP-hard to approximate within

n^c

for some fixed constant

c>0

under a deterministic reduction?

A deterministic polynomial-time many-one reduction from 3SAT proves that Euclidean $\mathrm{GapCVP}_{n^{1/400}}$ is NP-hard. The same construction yields $n^{1/200}$ hardness for binary nearest codeword and syndrome decoding and $n^{1/(200p)}$ hardness in fixed rational $\ell_p$ norms.

Details and sources

AI contribution

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending

Lean certificate for the closest-vector reduction Formalization scope, declarations, axioms, and zero-sorry audit Independent proof-checking instructions

Publication

OpenAI manuscript, reasoning walkthrough, public Lean certificate, manifest, and comparator package

Preprint / manuscript

OpenAI announcement and result summaries Reconstructed reasoning walkthroughs Sébastien Bubeck’s release discussion

Activity evidence

A documented editorial estimate based on CVP’s foundational role in lattice algorithms, computational complexity, and post-quantum cryptography.

This resolves the existence of an unconditional fixed-polynomial approximation-hardness factor through a direct reduction. It does not determine the optimal exponent, and it neither invokes the PCP theorem nor assumes the Projection Games Conjecture.

Fixed-polynomial hardness barrier crossed

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Astra (internal version)
Verification: Sorry-free Lean under standard axioms; OpenAI agent-reviewed; independent specialist review pending
Open for: Longstanding lattice-approximation hardness barrier
Research activity*: 5/5

10 Jun 2026Computability theory and model theory

Nonuniform definability of automorphisms

Problem statement

Does there exist a computably represented countable structure

A

that is computably AUT-countable on a cone, but for every finite parameter tuple

\bar a

has an automorphism

\pi

that is not

\Sigma^\mathrm{in}_1

-definable from

\bar a\cup\pi(\bar a)

Three systems produced mathematically correct constructions. The First Proof editors rated all three as requiring only minor revisions, chiefly because of missing citations rather than gaps in the argument.

Details and sources

AI contribution

ProofCouncil, the UCLA Moonshot harness, and ChatGPT 5.5 Pro independently generated solutions in the controlled one-shot benchmark.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Double-blind expert review; minor revisions

Publication

First Proof Second Batch report, complete submissions, logs, and referee reports

Preprint / manuscript

First Proof benchmark and source files Problems, submissions, reviews, and logs

Activity evidence

The problem connects active work on computable structure theory, automorphism groups, and definability; its authors reported that the example took several days to find.

This was a solved but unpublished research problem supplied by its authors, not a decades-old public conjecture. The mathematics passed expert review, while the reports flag serious attribution omissions in several submissions.

Research problem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro / ProofCouncil / UCLA Moonshot
Verification: Double-blind expert review; minor revisions
Open for: Unpublished research problem
Research activity*: 3/5

22 Jul 2026Average-case complexity and high-dimensional inference

Polynomial-time low-degree conjecture

Problem statement

Does failure of every low-degree polynomial test imply computational indistinguishability for broad permutation-invariant average-case problems?

The paper constructs permutation-invariant graph distributions with matching low-degree marginals yet a deterministic polynomial-time rank test that distinguishes them, disproving the standard binary low-degree prediction.

Details and sources

AI contribution

Multiple model runs generated the construction and proof components; the author checked, simplified, and organized the final argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked arXiv proof

Publication

Public arXiv manuscript and source; peer review pending

Preprint / manuscript

Edited reasoning reconstruction Research announcement

Activity evidence

The low-degree method is a major modern heuristic for computational thresholds in planted and inference problems.

The counterexample targets the standard binary formulation. It does not invalidate problem-specific low-degree lower bounds, and some explicit-samplability and quantitative variants remain open.

Standard binary formulation disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT-5.4 / GPT-5.5 / GPT-5.6
Verification: Author-checked arXiv proof
Open for: 8 years
Research activity*: 4/5

28 Jul 2026Post-quantum cryptanalysis

Improved key-recovery attack on HAWK

Problem statement

Can the best known key-recovery attack against the HAWK signature scheme be made substantially faster?

The project reduces the estimated work factor for HAWK-256 key recovery from about $2^{64}$ to a demonstrated attack around $2^{38}$ , with weaker improvements for larger parameter sets.

Details and sources

AI contribution

A multi-agent system developed the attack over roughly sixty hours with nontechnical human guidance; experts then reviewed the derivation and executable verifier.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

End-to-end implementation and coordinated expert review

Attack code and verifier

Publication

Public technical manuscript and reproducible attack repository

Preprint / manuscript

Technical manuscript

Activity evidence

Post-quantum cryptanalysis is exceptionally active, though HAWK itself is not a deployed standard.

HAWK is an undeployed former NIST candidate. The result is HAWK-specific, attacks remain exponential, and the work has not yet been peer reviewed; it does not imply a break of deployed post-quantum standards.

Research announcement

Best-known HAWK attack substantially improved

Problem origin: Origin not yet traced
System: Claude Mythos Preview
Verification: End-to-end implementation and coordinated expert review
Research activity*: 5/5

28 Jul 2026Symmetric-key cryptanalysis

Möbius-Bridge attack on seven-round AES-128

Problem statement

Can the best known attack on seven-round AES-128 be improved at the standard chosen-plaintext data budget?

Under the same $2^{105}$ chosen-plaintext budget, the proposed Möbius-Bridge technique improves the best published attack on seven-round AES-128 by an estimated factor of roughly $200$ – $800$ .

Details and sources

AI contribution

The multi-agent run generated about one billion output tokens. Human cryptographers spent hundreds of hours validating, correcting, and reconstructing the final derivation.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Human-reviewed derivation and small-instance checks

Supporting code

Publication

Public technical manuscript, code, and edited reasoning reconstruction

Preprint / manuscript

Technical manuscript

Activity evidence

AES cryptanalysis is among the most scrutinized areas of symmetric-key research, even when results concern reduced rounds.

This concerns seven of AES-128’s ten rounds and remains computationally impractical. Full AES-128 is unaffected, the full-scale attack was not executed, and the result has not yet been peer reviewed.

Reduced-round attack improved

Problem origin: Origin not yet traced
System: Claude Mythos Preview
Verification: Human-reviewed derivation and small-instance checks
Research activity*: 5/5

20 Jul 2026Authenticated encryption and cryptanalysis

Full key-recovery attack on SpoC-128

Problem statement

Is there a practical full-strength key-recovery attack on the unmodified SpoC-128 authenticated-encryption scheme?

Two chosen-nonce oracle queries recover the full $128$ -bit key of unmodified SpoC-128. An empty-message query skips initialization, and a one-block query leaks a state that can be inverted back to the key.

Details and sources

AI contribution

Mythos 5 and Sonnet 5 independently found the same previously unreported attack inside CryptanalysisBench.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Executable security-game verification and paper audit

Executable benchmark and attack artifacts

Publication

Public arXiv paper and reproducible benchmark repository

Preprint / manuscript

CryptanalysisBench paper

Activity evidence

SpoC was evaluated in the NIST Lightweight Cryptography process, with prior public analysis limited to reduced variants or much higher data.

The attack defeats the complete 18-step reference scheme through a flaw in the mode, not the underlying permutation. SpoC was a NIST lightweight-cryptography candidate rather than a deployed standard; peer review is pending.

Full unmodified scheme broken

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Claude Mythos 5 / Claude Sonnet 5
Verification: Executable security-game verification and paper audit
Open for: No prior full-strength key-recovery attack located
Research activity*: 4/5

20 Jul 2026Post-quantum KEM cryptanalysis

KINDI CCA-security proof

Problem statement

Is KINDI CCA secure without a re-encryption check, as claimed by its published uniqueness lemma?

A decryption-reaction attack recovers the KINDI secret key. It refutes the published lemma claiming that a decapsulated key has a unique valid ciphertext: small ciphertext perturbations can preserve the same key.

Details and sources

AI contribution

Mythos 5 found the attack while solving the full-strength KINDI task in CryptanalysisBench, exposing a previously unreported error in the submitted CCA-security proof.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Executable security-game verification and paper audit

Executable benchmark and attack artifacts

Publication

Public arXiv paper and reproducible benchmark repository

Preprint / manuscript

CryptanalysisBench paper

Activity evidence

KINDI is historical rather than deployed, but published CCA-security claims for post-quantum KEMs are a significant cryptographic audit target.

KINDI was an early NIST post-quantum candidate and is not deployed. The flaw is a familiar failure mode normally prevented by re-encryption in a Fujisaki–Okamoto transform; external peer review is pending.

Published security lemma disproved

Claimed outcome: Disproved
Problem origin: Origin not yet traced
System: Claude Mythos 5
Verification: Executable security-game verification and paper audit
Open for: No prior publication of this proof error located
Research activity*: 4/5

28 Jul 2026Computational social choice

Kemeny rank aggregation for three voters

Problem statement

Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings?

A reduction from MAX CUT proves that computing a Kemeny-optimal ranking is NP-complete for exactly three voters. Together with earlier results, every fixed number of voters $n\geq3$ is now hard, while $n=2$ remains polynomial-time solvable.

Details and sources

AI contribution

GPT-5.6 Sol Ultra autonomously found an initial five-voter reduction and then the three-voter breakthrough. Claude Fable 5 helped explore simplifications, and Peters substantially simplified and wrote the final argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean-checked reduction; author-written preprint

Dedicated Lean formalization Broader hardness-reductions library

Publication

Public arXiv paper and dedicated Lean formalization

Preprint / manuscript

Independent follow-up proof and broader dichotomies

Activity evidence

Dwork and collaborators asked the question in 2001; later computational-social-choice papers repeatedly described it as famous and long-standing.

The formal development checks the reduction used for the three-voter NP-hardness theorem. An independent 30 July follow-up proves the same three-ranking threshold and broader dichotomies; the AI-assisted manuscript remains a first arXiv version and has not yet been peer reviewed.

Twenty-five-year complexity question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra / Claude Fable 5
Verification: Lean-checked reduction; author-written preprint
Open for: 25 years
Research activity*: 5/5

28 Jul 2026Exact exponential-time graph algorithms

Fixed- $k$ graph coloring below $2^n$ time

Problem statement

For every fixed

k>6

, can

k

-coloring be solved in worst-case time

(2-\varepsilon_k)^n

, strictly faster than the general

2^n

chromatic-number algorithm?

For every fixed $k$ , randomized $k$ -coloring runs in $(2-\varepsilon_k)^n$ time for some $\varepsilon_k>0$ . The theorem also covers list coloring over every fixed palette.

Details and sources

AI contribution

Or Zamir supplied the proof framework and all main ideas. After earlier failures, the model completed a bounded-palette interpolation subargument from a focused lead.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-verified arXiv proof

Publication

Public arXiv manuscript with a detailed AI-use appendix

Preprint / manuscript

Pinned separating-family verifier

Activity evidence

Graph coloring is a foundational NP-complete problem with a large exact- and fine-grained-algorithms literature.

The author explicitly says all main proof ideas were his. He also warns that the model reproduced an existing all-subgraph-coloring proof without attribution, so this is indexed as human-led collaboration rather than autonomous discovery.

Sub-

2^n

algorithm proved for every fixed

k

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT-5.6 Sol
Verification: Author-verified arXiv proof
Open for: Long-standing exact-algorithms gap
Research activity*: 5/5

28 Jul 2026Parameterized automata and formal languages

The $4^k$ barrier for the $k$ -distinct language

Problem statement

Can the

k

-distinct language be recognized by an acyclic NFA of size

c^kn^{O(1)}

for some

c<4

A deterministic acyclic NFA construction recognizes repetition-free length- $k$ words with size at most $2^{1.96992k}n^{O(1)}<3.918^kn^{O(1)}$ , improving the previous $4^{k+o(k)}n^{O(1)}$ upper bound.

Details and sources

AI contribution

The author used the models for proof checking and simplification, parameter optimization, literature review, figures, and organization, then independently reviewed every argument, citation, program, and calculation.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Proof + interval verification + pinned code

Publication

Public arXiv manuscript, interval-verification appendix, and pinned verifier repository

Preprint / manuscript

Activity evidence

The author’s public 2014 question developed into a specialist line in parameterized automata constructions.

The work breaks the longstanding $4^k$ construction barrier but does not determine the minimum NFA size. The disclosed AI role is supporting rather than a claim of autonomous discovery.

Original public problem

Upper-bound barrier broken

Problem origin: Human-source problem
System: ChatGPT / Codex 5.4–5.6 Pro + Gemini 3.1 Pro
Verification: Proof + interval verification + pinned code
Open for: 12 years
Research activity*: 3/5

20 Jul 2026Streaming algorithms and online matching

Single-pass semi-streaming matching

Problem statement

Can a single-pass semi-streaming algorithm beat the naive greedy

1/2

approximation for maximum matching?

No deterministic or randomized single-pass semi-streaming algorithm can approximate maximum matching better than $1/2$ . The same argument fixes the optimal competitive ratio for online matching with preemption at $1/2$ .

Details and sources

AI contribution

The authors supplied the exact linear-program formulation for a proof step; GPT-5.6 Sol solved that optimization problem and returned feasible values. Their final proof directly verifies the values and differs from the generated derivation.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Three-author proof; peer review pending

Publication

Public revised arXiv manuscript with precise AI disclosure

Preprint / manuscript

Activity evidence

A central question in graph streaming with a large lower-bound and matching-approximation literature.

The decisive blueprint framework is human-authored; the AI role is a bounded but mathematically substantive optimization step. The theorem closes a question open since the semi-streaming model was introduced.

Greedy half-approximation proved optimal

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Three-author proof; peer review pending
Open for: Outstanding for more than two decades
Research activity*: 5/5

22 Jul 2026Automata theory and synchronization

Černý conjecture for one-cluster automata

Problem statement

Does every

n

-state synchronizing one-cluster automaton admit a reset word of length at most

(n-1)^2

For a synchronizing one-cluster automaton, a reset word of length at most $(m-1)(n-1)+m\ell\leq(n-1)^2$ is constructed. The paper also proves the positive-level relative-extending-word conjecture and gives sharp parameter examples.

Details and sources

AI contribution

The proof was developed through interaction with Codex in ultra reasoning mode and then mathematically verified by the author.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-verified arXiv proof

Publication

Public arXiv manuscript and TeX source; no peer review or formalization located

Preprint / manuscript

Activity evidence

The general quadratic reset-word bound dates to 1964, while the one-cluster strengthening addressed here was posed in 2016 and has an extensive dedicated literature.

This proves the Černý bound for the one-cluster class with positive level. It does not prove the general Černý conjecture for arbitrary synchronizing automata.

One-cluster case proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Codex / GPT-5.6 Sol Ultra
Verification: Author-verified arXiv proof
Open for: 10 years
Research activity*: 4/5

24 Jul 2026Private information retrieval and complexity

Minimum sparsity of $S$ -decoding polynomials

Problem statement

Can an

S

-decoding polynomial modulo a suitable product of

k

primes attain the lower-bound minimum of

k+1

nonzero coefficients?

A construction with exactly $k+1$ nonzero coefficients matches the known lower bound for special products of $k$ primes, resolving the Ghasemi–Kopparty sparsity problem and yielding exponentially fewer-server PIR.

Details and sources

AI contribution

The main constant-server result and its proof were discovered in a GPT-5.5 Pro conversation prompted by Gupte and Ragavan.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked arXiv proof + empirical validation

Publication

Public arXiv paper and TeX source; peer review pending

Preprint / manuscript

Activity evidence

A recent but consequential problem in the matching-vector PIR framework, with immediate conditional and unconditional communication-complexity consequences.

The minimum-sparsity construction is unconditional and the resulting server bound is unconditional for $s\leq15$ . The general constant- $s$ PIR theorem assumes the paper’s stated number-theoretic conjecture, implied by either the generalized repunit conjecture or Schinzel’s hypothesis H.

Conditional resolution; unconditional through

s=15

Problem origin: Origin not yet traced
System: GPT-5.5 Pro
Verification: Author-checked arXiv proof + empirical validation
Open for: Open problem posed at ITCS 2026
Research activity*: 4/5

Algorithms & optimization

3 Sep 2026Approximation algorithms and polyhedral combinatorics

Fiorini's extreme-point conjecture for feedback vertex set

Every basic feasible solution of the strong-density relaxation for feedback vertex set has a variable at least $1/2$ . A related edge relaxation has the same property and yields polynomial-time iterative-rounding $2$ -approximations.

Details and sources

AI contribution

The authors fed their research notes to Claude Opus 4.7 and Gemini 3.1 Pro over multiple rounds; the systems supplied two key ideas that unlocked the proof.

Problem origin

Samuel Fiorini posed the iterative-rounding extreme-point problem in a 2021 Oberwolfach report.

Verification

Three-author checked preprint; external review pending

Publication

Public arXiv proof manuscript

Preprint / manuscript

Pinned Lean proof and source repository Palomar formalization record

The conjectured extreme-point property is proved. Efficient separation for the original strong-density relaxation remains unknown, so the paper uses a related edge relaxation or an extended formulation for its algorithms.

Extreme-point conjecture proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: Claude Opus 4.7 + Gemini 3.1 Pro
Verification: Three-author checked preprint; external review pending

2 Sep 2026Randomized numerical linear algebra

Optimal SparseStack oblivious subspace embedding

For fully independent SparseStack, $m=O((d+\log(d/\delta))/\varepsilon^2)$ rows and $s=O(\log(d/\delta)/\varepsilon)$ nonzeros per column suffice for an oblivious subspace embedding of a fixed $d$ -dimensional subspace.

Details and sources

AI contribution

Diar Heidary states that ChatGPT 5.6 Sol and Claude Fable 5 derived the mathematical arguments under his direction; ChatGPT 5.6 Sol and Claude Fable 5.1 collaborated on the manuscript. He accepts responsibility for the claims and correspondence with Lean.

Problem origin

Nelson and Nguyen proposed the target parameters in 2013, and the question for the fully independent SparseStack distribution was later recorded as Problem 5.4 in a public problem list.

Verification

Pinned Lean proof and Palomar registration; review pending

Claim audit

The author believes the formal statements match the manuscript, but no independent specialist review or peer-reviewed publication was located.

Publication

Public arXiv paper, pinned Lean project, and Palomar record

Preprint / manuscript

The theorem is for the fully independent SparseStack distribution and a fixed subspace, or a random subspace independent of the sketch. It does not cover adaptive subspaces, limited-independence implementations, or every sparse-OSE construction.

Target SparseStack parameters proved and formalized

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol + Claude Fable 5/5.1
Verification: Pinned Lean proof and Palomar registration; review pending
Claim audit: Issue documented

2 Sep 2026Algorithms & optimization

Weighted EF1 allocations for additive mixed goods and chores

For additive mixed-manna valuations and arbitrary positive entitlements, a weighted envy-free-up-to-one-item (WEF1) allocation always exists and can be computed in polynomial time.

Details and sources

AI contribution

Zehan Lin, Shengxin Liu, Biaoshuai Tao, and Shengwei Zhou state that they developed the proof ideas independently; GPT assisted exploration and organization of the proof constructions. This is supporting research assistance, not an autonomous AI discovery.

Problem origin

Garg and Sharma's 2024 paper explicitly left general WEF1 existence for weighted mixed manna open after proving the two-agent case and the weaker WEF1T guarantee. The new paper's Question 1 and Theorem 3.9 address that same problem.

Verification

Author-prepared proof; independent review pending

Publication

Public research preprint

Preprint / manuscript

Original open question and two-agent result

The guarantee does not assert WEF1 together with fractional Pareto optimality. The paper separately proves WEF1T plus fractional Pareto optimality and randomized guarantees; these are not counted as additional open-problem resolutions here. No public formalization or code was located; independent review remains pending.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol
Verification: Author-prepared proof; independent review pending

2 Sep 2026Algorithms & optimization

Pairwise-independent correlation gap in dimension four

The remaining four-variable case has sharp correlation gap $4/3$ for monotone submodular functions, using cone certificates and 2,745 Bernstein coefficient systems.

Details and sources

AI contribution

Gustav Malmqvist's human-AI exploration suggested the certificate strategy; Arjun Ramachandra corrected and refined it and verified the computational results.

Problem origin

The manuscript identifies the prior human problem and its literature; the scope of the result is recorded below.

Verification

Author-reported computational certificates; repository unavailable at audit

Publication

Public research preprint

Preprint / manuscript

Original five-variable counterexample and open case

The universal $4/3$ conjecture was already disproved for five or more variables. The paper also derives the asymptotic worst-case value $e/(e-1)$ , but the explicit AI disclosure concerns the four-variable proof. Its printed certificate-repository URL returned 404 on 4 September; this audit did not replay those certificates.

Claimed resolution in a public preprint

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT 5.6 and Claude Fable 5
Verification: Author-reported computational certificates; repository unavailable at audit

19 Aug 2026Nonsmooth stochastic optimization

Last-iterate convergence for definable mini-batch stochastic approximation

Problem statement

Under definability and the stated vanishing-step hypotheses, must the entire mini-batch stochastic-approximation sequence converge rather than merely have connected accumulation sets?

Two convex piecewise-affine semialgebraic summands on $\mathbb R$ and a deterministic nonincreasing step-size sequence $\alpha_k=o(1/\log k)$ produce bounded mini-batch iterates whose accumulation set is the full interval $[-1,1]$ .

Details and sources

AI contribution

The author reports using the models for mathematical brainstorming, development, and drafting, then independently checking every mathematical statement and reference.

Problem origin

The convergence assertion was posed in Remark 12 of Bolte and Pauwels' 2021 paper on conservative fields and automatic differentiation.

Verification

Author-verified complete counterexample proof; independent review pending

Publication

Public counterexample preprint

Preprint / manuscript

Exact certificate and replay repository

Activity evidence

A documented editorial estimate based on correcting a general convergence claim in nonsmooth stochastic optimization with an explicit low-dimensional construction.

The counterexample targets the exact mini-batch recursion and step-size regime in the conjecture. It does not rule out convergence under stronger assumptions such as square-summable steps.

Bolte--Pauwels convergence conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: ChatGPT 5.6 Sol (OpenAI) and Gemini 3.1 Pro DeepThink (Google)
Verification: Author-verified complete counterexample proof; independent review pending
Open for: Bolte--Pauwels (2021), Remark 12
Research activity*: 3/5

14 Aug 2026Alternating direction methods and convex optimization

Three-block ADMM with the identity matrix as its third constraint block

Problem statement

Must direct three-block ADMM converge when the third constraint block is the identity matrix and the first two objectives are strongly convex quadratics?

Two exact rational constructions disprove universal convergence in the identity-third-block subclass: a period- $66$ nonconvergent orbit and a separate locally attracting period- $23$ cycle that can be converted to an all-identity instance.

Details and sources

AI contribution

The authors report a human-guided workflow in which GPT-5.6 Sol proposed the period-66 construction and exact certificate, while Kimi K3 independently produced the period-23 construction. Humans selected the problem, constrained the search, audited the literature, and promoted only candidates passing exact checks.

Problem origin

The paper's literature audit identifies convergence of direct three-block ADMM with an identity third constraint block as a previously unresolved human-posed subclass: neither a proof nor a counterexample was known.

Verification

Exact rational certificates with independent implementations; no external peer review

Publication

Public arXiv manuscript and reproducible exact-certificate repository

Preprint / manuscript

Activity evidence

A documented editorial estimate based on resolving a clearly delimited convergence question with two exact constructions and public replay scripts, tempered by the lack of external review or an immutable release.

The examples settle universal convergence for the specified identity-third-block subclass. They do not address every corrected ADMM variant or establish global attraction. The repository explicitly describes agreement among its implementations as internal reproducibility, not independent peer review, and had no immutable release tag at audit time.

Previously unresolved convergence subclass answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.6 Sol via Codex; Kimi K3 via Kimi Code
Verification: Exact rational certificates with independent implementations; no external peer review
Open for: Previously unresolved after counterexamples for broader three-block ADMM
Research activity*: 4/5

11 Aug 2026First-order convex optimization

Lower bound for stepsize-only acceleration of gradient descent

Problem statement

How fast can the last iterate of plain gradient descent converge when acceleration is attempted only through a predetermined nonnegative stepsize schedule?

For smooth convex objectives, predetermined nonnegative stepsize schedules can have last-iterate convergence no better than $\Omega(T^{-1.9319})$ , giving rigorous evidence that stepsizes alone cannot attain the general first-order optimum $O(T^{-2})$ .

Details and sources

AI contribution

The authors state that GPT-5.6 Sol Pro developed the proof under their guidance. They selected and framed the problem and take responsibility for the manuscript.

Problem origin

The work addresses a human literature question about the best last-iterate rate attainable by plain gradient descent using only predetermined nonnegative stepsizes.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the active literature on acceleration without momentum and the remaining quantitative exponent gap.

This is a substantive lower-bound advance, not a complete characterization. The known upper-rate exponent $\log_2(1+\sqrt2)$ and the new lower-bound exponent $1.9319$ do not match.

Lower bound improved; exponent gap remains

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Pro
Verification: Author-checked proof preprint; external review pending
Research activity*: 4/5

10 Aug 2026Stochastic first-order complexity

Bounded-noise lower bound for smooth nonconvex stochastic optimization

Problem statement

Does almost-surely bounded stochastic-gradient error permit a better minimax rate than bounded variance for smooth nonconvex optimization?

In the fresh-sample $K=1$ model, every randomized adaptive algorithm needs $\Omega(\Delta L/\varepsilon^2+\Delta L\sigma^2/\varepsilon^4)$ queries to find an $\varepsilon$ -stationary point, matching the standard upper bound even when gradient noise is almost surely bounded.

Details and sources

AI contribution

The author reports that the model independently generated the proof in a two-hour session after receiving the prompt. The human author checked the proof and revised and polished the manuscript.

Problem origin

Arjevani, Carmon, Duchi, Foster, Srebro, and Woodworth raised the bounded-oracle-error complexity question in the prior human literature.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the mature stochastic-optimization lower-bound literature and the exact match to the standard upper rate.

The theorem is for the $K=1$ fresh-sample oracle model and closes the cited bounded-error versus bounded-variance rate question in that setting.

Prior lower-bound paper

2023 complexity question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra via Codex
Verification: Author-checked proof preprint; external review pending
Research activity*: 4/5

7 Aug 2026Social choice and metric committee selection

Pessimal elections for approximately dominating sets

Problem statement

Is the

O(1/\varepsilon^2)

size bound for approximately dominating committees asymptotically necessary?

There are elections in which every approximately dominating committee needs $\Omega(1/\varepsilon^2)$ members, matching the known $O(1/\varepsilon^2)$ upper bound up to constants.

Details and sources

AI contribution

The model produced the construction and proof after receiving the earlier paper and an open-problem prompt. The authors supplied one streamlining idea, verified the argument, and rewrote the manuscript.

Problem origin

The lower-bound question was stated in the authors' earlier work and recorded as Conjecture 3.14 in a human-written survey.

Verification

Author-checked proof preprint; external review pending

Publication

Public proof preprint

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the prior conjecture, recent survey treatment, and links to representative committees and metric social choice.

The result closes the asymptotic dependence on the approximation parameter; it does not determine the best constant.

Asymptotic committee-size question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra
Verification: Author-checked proof preprint; external review pending
Research activity*: 3/5

6 Aug 2026Fair division under allocation constraints

Balanced EF1 and fractional Pareto-optimal allocations

Problem statement

Does every indivisible-goods instance with nonnegative additive valuations admit a balanced allocation that is EF1 and fractionally Pareto optimal within the balanced allocation polytope?

Every instance with nonnegative additive valuations admits a balanced allocation that is envy-free up to one good and fractionally Pareto optimal among balanced fractional allocations, removing the prior restrictions to personalized bivalued valuations or at most two valuation types.

Details and sources

AI contribution

The authors report that Codex derived all mathematical proofs and counterexamples from their research directions, literature connections, and strategies. The authors verified and simplified the arguments, with further assistance from GPT-5.6 Sol and Claude Fable 5.

Problem origin

The compatibility of fairness and efficiency under balancedness is an established human research question; earlier work handled restricted valuation classes and weaker guarantees.

Verification

Author-verified proof preprint; external review pending

Publication

Public proof preprint with detailed AI acknowledgement

Preprint / manuscript

Algorithm and proof preprint

Activity evidence

A documented editorial estimate based on the broad fair-division literature, the practical role of EF1, and the recent sequence of compatibility and impossibility results under constraints.

The theorem settles the additive-valuation existence question under balancedness. Stronger questions for general matroid constraints, richer valuations, chores, and efficient computation remain explicitly open in the paper.

General additive-valuation existence question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Codex (GPT-5.6 Sol Max) / Claude Fable 5
Verification: Author-verified proof preprint; external review pending
Open for: General additive case left open after restricted 2026 results
Research activity*: 4/5

3 Aug 2026Randomized algorithms and counting complexity

FPRAS for two-terminal network reliability

Problem statement

Does two-terminal network reliability admit an FPRAS on general directed and undirected graphs?

Two-terminal reliability on general directed and undirected graphs admits a fully polynomial-time randomized approximation scheme. The companion two-terminal unreliability problem is also shown to be $\#\mathrm{BIS}$ -hard.

Details and sources

AI contribution

The authors state that GPT-5.6 Sol Ultra discovered the key algorithmic idea. They developed the complete algorithm, reductions, mixing analysis, and public proof manuscript.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-written proof; independent review pending

Publication

Public 31-page preprint with complete algorithm and proof

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the question's 1994 survey formulation, its connection to classical $\#P$ -complete reliability problems, and sustained work on randomized counting.

This resolves the relative-approximation question for two-terminal reliability on general graphs, explicitly posed in Kannan's 1994 survey. It does not make exact reliability computation efficient, and relative approximation of the complementary unreliability quantity remains hard under the paper's stated reduction.

General approximation question answered positively

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.6 Sol Ultra
Verification: Author-written proof; independent review pending
Open for: 32 years
Research activity*: 4/5

29 Jul 2026Distribution testing and randomized algorithms

Linear-time approximation of product-distribution TV distance

Problem statement

Can the total variation distance between two explicitly specified product distributions be multiplicatively approximated in time linear in the input size?

For explicitly given product distributions $P,Q$ on $[q]^n$ , there is a randomized $(1\pm\varepsilon)$ multiplicative approximation to $d_{\mathrm{TV}}(P,Q)$ in $O(qn\varepsilon^{-2}\log(1/\delta))$ time; a matching $\Omega(qn)$ marginal-query lower bound makes the input-size dependence optimal.

Details and sources

AI contribution

The authors report that a single adapted prompt produced the linear-time algorithm after their own roughly $O(qn^{1.3})$ Bernoulli-case approach. Further interaction surfaced the filtered Monte Carlo reference; the authors then wrote or rewrote the proofs and assume responsibility for them.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; independent review pending

Publication

Public seven-page preprint with complete proofs

Preprint / manuscript

Algorithm and proof preprint

Activity evidence

A documented editorial estimate based on several recent approximation algorithms for product-distribution distances and a now-matched input-size lower bound.

The result improves the previous quadratic or roughly $n^{3/2}$ dependence on the number of coordinates and proves an optimal linear input-size dependence for fixed accuracy and confidence. It is an algorithmic runtime advance rather than a resolution of the exact-evaluation problem, which remains #P-hard.

Optimal linear-time algorithm proved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.6 Sol Ultra
Verification: Author checked; independent review pending
Open for: Runtime gap left by earlier approximation algorithms
Research activity*: 3/5

20 Jul 2026Stochastic multi-objective optimization

Near- $T^{-1}$ convergence for stochastic multi-gradient descent

Problem statement

Under the standard smoothness and bounded-variance assumptions, how fast can vanilla stochastic multi-gradient descent drive the squared Pareto-stationarity measure toward zero?

For smooth nonconvex stochastic multi-objective problems with unbiased, variance-bounded stochastic gradients, constant-step stochastic multi-gradient descent with linearly growing mini-batches attains a squared Pareto-stationarity rate $\widetilde O(T^{-1})$ , improving the previous $\widetilde O(T^{-1/4})$ guarantee under the same setting.

Details and sources

AI contribution

The model generated the initial proof strategy in response to an author-written graduate homework-solution prompt. The author verified, reorganized, and wrote the argument; the paper documents the prompt and proof-discovery process.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author checked; independent review pending

Publication

Public preprint with proof and discovery-process appendices

Preprint / manuscript

Convergence proof and prompt appendix

Activity evidence

A documented editorial estimate based on current work on stochastic MGDA, Pareto stationarity, and multi-objective learning.

This is a substantial convergence-theorem advance rather than a claim to settle the optimal rate in every stochastic multi-objective setting. The guarantee relies on the paper's stated smoothness, unbiasedness, variance, step-size, and batch-growth assumptions.

Convergence-rate gap sharply improved

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: ChatGPT 5.4 Thinking Extended
Verification: Author checked; independent review pending
Open for: Rate gap left by the 2024 analysis
Research activity*: 3/5

21 May 2026Approximation algorithms and matrix permanents

Optimal exponential approximation of positive-semidefinite permanents

Problem statement

What is the optimal exponential approximation ratio achievable in deterministic polynomial time for the permanent of a Hermitian positive-semidefinite matrix?

For a Hermitian positive-semidefinite matrix $A$ , a computable concave-programming quantity $\widehat P(A)$ satisfies $e^{-\gamma n}\widehat P(A)\leq\operatorname{per}(A)\leq\widehat P(A)$ . Together with existing hardness, this determines the best deterministic polynomial-time approximation ratio as $e^{(\gamma+o(1))n}$ , assuming $\mathrm P\ne\mathrm{NP}$ .

Details and sources

AI contribution

The two authors independently obtained the result through GPT-5.5 Pro Extended interactions: one was one-shot and the other a guided multi-turn exchange. Both verified the theorem and proof; Codex assembled and typeset the manuscript.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Both authors checked; independent review pending

Publication

Public two-author preprint with complete proof

Preprint / manuscript

Optimal approximation theorem and proof

Activity evidence

A documented editorial estimate based on the permanent's central role in counting complexity and a matched approximation-hardness threshold.

The optimality statement is asymptotic in the exponent and conditional on $\mathrm P\ne\mathrm{NP}$ . It closes the gap between the new algorithmic upper bound and the previously known $e^{(\gamma-\varepsilon)n}$ hardness threshold.

Optimal exponential ratio determined

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.5 Pro Extended + Codex
Verification: Both authors checked; independent review pending
Open for: Algorithm-versus-hardness approximation gap
Research activity*: 4/5

15 Jul 2026Analog computation and reaction networks

Chemical reaction networks and CRN-computable reals

Problem statement

Can the central equivalence and universality results connecting polynomial ODEs, chemical reaction networks, linear production protocols, and stochastic mean-field limits be assembled in one machine-checked framework?

Ripple assembles a sorry-free Lean 4 development of CRN-computable reals, polynomial-ODE and LPP compilation, stochastic mean-field limits, and deterministic and stochastic Turing-completeness. Version 2 retracts the paper’s earlier claim of a gap in Angluin et al. (2008): that issue was in Ripple’s formalization, not the original proof. Two other proof repairs remain, and the corrected development includes a checked construction showing that $\zeta(3)$ is CRN-computable.

Details and sources

AI contribution

The authors used the models to translate, repair, and extend the mathematical development inside an iterative Lean verification loop.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Lean checked; no sorry

Publication

Public arXiv paper and Lean repository

Preprint / manuscript

Activity evidence

The work connects several active literatures in analog computation, reaction-network theory, and mechanized mathematics, but formalizes a focused technical program.

This is a formalization and repair record, not a claim that every theorem in the framework was first discovered by AI. The July 2026 v2 correction also fixes the attribution of the $\zeta(3)$ construction. The paper reports only the foundational axioms already used by Mathlib.

Theory formalized and gaps repaired

Problem origin: Origin not yet traced
System: Claude Opus 4.6–4.8 / Claude Fable 5 / GPT-5.4–5.6
Verification: Lean checked; no sorry
Research activity*: 3/5

8 Apr 2026Learning dynamics

Exhaustive AdaBoost cycling question

Problem statement

For every finite training set, does exhaustive AdaBoost eventually converge to a finite cycle of weak classifiers and weight vectors?

Wang constructs a finite exhaustive-AdaBoost instance whose orbit never becomes periodic, resolving a question posed by Rudin, Schapire, and Daubechies in 2012.

Details and sources

AI contribution

The author credits both models as collaborators in developing the block-product counterexample and its proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Exact rational certificate

Publication

Public computer-assisted arXiv proof

Preprint / manuscript

Activity evidence

A standing theoretical-machine-learning question from COLT 2012 with a sustained specialist thread on AdaBoost dynamics.

Every asserted computation is certified with exact rational arithmetic; the nonperiodicity argument uses an irrational logarithmic ratio.

Question answered negatively

Claimed outcome: Disproved
Problem origin: Human-source problem
System: GPT-5.4 Pro / Claude Opus 4.6
Verification: Exact rational certificate
Open for: 14 years
Research activity*: 3/5

5 Jul 2026Discrepancy theory and algorithms

Optimal online discrepancy in linear time

Problem statement

Given online vectors

v_t\in\mathbb{R}^d

with

\lVert v_t\rVert_2\leq1

, can signs

\varepsilon_t\in\{-1,1\}

be chosen in

O(dT)

time so that every prefix has

\ell_\infty

discrepancy

O(\sqrt{\log T})

with high probability?

Aden-Ali gives an $O(dT)$ -time online algorithm attaining the optimal $O(\!\sqrt{\log T}\!)$ prefix-discrepancy bound, replacing an earlier algorithm exponential in both $T$ and $d$ .

Details and sources

AI contribution

The paper states that the model conversation discovered both the algorithm and the main proof after the author supplied the research prompt.

Problem origin

The problem's first formulation has not yet been independently traced to a primary source.

Verification

Author-checked preprint

Publication

Public eight-page arXiv proof

Preprint / manuscript

Activity evidence

Online vector balancing and discrepancy minimization are internationally active areas; the target matched a known optimal bound but required an efficient construction.

This closes a computational-efficiency gap around an already optimal discrepancy bound; it is not a new improvement to the asymptotic bound itself.

Runtime gap closed

Claimed outcome: Proved
Problem origin: Origin not yet traced
System: GPT-5.5 Pro Extended
Verification: Author-checked preprint
Open for: Long-standing runtime gap
Research activity*: 4/5

28 May 2026Randomized graph algorithms

Nearly uniform sampling of directed Eulerian tours

Problem statement

Does the proposed flip–repair Markov chain mix rapidly enough to yield a nearly uniform directed-Eulerian-tour sampler in

\widetilde O(m^{3/2})

worst-case time?

Anari gives a worst-case $\widetilde O(m^{3/2})$ sampler for nearly uniform directed Eulerian tours, breaking the previous $mn$ -type barrier on sparse graphs.

Details and sources

AI contribution

The author devised the algorithmic plan and conjectured the key mixing theorem; GPT-5.5 Pro Extended supplied its linear-algebra proof, while Codex assisted manuscript assembly.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked preprint

Publication

Public 42-page arXiv proof

Preprint / manuscript

Activity evidence

Eulerian-tour sampling sits in an active algorithms-and-probability literature, while the exact mixing statement was newly isolated by the author.

The AI-resolved object is the author’s mixing conjecture inside a broader new algorithm, not a previously named community conjecture.

Conjectured mixing theorem proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro Extended / Codex
Verification: Author-checked preprint
Open for: Research bottleneck posed during the project
Research activity*: 3/5

4 Apr 2026Convex–concave optimization

Last-iterate rate for anchored gradient descent–ascent

Problem statement

For a monotone

K

-Lipschitz saddle operator arising from a smooth convex–concave min–max problem, can anchored gradient descent–ascent be scheduled so that its exact last-iterate squared-gradient residual is

O(1/t)

A new anchoring schedule gives squared-gradient residual $O(1/t)$ for smooth convex–concave min–max problems, closing the rate gap left by the 2019 analysis.

Details and sources

AI contribution

The model and human researchers jointly discovered the schedule and the discrete-recurrence argument; Nexus produced the checked formal proof.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Lean checked

Research paper 2019 precursor

Publication

Standalone public preprint and Lean source

Preprint / manuscript

Activity evidence

A sustained specialist line in minimax optimization with published precursor and follow-up analyses.

The prior method obtained exponents approaching but not reaching the exact $O(1/t)$ squared-residual rate.

Optimal rate attained

Claimed outcome: Proved
Problem origin: Human-source problem
System: AlphaProof Nexus
Verification: Lean checked
Open for: 7 years
Research activity*: 3/5

14 Jul 2026Derivative-free convex optimization

Near-quadratic value-oracle lower bound for convex optimization

Problem statement

How many function-value queries are necessary for deterministic optimization of a Lipschitz convex function in high dimension?

The work proves a deterministic value-oracle lower bound $\Omega(d^2/\log(d+1))$ at accuracy $\Theta(d^{-1/2})$ , nearly matching the $O(d^2\log^2d)$ upper bound.

Details and sources

AI contribution

Two public model conversations generated the lower-bound construction. Kerger checked and refined it, then released a Lean development for a sharper deterministic theorem.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Headline lower bound Lean checked

Publication

Public arXiv manuscript, conversations, and Lean repository

Preprint / manuscript

First discovery conversation Second discovery conversation Author announcement

Activity evidence

The value-oracle complexity gap has persisted through decades of work on derivative-free convex optimization.

The Lean code verifies the deterministic lower bound at the stated accuracy. It does not formalize the Protasov upper bound, the combined asymptotic summary, randomized or smooth cases, or every corollary in the paper.

Polynomial gap closed up to logarithms

Problem origin: Human-source problem
System: GPT-5.6 Sol Pro
Verification: Headline lower bound Lean checked
Open for: 30 years
Research activity*: 4/5

16 Jun 2026Stochastic-gradient optimization

SS–RS–GD inequalities

Problem statement

For well-conditioned symmetric quadratic losses, must the expected single-shuffle, random-reshuffle, and gradient-descent operators satisfy

\lVert W_{ss}\rVert\leq\lVert W_{rs}\rVert\leq\lVert W_{gd}\rVert

Explicit well-conditioned positive-semidefinite matrices disprove $\lVert W_{ss}\rVert\leq\lVert W_{rs}\rVert$ , already for $n=3$ , $K=2$ , and $d=4$ . The companion inequality $\lVert W_{rs}\rVert\leq\lVert W_{gd}\rVert$ is proved in the stated well-conditioned regime.

Details and sources

AI contribution

Peng reports that the proof was found by GPT-5.5 Pro Extended under his prompting.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-checked arXiv proof

Publication

Public arXiv manuscript and TeX source; peer review pending

Preprint / manuscript

Research paper and interaction report

Activity evidence

A focused COLT 2021 open problem in the active theory of random reshuffling and finite-sum optimization.

This resolves the paired COLT 2021 open question: the SS–RS half is false, while the RS–GD half holds under the paper’s explicit spectral conditioning.

One inequality disproved; the other proved

Claimed outcome: ProvedDisproved
Problem origin: Human-source problem
System: GPT-5.5 Pro Extended
Verification: Author-checked arXiv proof
Open for: 5 years
Research activity*: 3/5

8 Jun 2026Stochastic scheduling and queueing control

Three-server heterogeneous-queue threshold policy

Problem statement

Does the threshold-policy structure of the Lin–Kumar two-server queue remain optimal once a third heterogeneous server is introduced?

A threshold policy is optimal for a queue with one fast server and two identical slow servers, resolving the first case beyond the classical one-fast/one-slow system.

Details and sources

AI contribution

The six authors credit GPT-5.5 Pro with the core technical ideas, then verified and rewrote the proofs; three key lemmas were also checked in Lean 4.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author-verified proof; three key lemmas Lean checked

Publication

Public arXiv manuscript with an interaction report and partial Lean verification

Preprint / manuscript

Activity evidence

The base problem dates to Lin–Kumar’s 1984 queueing-control model; extension beyond two servers had remained open.

The result handles one fast and two identical slow servers. Optimality for arbitrary numbers or heterogeneous collections of slow servers remains open, and no standalone public Lean repository was located.

First nontrivial multi-server case proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: GPT-5.5 Pro
Verification: Author-verified proof; three key lemmas Lean checked
Open for: 42 years
Research activity*: 4/5

18 Jul 2026VLSI routing and network-design algorithms

Terminal-only Manhattan cost–radius spanning trees

Problem statement

Given Manhattan terminals, a root, a total-length budget, and a source-radius budget, is deciding whether a spanning tree meets both bounds polynomial-time solvable?

The rooted terminal-only Manhattan cost–radius spanning-tree decision problem is weakly NP-complete. A height-partition construction also gives a continuous cost–radius tradeoff and a balanced $(2,2)$ bicriteria guarantee.

Details and sources

AI contribution

Five deliberately conflicting Codex investigations pursued hardness, exact algorithms, approximation, and implementation. Keren Zhu reconstructed and verified the surviving proofs, preserved falsifying counterexamples, and evaluated the final solver against a frozen benchmark.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Reconstructed proof, executable solver, and public audit bundle

Solver and reproducibility materials

Publication

Public arXiv manuscript, task specifications, code, benchmarks, and checkers

Preprint / manuscript

Activity evidence

A documented editorial estimate based on the 1992 open complexity question and decades of practical Prim–Dijkstra and shallow-light routing work.

The result answers the precise terminal-only Manhattan question left open in 1992. The reduction establishes weak, not strong, NP-hardness; incomplete Lean developments are explicitly excluded from the formal-verification claim.

Complexity classified; bicriteria guarantee proved

Claimed outcome: Proved
Problem origin: Human-source problem
System: OpenAI Codex (GPT-5.6 Sol Ultra)
Verification: Reconstructed proof, executable solver, and public audit bundle
Open for: 34 years
Research activity*: 4/5

15 Jul 2026Online discrepancy and vector balancing

Online Spencer vector-balancing question

Problem statement

Can online vector balancing in the Spencer setting achieve the optimal order of prefix discrepancy by an efficient algorithm?

A compactly supported Metropolis fixed-point walk gives an efficient online prefix-discrepancy algorithm, resolving the Spencer-setting question and extending the offline Beck–Fiala bound to $d\geq\log(T)^{1+o(1)}$ .

Details and sources

AI contribution

The model generated the proof over several rounds of high-level author prompting. Altschuler and Tikhomirov manually checked, rewrote, and took responsibility for the final argument.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Authors manually checked and rewrote the proof

Publication

Public arXiv manuscript and TeX source; peer review pending

Preprint / manuscript

Resolution case study Original conjecture

Activity evidence

The result sits in the active Beck–Fiala, Spencer, and online Komlós programs and improves the sparsity range immediately after a major offline breakthrough.

The paper resolves the online Spencer question and reaches the stated near-logarithmic sparsity range. It does not prove the full offline Beck–Fiala conjecture for every $d$ .

Online question resolved

Claimed outcome: Proved
Problem origin: Human-source problem
System: ChatGPT-5.6 Pro
Verification: Authors manually checked and rewrote the proof
Open for: 3 years
Research activity*: 4/5

3 Feb 2026Online submodular optimization

Copying-versus-moving conjecture for online submodular welfare

Problem statement

Is the expected marginal gain from copying an item to the end of a random-order stream always at most the gain from moving its original occurrence there?

An explicit instance with three items and two agents shows that copying an item to the end of a random-order stream can have larger expected marginal gain than moving its original occurrence there.

Details and sources

AI contribution

A zero-shot Gemini run selected the question, built the counterexample, and carried out the six-permutation arithmetic. The section authors independently checked the calculation.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Counterexample independently checked by authors

Publication

Public Gemini case-study paper with the complete counterexample; no formal artifact located

Preprint / manuscript

Activity evidence

The conjecture was posed in 2017 as a possible route to improving random-order online submodular-welfare guarantees.

The counterexample invalidates the proposed copying-versus-moving step toward a $0.567$ competitive ratio. It does not determine the tight approximation ratio of random-order greedy.

Google DeepMind research overview

Conjecture disproved

Claimed outcome: Disproved
Problem origin: Human-source problem
System: Gemini Deep Think
Verification: Counterexample independently checked by authors
Open for: 9 years
Research activity*: 3/5

15 Apr 2026Approximation algorithms

Avidor–Zwick Max-Cut question for triangle-strengthened SDP

Problem statement

For fixed

d

, can every

d

-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than

\alpha_{GW}

For every fixed dimension $d$ , a polynomial-time rounding algorithm beats the Goemans–Williamson ratio for a $d$ -dimensional feasible solution of the Max-Cut SDP strengthened by triangle inequalities, achieving $\alpha_{GW}+2^{-O(d)}$ .

Details and sources

AI contribution

Gemini first supplied the key geometric anti-concentration lemma. The authors then used ChatGPT and Gemini to sharpen it to the optimal exponential scale, edited three proof routes, and checked the final paper.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Authors edited and checked the proofs

Publication

Dedicated public arXiv manuscript plus the broader Gemini research case study

Preprint / manuscript

Dedicated research paper Gemini case-study paper

Activity evidence

The low-dimensional Max-Cut question dates to Avidor and Zwick’s 2005 work and lies within the foundational Goemans–Williamson approximation program.

The theorem requires triangle inequalities in addition to the standard Max-Cut SDP. The original Avidor–Zwick question for the unstrengthened basic SDP remains open.

Triangle-strengthened question answered

Claimed outcome: Proved
Problem origin: Human-source problem
System: Gemini Deep Think / ChatGPT-5.2 Extended Pro / Gemini 3 Pro
Verification: Authors edited and checked the proofs
Open for: 21 years
Research activity*: 4/5

9 Jul 2026Planar graph algorithms

Minimum edge-outerplanar embedding

Problem statement

Can the minimum edge-outerplanarity of a finite loopless planar graph—minimized over all planar embeddings—be computed in polynomial time?

A polynomial-time algorithm now computes a planar embedding of minimum edge-outerplanarity for every finite loopless planar graph, resolving Bentz’s 2009 question.

Details and sources

AI contribution

GPT-5.5 Pro produced the initial proof in a solver–verifier pipeline; Hantao Yu then manually checked and polished it.

Problem origin

The indexed source traces the problem to human-authored literature or a pre-existing problem registry.

Verification

Author checked and polished

Solver–verifier pipeline

Publication

Complete public arXiv proof and open solver–verifier pipeline

Preprint / manuscript