whataifound.org: What AI Has Actually Contributed

722 manuscripts in 372 result families from an unreleased OpenAI model

Wed, 07 Oct 2026 00:00:00 +0000

OpenAI published 722 manuscripts grouped into 372 claimed results across number theory, algebraic geometry, analysis, combinatorics, theoretical computer science and mathematical physics, all produced by an unreleased internal model, with Lean formalizations attached to 235 of the families and none yet checked by anyone outside the company. The repository's README says the model was posed about 4,000 open problems during an evaluation, that each result used on average three hours of ChatGPT Pro thinking compute, and that the output was aggregated into families and filtered for significance; a family groups a principal result with companions, consequences or alternative proofs. The families claimed include the full Birch-Swinnerton-Dyer formula in Selmer corank at most one, the quasi-Riemann hypothesis, a negative answer to Hilbert's tenth problem over the rationals, the Unique Games Conjecture, L = RL = BPL, matrix multiplication with exponent at most 9/4, Kakeya in three and four dimensions, nonamenability of Thompson's group F, isomorphism of the free group factors and Kadison's similarity conjecture, alongside dozens of counterexamples to named conjectures. Lean 4 formalizations are linked from 235 families; the formalization catalogue lists 162 papers whose main result is formalized and Comparator challenge files for checking the statements against permitted axioms. Ten families also come with abridged summaries of the model's reasoning. The README names two exceptions to the fixed procedure, the work on a zero-free region for the Riemann zeta function and the Hodge conjecture for CM abelian varieties, and says the write-up of the ℜs > 11/12 zero-free region was edited by humans for readability. Lab: OpenAI. Model: Unreleased internal OpenAI model, not named in the release. Verification: Claimed. Autonomy: AI-led.

Quasi-Riemann hypothesis: no zeros of zeta or any Dirichlet L-function with real part above 7/8

Wed, 07 Oct 2026 00:00:00 +0000

A Lean-formalized proof that neither the Riemann zeta function nor any Dirichlet L-function has a zero with real part greater than 7/8, the first fixed zero-free half-plane short of the line where real part equals 1, which also rules out Landau-Siegel zeros but leaves the Riemann hypothesis itself open. The main theorem states that every finite-order Hecke L-function over ℚ(√(−3)) and every Dirichlet L-function, including ζ(s), is zero-free in ℜs > 7/8, the principal pole at s = 1 excepted. The quasi-Riemann hypothesis asks for any fixed σ₀ < 1 with no zeros to the right of it; the classical zero-free regions of de la Vallée Poussin and Vinogradov-Korobov shrink toward ℜs = 1 with height, and zero-density estimates such as Guth and Maynard's bound how many zeros there are rather than excluding them. A fixed half-plane gives the prime number theorem with error O(x^(7/8 + ε)). The paper works through Kubota's metaplectic theory and Patterson's cubic theta series, with sextic large-sieve estimates extending Goldmakher, Louvel and Blomer, using Dunn and Radziwiłł's unconditional cusp expansions. A companion manuscript reaches the weaker half-plane ℜs > 11/12 by a different proof, and a third proves (1 − β)log q ≥ c for every real zero β of every primitive real character of conductor q ≥ 3. The Lean statement for zeta is a single line against Mathlib's own definition, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re, with separate Comparator challenges for the Dirichlet and Hecke versions and for the Siegel-zero bound, each permitting only propext, Quot.sound and Classical.choice. Lab: OpenAI. Model: Unreleased internal OpenAI model, not named in the release. Verification: Claimed. Autonomy: Collaborative.

Kakeya maximal conjecture in three dimensions and the Kakeya set conjecture in four

Wed, 07 Oct 2026 00:00:00 +0000

Two manuscripts claim that every set in four-dimensional space containing a unit segment in every direction has full Hausdorff dimension, and that the Kakeya maximal function estimate holds in three dimensions, extending Wang and Zahl's 2025 resolution of the three-dimensional set conjecture. The four-dimensional paper proves that every K ⊂ ℝ⁴ containing a unit segment in every direction has dim_H K = 4, with no measurability assumed on the set or on the choice of segments. Its introduction gives the best earlier lower bound as 3.059, from Katz and Zahl's planebrush argument, against Wolff's 3 from 1995; Rai Choudhuri's 13/4 covers only sticky sets. The three-dimensional paper proves the stronger maximal form: for every ε > 0 the radius-δ tube maximal operator maps L³(ℝ³) to L³(S²) with norm O_ε(δ^(−ε)). It builds on the streamlined Guth, Wang and Zahl proof of the set estimate and addresses the gap Wang and Zahl state after their own theorem, replacing a density loss λ^(K(ε)) by λ³. Scientific American's report names only the four-dimensional result. Lab: OpenAI. Model: Unreleased internal OpenAI model, not named in the release. Verification: Claimed. Autonomy: AI-led.

Optimal packings of one to seven unit squares in a circle, proved in Lean

Tue, 06 Oct 2026 00:00:00 +0000

Machine-checked proofs of the smallest circle that holds n unit squares, and of every packing that attains it, for n = 1 to 7; for five, six and seven squares no earlier proof of optimality was known. The question is the least radius of a disk that holds n non-overlapping unit squares, each free to move and rotate. The library proves the radii √2/2, √5/2, 5√17/16, √2, √(5/2), √(q^*) with q^* a quartic irrational near 2.851, and √13/2. The optimal packing is unique up to rotation and relabelling for n ≤ 6; for n = 7 the optimal packings form a family in which the three middle squares slide along their column. For three to seven squares the proofs place a small circle about the disk centre and count how much of it each square avoiding the centre must hold; for seven squares a pair theorem on the directions of neighbouring squares is by far the longest part. The repository reports that the arc idea came from Claude, that ChatGPT first applied it to five squares, and that the models wrote the proofs and all of the Lean code, with the owner directing and reviewing. Lab: Independent. Model: ChatGPT (GPT-6 Pro) and Claude Opus 5 and 5.5. Verification: Formally verified. Autonomy: AI-led.

Closed form for the anisotropic Watson integral of the cubic lattice

Tue, 06 Oct 2026 00:00:00 +0000

The return probability of a random walk on the cubic lattice with three unequal hopping rates, open since Watson's 1939 solution of the symmetric case, is given in closed form through periods of a genus-two curve, with a proof that no formula in elliptic integrals exists for generic rates. The lattice Green's function W(α₁, α₂, α₃;w) = 1/π³∫_([0, π]³)d³k/(w − α₁cos k₁ − α₂cos k₂ − α₃cos k₃) gives the return probability R = 1 − 1/(w^(⋆)W) at the band edge w^(⋆) = α₁ + α₂ + α₃. Closed forms were known when one rate vanishes or when two or three are equal (Watson 1939, Joyce 1973, Delves and Joyce 2001), each an algebraic function times at most two complete elliptic integrals. The authors' summary states that for generic rates wW satisfies a fifth-order differential equation whose group has connected part SO(5), so no such elliptic formula exists, and gives W instead through a minor of the period matrix of a genus-two curve built from the rates. At rates in the ratio 1:5:7 the walker returns with probability about 0.4394, against 0.3405 at equal rates. The result is one of roughly fourteen in a guest essay by Schwartz on Anthropic's blog describing work with Claude through his BootLoops harness. Lab: Independent. Model: Claude (the announcement names Claude Fable 5). Verification: Claimed. Autonomy: AI-assisted.

Six-task counterexample to the Kernel Conjecture of pinwheel scheduling

Tue, 06 Oct 2026 00:00:00 +0000

A six-task pinwheel scheduling instance with periods (3, 4, 5, 20, 22, 36) can be scheduled, but not once its longest period is cut to 32, which refutes the Kernel Conjecture and the equivalent 2ᵏ Conjecture of Gąsieniec, Smith and Wild. In pinwheel scheduling one task is performed per day, and a task with period aᵢ must be performed at least once in every aᵢ consecutive days. Gąsieniec, Smith and Wild conjectured that every schedulable instance with k tasks is dominated by a schedulable one, with no period longer, whose largest period is at most 2^(k − 1), and proved this equivalent to the claim that a loosely schedulable instance always has a schedule with a free day at least every 2ᵏ days. For six tasks the cap is 32. A 36-day repeating word schedules (3, 4, 5, 20, 22, 36), while finite certificates show that (3, 4, 5, 20, 22, b) cannot be scheduled for b = 32 or b = 35, so the threshold for the sixth period is exactly 36 and every dominated instance capped at 32 fails by monotonicity. Each certificate lists every reachable state with a rank that strictly decreases along every legal move, which rules out non-periodic schedules too. The repository carries a Lean 4 formalization of the certificate argument and of both endpoints, built against the standard library alone. Lab: Independent. Model: GPT-6 Astra (in OpenAI Codex). Verification: Formally verified. Autonomy: AI-led.

Six-particle MHV amplitude in planar N=4 super Yang-Mills computed at nine loops

Thu, 01 Oct 2026 00:00:00 +0000

Claude computed the six-particle maximally-helicity-violating amplitude in planar N = 4 super Yang-Mills theory at nine loops, one loop beyond the previous record, from a single prompt naming the problem. Two Anthropic physicists gave Claude one prompt, "The problem is to compute the Six-particle (hexagon) amplitude in planar N=4 SYM at nine loops", and afterwards only told it to keep working. It did the calculation two ways: the original hexagon bootstrap, and the indirect route of bootstrapping the three-point form factor of the chiral stress-tensor multiplet, mapping it to the amplitude by antipodal duality and lifting it with pentagon operator product expansion data. The bootstrap ran in Python with SymPy on about $100 of compute, 96 CPUs for a week. The data page records the checks Claude ran before handing the result over: rational reconstruction at three 31-bit primes, agreement between independent prime computations, a reproduction of the published eight-loop amplitude on 1,000 random words, and symmetry and vanishing tests. Lance Dixon, who set the eight-loop record, then validated the result, mostly by mapping the amplitude back to the form factor. The symbol and function files are public on Zenodo under CC BY 4.0. Lab: Anthropic. Model: Claude Fable 5.1 (in Claude Science). Verification: Author verified. Autonomy: AI-led.

Erdős Problem #1220 formalized as not provable in ZFC, from a 1987 Shelah-Stanley consistency result

Mon, 28 Sep 2026 00:00:00 +0000

A Lean 4 formalization shows that ZFC cannot prove a positive answer to Erdős Problem #1220 on partition relations for singular cardinals, by formalizing Shelah and Stanley's 1987 forcing construction, with AI models assisting on the Lean code. Erdős Problem #1220, from Erdős and Hajnal in 1971, asks whether λ → (λ, ℵ₁)² holds for every singular cardinal λ with λ and cf(λ) both ℵ₀-inaccessible. The repository proves, inside Mathlib's first-order logic, that ZFC does not prove the affirmative sentence, and a second target certifies that the sentence is equivalent in Mathlib's ZFSet to a Mathlib-level statement of the problem as posed on erdosproblems.com. The construction follows Shelah and Stanley (Sh:258, section 3) with the ground-model witness λ = ℶ_(c⁺), and builds a Boolean-valued model using the Lean 4 port of Flypitch vendored through Elliot Glazer's Erdős #501 development. Only non-provability is formalized: consistency of the positive answer, and so full independence, is outside its scope. Lab: Independent. Model: Claude Opus 5.5 (Claude Code); Astra (OpenAI Codex). Verification: Author verified. Autonomy: AI-assisted.

Hamilton cycles in connected Cayley graphs of polylogarithmic degree, toward the Lovász conjecture

Mon, 28 Sep 2026 00:00:00 +0000

Every connected Cayley graph on n vertices with degree at least C(log n)¹³/log log n has a Hamilton cycle, lowering the degree threshold in the Cayley-graph form of the Lovász conjecture from a power of n to a power of log n; the manuscript was drafted by GPT 6 Pro and the proof formalized in Lean by Claude agents. Lovász asked in 1969 whether every connected vertex-transitive graph has a Hamilton path; the Cayley-graph form asks whether every connected Cayley graph on at least three vertices is Hamiltonian. For arbitrary connected Cayley graphs this was known for linear degree (Christofides, Hladký and Máthé, 2014) and for degree at least n^(1 − c) (Bedert, Draganić, Müyesser and Pavez-Signé, arXiv:2603.08675). The research draft dated 25 September 2026 lowers the threshold to C(log n)¹³/log log n through local absorption, signed rounding, a weighted partition and a connecting system. The repository formalizes the whole draft as a DAG of about 55 lemmas in roughly 36,500 lines, including classical inputs the paper uses without proof, and states the theorem in a Mathlib-only Challenge.lean using Mathlib's own Cayley graph and Hamiltonicity definitions. Lab: Independent. Model: GPT 6 Pro (manuscript draft); Claude Opus 5.5 agents in Grok Build (Lean formalization). Verification: Claimed. Autonomy: Collaborative.

Four Color Theorem formalized in Lean 4 by AI agents, in two independent developments

Mon, 28 Sep 2026 00:00:00 +0000

Two independent Lean 4 proofs of the Four Color Theorem, each following Gonthier and Werner's Coq proof and each written almost entirely by a Claude model under one person's direction, resting only on Lean's three standard axioms. Chris Emery's corun1024/4ct, public on 2026-09-16, proves FourColor.fourColorTheorem from a 195-line statement file that transcribes Gonthier's real-plane statement and proves its elementary topological notions equal to Mathlib's IsOpen, closure and IsPreconnected. Its README reports 115,342 declarations checked with no sorry and no extra axiom; its reducibility certificates are produced by C and Python tools and then checked by the Lean kernel. Robert D. Barish's port, public on 2026-09-26, transcribes Coq's realplane.v one for one in a 220-line Challenge.lean and computes all 633 reducibility certificates inside Lean during the build, so the Lean toolchain alone checks it. Its README reports a local rehearsal of Palomar's pipeline in which leanchecker, nanoda and con-ron accept the proof, and it carries a Palomar registration. Neither development uses native_decide. Lab: Independent. Model: Claude Fable 5.1 and Claude Opus 5 (Emery); Claude Opus 5.5 (Barish). Verification: Claimed. Autonomy: AI-led.

Exponential lower bound for the bit pigeonhole principle in unrestricted resolution over parities

Mon, 28 Sep 2026 00:00:00 +0000

Any refutation of the bit pigeonhole principle in resolution over parities, a proof system whose lines are disjunctions of linear equations mod 2, must be exponentially long even with no restriction on its depth or regularity; the mathematics was produced by AI models and the main theorem is checked in Lean. Resolution over parities, Res(⊕), extends resolution by letting clauses be disjunctions of affine equations over 𝔽₂. Superpolynomial size lower bounds were previously known only for restricted refutations: tree-like, regular, or of bounded depth. The paper claims that every DAG-like Res(⊕) refutation of the bit pigeonhole principle with n + 1 pigeons and n = 2ˡ holes has more than exp(n/(32768 l²)) clauses for every l of at least 32. The argument translates a refutation into a low-degree polynomial calculus refutation over extension variables in the style of Buss, Impagliazzo, Krajicek, Pudlak, Razborov and Sgall, removes the extension variables with one substitution, and closes with a Razborov-style degree lower bound proved through the homology of chessboard complexes. The main theorem, a general sufficient condition for Res(⊕) size lower bounds, and their dependencies are formalized in Lean 4 in the author's repository. Lab: Independent. Model: GPT-6 Astra (mathematics, version 1 formalization and drafting); Claude Fable 5.1 and Claude Opus 5 (revision 1 formalization). Verification: Claimed. Autonomy: AI-led.

Norm-variation of multiple ergodic averages, including Tao's norm-convergence theorem, autoformalized in Lean in one week

Mon, 28 Sep 2026 00:00:00 +0000

A Lean 4 proof of about 107,000 lines of a new norm-variation estimate for multiple ergodic averages of commuting transformations, which strengthens Tao's 2008 norm-convergence theorem, was produced by coding agents in one week from a human-written blueprint and hand-formalized statements. The companion arXiv paper (2608.27321) is a blueprint for a forthcoming traditional paper by the same authors. It proves r-variation bounds in L² for multiple ergodic averages of n commuting measure-preserving transformations, strengthening Tao's norm-convergence theorem and, per its abstract, resolving an open question, through an explicit real-variable estimate for twisted multilinear averages tied to singular Brascamp-Lieb inequalities. In the repository the statements of the three main theorems, nCT.main_ergodic_theorem, nCT.tao_norm_convergence and nCT.main_twisted_theorem, were formalized by hand, and the proofs were autoformalized over one week by coding agents under human supervision, about 107,000 lines excluding comments and blank lines. That includes prerequisites not in Mathlib: multilinear complex interpolation, the Calderón transference principle and properties of Wiener space functions. Lab: Independent. Model: Coding agents, models not stated. Verification: Author verified. Autonomy: AI-led.

Agent-run genome survey identifies array-associated reverse transcriptases in jumbo phages

Thu, 24 Sep 2026 00:00:00 +0000

Claude agents surveying reverse transcriptase genes across 1.9 billion metagenomic protein clusters flagged a family of jumbo-phage reverse transcriptases paired with an array of repeated non-coding DNA units and a dedicated partner gene, an arrangement the authors say fits no described reverse transcriptase class. A harness of Claude Code agents, each task split between a worker and a reviewing supervisor, ran from a human-written brief asking for new partner-gene associations of reverse transcriptases. Over 119 tasks, 949 agent sessions and 21.5 hours without human intervention, the agents built their own profile models, recovered about 200,000 reverse transcriptase clusters and filed 19 reports. The defining observation was off-brief: a worker reading the raw DNA upstream of a phage reverse transcriptase noticed a tandem repeat array, checked it against known elements including CRISPR arrays, and reported it. The authors then characterized the family in interactive sessions, finding 95 related clusters, 28 with arrays of 3 to 21 repeats of 15 to 49 nucleotides separated by 120 to 220 nucleotide spacers, a clade beside retrons, and no nearby cas genes. Published infection RNA-seq for Staphylococcus phage SA1 shows the arrays highly expressed, and the authors' own small-RNA sequencing of the SA1 array expressed in E. coli shows it processed into discrete unit RNAs. Lab: Anthropic. Model: Claude Mythos 5. Verification: Claimed. Autonomy: Collaborative.

A 3-generator 9-relator group with unsolvable word problem, and smaller Adian-Rabin families

Thu, 24 Sep 2026 00:00:00 +0000

A group given by 3 generators and 9 relations has an unsolvable word problem, beating the 12-relation record Borisov set in 1969, and it yields families of 4-generator 11-relator and 2-generator 10-relator presentations for which no algorithm can decide whether the group is trivial. How small a finite presentation can be while still being algorithmically undecidable is a quantitative question that runs from Boone's 32-relator group through Collins to Borisov's 12. Starting from Borisov's group, the construction adds two HNN extensions and then removes seven redundant relators and six generator-relator pairs by Tietze transformations, leaving 3 generators and 9 relators. Feeding that group into Tancer's and Gordon's reductions, with two further Tietze eliminations in Tancer's case, gives the two Adian-Rabin families, improving Tancer's deficiency-9 family and Gordon's 13-relator family. By Markov's argument these give Corollary 1.5, that a connected sum of n copies of S2 x S2 is topologically unrecognizable for n at least 7 and smoothly unrecognizable for n at least 9. The three algebraic theorems, with every dependency down to Post's machine-to-Thue construction and Matiyasevich's three-rule Thue system, are formalized in about 28,000 lines of Lean. Lab: Independent. Model: GPT-5.6 Sol Pro (construction); GPT-5.6 Sol in Codex (Lean formalization). Verification: Author verified. Autonomy: AI-led.

Counterexample to Talagrand's operator cotype problem

Tue, 22 Sep 2026 00:00:00 +0000

An operator is exhibited whose Rademacher cotype is not controlled by the maximum of its Gaussian cotype and its (q,1)-summing norm, answering Talagrand's question in the negative already at q equals 2. Talagrand asked whether the Rademacher cotype of an operator is universally controlled by the maximum of two quantities: its Gaussian cotype and its (q,1)-summing norm. The paper answers no, and does so at the smallest interesting parameter, q equals 2, rather than only for large q. The AI disclosure is a single sentence and it is the last line of the abstract: the counterexample was discovered by ChatGPT (GPT-5.6). That places the decisive step, producing the object itself, with the model, and leaves the write-up with the author. Lab: Independent. Model: ChatGPT (GPT-5.6). Verification: Claimed. Autonomy: AI-led.

Huang-Jiang-Oblomkov conjecture proved for every torus-knot singularity, with a Lean formalization conditional on two literature inputs

Tue, 22 Sep 2026 00:00:00 +0000

The geometric extension of the Rogers-Ramanujan and Andrews-Gordon identities conjectured by Huang, Jiang and Oblomkov is proved for every torus-knot singularity with coprime exponents, via a stronger finite identity. The conjecture of Huang, Jiang and Oblomkov gives a geometric extension of the Rogers-Ramanujan and Andrews-Gordon identities for the torus-knot singularity Xᵃ = Yᵇ with coprime exponents. The paper establishes a threefold equality between a normalized count of pairs of commuting nilpotent matrices over a finite field satisfying Aᵃ = Bᵇ, the HJO q-series, and an explicit infinite product. The main result is stated as a stronger finite identity, that the rank N HJO sum equals a q-Pochhammer factor times the generating function for balanced cylindric partitions with entries bounded by N, with the conjecture following in the limit. The proof is human mathematics: it combines the compositional rational shuffle theorem of Bergeron, Garsia, Leven and Xin and of Mellit with a multiplicativity theorem for slope operators and a determinantal model, linked by a common q-difference equation. The abstract's AI disclosure covers the formalization only: the finite identity and the HJO conjecture have been formalized in Lean by AxiomProver, conditional on two stated literature inputs. Lab: Axiom Math. Model: AxiomProver. Verification: Claimed. Autonomy: AI-assisted.

Browning-Sawin conjecture on random sign-coefficient hypersurfaces proved, with a Lean formalization assuming existing literature

Tue, 22 Sep 2026 00:00:00 +0000

Random hypersurfaces with coefficients drawn uniformly from plus and minus one are shown to be smooth with probability tending to one as the degree grows, with a quantitative rate. Browning and Sawin conjectured that random hypersurfaces with sign coefficients are smooth with probability tending to one as the degree grows. The paper proves this and adds a rate: for each n at least 1, a degree d form in n + 1 variables with independent uniform coefficients in plus and minus one defines a singular complex hypersurface with probability Oₙ(d^(−1/2)). Positive-dimensional singular loci are shown to occur with exponentially small probability, and for n at least 3 the same exponential bound is given for failure of absolute irreducibility. The abstract's AI disclosure is one sentence and covers the formalization: these results have been formalized in Lean by AxiomProver assuming existing literature. Lab: Axiom Math. Model: AxiomProver. Verification: Claimed. Autonomy: AI-assisted.

Finite-time blowup for the IPM equation with uniformly space-time smooth forcing

Tue, 22 Sep 2026 00:00:00 +0000

The incompressible porous medium equation on the two-dimensional torus is shown to develop a finite-time singularity under a force that is smooth in space and time together, extending a result previously available only for spatially smooth forcing. Cordoba and Martinez-Zoroa proved finite-time blowup for the incompressible porous medium equation with a spatially smooth force. This paper adapts their technique to a force that is uniformly smooth in space and time together, which is the stronger requirement. It exhibits a smooth odd initial density, a smooth odd force on the unit time interval crossed with the two-dimensional torus, and a classical solution on the half-open interval whose density gradient and spatial velocity gradient both diverge in the supremum norm as time approaches one, while the density itself still converges in every Holder class below Lipschitz. It is one of three companion forced-blowup results from the same group covering IPM, Boussinesq and three-dimensional Euler; the other two are not yet in this registry. Lab: Independent. Model: Not named; the arXiv comment field states only that the proof is LLM-assisted. Verification: Claimed. Autonomy: AI-assisted.

Existence of the core in approval-based committee elections

Tue, 22 Sep 2026 00:00:00 +0000

Every approval-based multi-winner election is shown to admit a committee in the core, settling the main open question in the area, and one can be found in polynomial time. The core is a stability and group-fairness notion for multi-winner elections: a committee is in the core when no group of voters large enough to deserve a different outcome all prefer one. Whether a core committee always exists was the main open question in the theory of approval-based multi-winner elections. The paper shows one always does. The proof introduces a voting rule that optimizes an entropy-like objective over committees and payment systems together, and shows every local optimum of that objective lies in the core, which also yields a polynomial-time algorithm. The arXiv comment field carries the AI disclosure in one line: the proof was obtained with GPT-6 Astra. Lab: Independent. Model: GPT-6 Astra. Verification: Claimed. Autonomy: Collaborative.

Improved lower bound for the complex Grothendieck constant

Tue, 22 Sep 2026 00:00:00 +0000

The complex Grothendieck constant is shown to exceed 1.35584631827168, closing more than a quarter of the gap between the long-standing Davie lower bound and the Haagerup upper bound. The classical complex Grothendieck constant is bracketed between Davie's lower bound and Haagerup's upper bound, and the interval has been stubborn. This paper reports a new lower bound of 1.35584631827168 and states that it closes more than one quarter of that gap. The abstract records two separate facts about how it was produced: the numerical part of the proof is rigorously verified by interval arithmetic, and the lower bound and the proof were discovered by the Odin Automatic AI Research Agent. Lab: Independent. Model: Odin Automatic AI Research Agent. Verification: Author verified. Autonomy: AI-led.

Nivat's conjecture claimed in its sharp form, with the submitter stating he cannot check it

Tue, 15 Sep 2026 00:00:00 +0000

A Lean development claims Nivat's 1997 conjecture in full: if a two-dimensional configuration has at most mn distinct patterns on some m by n rectangle, it has a nonzero period. Nivat conjectured in 1997 that low rectangular pattern complexity forces periodicity in a two-dimensional configuration. Partial results are known under stronger complexity bounds; the sharp mn form has stood. The repository separates Challenge.lean, which states the result in mathlib terms alone, from Solution.lean, which proves it, and carries a Palomar registration. The submitter's disclaimer is the unusual part and is quoted in the caveats: he says the proof is entirely the model's and that he is not qualified to check it. Lab: Independent. Model: GPT-6 Pro (proof); GPT-6 Astra Ultra through Codex (Lean formalization). Verification: Claimed. Autonomy: AI-led.

Bollobas-Nikiforov conjecture claimed in full, in a weighted form

Tue, 15 Sep 2026 00:00:00 +0000

Five named mathematicians with AI assistance claim the 2007 Bollobas-Nikiforov conjecture for every noncomplete graph, bounding the sum of the squares of the two largest adjacency eigenvalues, with a Lean formalization. Bollobas and Nikiforov conjectured in 2007 that every noncomplete graph on at least two vertices satisfies a bound on the sum of the squares of its two largest adjacency eigenvalues, in terms of its edge count and clique number. The repository formalizes the conjecture together with a weighted spectral inequality and a completely positive matrix theorem that prove it, and carries a Palomar registration. The AI roles are split and disclosed individually: ideation and the manuscript from GPT-6 Astra, the Lean development completed by Grok 4.6 with minimal direction, and submission packaging by a Claude agent. Lab: Independent. Model: GPT-6 Astra (ideation and manuscript); Grok 4.6 (Lean formalization); Claude Fable 5.1 (submission packaging). Verification: Claimed. Autonomy: Collaborative.

Disproof of the Ibragimov-Iosifescu conjecture for phi-mixing sequences

Tue, 15 Sep 2026 00:00:00 +0000

A Lean development constructs a stationary phi-mixing sequence with finite second moments whose normalized partial sums fail to satisfy the central limit theorem, disproving a 1971 conjecture. Ibragimov conjectured that a strictly stationary phi-mixing sequence with finite second moments and diverging partial-sum variance satisfies the central limit theorem; Iosifescu's stronger form asks for convergence of the rescaled partial-sum process to Brownian motion. The construction uses bounded causal feedback to suppress ordinary fluctuations while rare large spikes retain the variance, giving variance divergence together with collapse of the normalized sums along a subsequence. Disproving Ibragimov's form rules out Iosifescu's, since that convergence would give the central limit theorem at time one. The recorded verification reports the standard three axioms only. Lab: OpenAI / Epoch AI. Model: GPT-6 Astra (pre-release). Verification: Claimed. Autonomy: AI-led.

Counterexample to Smale's mean value conjecture at K = 1

Tue, 15 Sep 2026 00:00:00 +0000

A Lean proof produced in an unsteered benchmark run disproves the K = 1 form of Smale's 1981 mean value conjecture, exhibiting a polynomial for which no critical point meets the bound. Smale proved in 1981 that for every complex polynomial f of degree at least two and every point z there is a critical point c with |f(z) − f(c)|/|z − c| ≤ K|f′(z)| for K = 4, and asked whether K can be lowered to 1. The constant has since been lowered to four minus order one over the degree, and the conjecture is known for small degrees and for polynomials whose roots are all real or all of one modulus. This development proves the negation of the Formal Conjectures statement. It came out of the same Epoch AI LeanOpenProblems evaluation run as the Koethe and Erdos-Sos entries already in the registry, and Comparator checks the compared statement against the challenge file under the three standard axioms. Lab: OpenAI / Epoch AI. Model: GPT-6 Astra (pre-release). Verification: Formally verified. Autonomy: Autonomous.

Bounded prime gaps improved to 212, with a Lean certificate of the deduction

Tue, 15 Sep 2026 00:00:00 +0000

Nine authors prove that infinitely many pairs of primes differ by at most 212, improving Stadlmann's 240, with the deduction certified in Lean by the group's own prover. The bounded-gaps line runs from Zhang in 2013 through Maynard's sieve to the Polymath8b bound of 246, which this registry already carries as a formalization entry, and then to Stadlmann's 240. This paper builds on Stadlmann's work to reach 212. The authors are explicit about the division of labour: the mathematics is theirs, and AxiomProver generated the Lean certificate of the deduction from natural-language specifications rather than finding the argument. Lab: Axiom Math. Model: AxiomProver. Verification: Claimed. Autonomy: Collaborative.

Prime gaps at most 186, conditional on three unproved Lean axioms

Tue, 15 Sep 2026 00:00:00 +0000

An OpenAI repository derives a bound of 186 on infinitely many prime gaps in Lean, from three inputs it states as axioms rather than proving. This is numerically the strongest of three bounded prime-gap claims from the same window, ahead of the Axiom Math 212 recorded separately here and the 246 already in the registry, and it is the weakest evidentially. The development derives DHL[40,2], and so the bound, from three stated inputs. Those inputs are not proved in Lean: the repository's own README says the results remain conditional on three explicit input axioms and that the cited estimates and numerical computations have not been turned into Lean proofs. Lab: OpenAI. Model: GPT-6 Astra. Verification: Claimed. Autonomy: AI-led.

Finite-time blowup for Navier-Stokes with smooth forcing, Clay alternatives C and D

Thu, 10 Sep 2026 00:00:00 +0000

A Lean development proves that three-dimensional Navier-Stokes with a smooth forcing term breaks down in finite time, at every positive viscosity, on both Euclidean space and the periodic torus, which is two of the four statements the Clay problem description asks for a proof of, though not the unforced regularity question the phrase usually brings to mind. Fefferman's Clay problem description asks for a proof of one of four statements, offered as leeway to solvers while, in his words, retaining the heart of the problem. This work settles (C) and (D), the two breakdown alternatives, both of which permit a smooth external force meeting the decay conditions the description imposes: for every positive viscosity there exist smooth initial data and forcing admitting no global smooth solution of uniformly bounded kinetic energy on ℝ³, and smooth periodic data and forcing admitting no global smooth solution on ℝ³/ℤ³. The repository carries Lean 4 proofs of both, declared as navier_stokes_breakdown_R3 and navier_stokes_breakdown_periodic, each recorded with no sorry and depending on exactly propext, Classical.choice and Quot.sound. The Comparator reference statements were adapted from Google DeepMind's Formal Conjectures formalization of the Clay problem rather than written for this repository, and the challenge config enables the independent nanoda kernel. The same repository also carries two theorems on unforced Euler blowup, which are a separate result and are not covered by this entry. Lab: OpenAI. Model: Unnamed OpenAI model described as more capable than GPT-6 Astra for the proof; GPT-6 Astra through Codex for the Lean formalization. Verification: Claimed. Autonomy: AI-led.

Finite-time blowup for the unforced Euler equations from smooth compactly supported data

Thu, 10 Sep 2026 00:00:00 +0000

A Lean development proves that the three-dimensional incompressible Euler equations, with no forcing term, develop a singularity in finite time from smooth, compactly supported, divergence-free initial velocity on all of space. Whether the three-dimensional incompressible Euler equations can produce a singularity in finite time from smooth finite-energy data, with no forcing and no boundary, is one of the long-standing questions in mathematical fluid dynamics. The repository states the result twice: euler_breakdown_R3, that no global smooth unforced solution of uniformly bounded kinetic energy exists on ℝ³, and exists_compact_smooth_euler_singularity, which gives compactly supported smooth initial data with a positive finite maximal lifespan in the all-order Sobolev class, an unbounded limsup of the velocity's C¹ norm, and a divergent time integral of the vorticity's sup norm. That last condition is the Beale-Kato-Majda quantity, so the blowup is certified in the classical criterion rather than only as a loss of regularity. Both theorems are recorded with no sorry and dependence on exactly propext, Classical.choice and Quot.sound, and the Comparator config enables the independent nanoda kernel. OpenAI's account describes roughly 100 agents working about 50 hours on this result, against roughly 10,000 over 88 hours for the companion Navier-Stokes work. Euler is not one of the Clay Millennium problems; that list carries Navier-Stokes only. Lab: OpenAI. Model: Unnamed OpenAI model described as more capable than GPT-6 Astra for the proof; GPT-6 Astra through Codex for the Lean formalization. Verification: Claimed. Autonomy: AI-led.

Fermat's Last Theorem formalized end to end in Lean 4

Mon, 07 Sep 2026 00:00:00 +0000

A complete machine-checked proof of Fermat's Last Theorem in Lean 4, written by AI agents in eleven days, resting on Lean's three standard axioms and accepted by two independent kernels. The repository states the theorem over Lean's built-in naturals as fermat_last_theorem, and its default build target FinalCheck.lean derives Mathlib's own FermatLastTheorem from it, so the statement is not a bespoke restatement. The build fails unless the proof depends on exactly propext, Classical.choice and Quot.sound: no sorry, no added axiom, no native_decide. Three checks are recorded. A from-scratch build compiled all 60,475 modules; leanprover/comparator v4.33.0 checked the result against a Mathlib-only Challenge.lean and returned 'Your solution is okay!'; and nanoda 0.4.13, an independent Lean kernel written in Rust, accepted an export of the same environment with 1,052,234 declarations and no errors. The argument formalized is the classical one of Frey, Serre, Ribet, Wiles and Taylor-Wiles, and the development builds on the Imperial College London FLT project, flt-regular and Mathlib, with 106 files containing material from the first two. Lab: Anthropic. Model: Internal Anthropic research model, described in the announcement as roughly comparable to Claude Fable 5.1. Verification: Formally verified. Autonomy: AI-led.

Lean disproof of Krempa's matrix form of the Koethe conjecture

Mon, 07 Sep 2026 00:00:00 +0000

A machine-checked Lean proof refutes the matrix formulation that Krempa showed equivalent to the Koethe conjecture, found by a pre-release model with no human steering of the proof search. Koethe asked in 1930 whether every ring has a largest nil left ideal. Krempa showed in 1972 that this is equivalent to several other statements, among them a statement about matrix ideals, and it is that matrix form which is disproved here. The proof was produced in Epoch AI's LeanOpenProblems harness, in an evaluation run over the 222 research-open statements of the Wikipedia collection of Google DeepMind's Formal Conjectures, at a pinned benchmark commit. The benchmark file states the conjecture and its negation, both with sorry, and the model fills in exactly one. The construction builds a nil algebra from three weighted backward shifts over the algebraic closure of the two-element field. The compared theorem depends on no sorry and on no axioms beyond propext, Quot.sound and Classical.choice. Lab: OpenAI / Epoch AI. Model: GPT-6 Astra (pre-release). Verification: Formally verified. Autonomy: Autonomous.

Erdos-Sos conjecture proved in Lean, in a form marginally weaker than the classical statement

Mon, 07 Sep 2026 00:00:00 +0000

A machine-checked Lean proof establishes the Erdos-Sos conjecture in the benchmark's formalization, which in one parity case assumes one more edge than the classical statement requires. Erdos and Sos conjectured in 1962 that a graph on n vertices with more than (k − 1)n/2 edges contains every tree on k + 1 vertices. The Lean development proves the benchmark's direct formalization, which states the hypothesis as at least (k − 1)n/2 + 1 edges. The proof uses a permutation-word counting argument over ordered host-vertex configurations. It was produced in the same Epoch AI LeanOpenProblems evaluation run as the Koethe disproof, one autonomous attempt per statement, and the compared theorem carries no sorry outside the statement stubs and no axioms beyond the three standard ones. Lab: OpenAI / Epoch AI. Model: GPT-6 Astra (pre-release). Verification: Formally verified. Autonomy: Autonomous.

Disputed claim that Catalan's constant is irrational

Mon, 07 Sep 2026 00:00:00 +0000

A preprint claims a proof that Catalan's constant is irrational, using weighted tails in the Calegari-Dimitrov-Tang framework; a specific technical objection to the argument has been raised publicly and is unresolved. The irrationality of Catalan's constant is a long-standing open problem. The preprint adapts the method Calegari, Dimitrov and Tang used for the linear independence of 1, zeta(2) and L(2, chi_-3), replacing part of the arithmetic input with a construction of suitable weights and weighted tails. The acknowledgements set out an unusually specific division of labour: repeated attempts by the model failed, the author then supplied the weighting idea that made the approach work, and the model produced the numerical data. The paper states that 'the whole proof has passed the verification of Chatgpt 5.6 Solar', which is a machine reading rather than a human check. Lab: Nanjing University. Model: ChatGPT 5.6 Solar, named in the paper as the verifier; the model consulted during the work is named only as AI. Verification: Disputed. Autonomy: AI-assisted.

A quantum oracle separating QMA(2) from QMA

Mon, 07 Sep 2026 00:00:00 +0000

A quantum oracle is constructed relative to which QMA(2) differs from QMA, resolving Watrous's no-disentanglers conjecture, from a proof idea the authors state was generated by a language model. The paper exhibits a quantum oracle relative to which QMA is not equal to QMA(2), and derives the no-disentanglers conjecture of Watrous: for every choice of error parameters summing to less than one, any disentangler requires input size exponential in the number of output qubits. The proof combines the unitarily invariant polynomial method of She and Yuen with a construction based on the symmetric and antisymmetric subspace projectors, reducing the QMA lower bound to the approximate degree of OR. The paper carries a section headed 'Tool and computational resource disclosure' which argues that AI-assisted research should record how findings were obtained, and then discloses the model's role. Lab: MIT / Columbia / University of Washington. Model: ChatGPT 5.6 Sol. Verification: Author verified. Autonomy: AI-assisted.

Confirmation of the Daykin-Frankl conjecture from a language-model proof

Mon, 07 Sep 2026 00:00:00 +0000

A four-page note verifies and communicates a language-model-generated proof of the 1983 Daykin-Frankl conjecture on incomparable elements in convex subsets of the Boolean lattice. Daykin and Frankl conjectured in 1983 that a convex subset P of the n-dimensional Boolean lattice contains at least |P| times the central binomial coefficient divided by 2 to the n pairwise incomparable elements. The note's abstract states its provenance in one sentence: 'We verify and communicate an LLM-generated proof of this conjecture.' The argument proves a stronger product inequality for convex subsets and recovers the original conjecture as a special case, by an induction on dimension resting on a convexity lemma. Lab: Independent. Model: GPT-5.6 Sol Pro. Verification: Author verified. Autonomy: AI-led.

A smooth counterexample to the Trautman conjecture

Mon, 07 Sep 2026 00:00:00 +0000

A strongly pseudoconvex CR three-manifold is constructed that carries a nowhere-zero closed section of its canonical bundle yet is not locally embeddable, disproving the Trautman conjecture. The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in complex two-space. The paper modifies a standard construction of nonembeddable smooth strongly pseudoconvex CR three-manifolds so that the closed-section condition is preserved, producing a strongly pseudoconvex counterexample. The construction proceeds through Hamiltonian perturbations of the Heisenberg CR structure, bump functions on shrinking solid tori and weighted moment identities, and is then globalized to the three-sphere. Lab: Oklahoma State University. Model: ChatGPT Plus, version not named. Verification: Author verified. Autonomy: AI-assisted.

Lean proof that the percolation probability vanishes at the critical point in every dimension

Mon, 07 Sep 2026 00:00:00 +0000

A Lean development proves that nearest-neighbour Bernoulli bond percolation on the integer lattice has no infinite cluster at its critical parameter in every dimension at least two, which would settle the dimensions 3 to 10 that were open. The development proves Kozma and Nitzan's Conjecture 3 (arXiv:2401.12397), which by their Theorem 6 gives that the percolation probability vanishes at the critical point on the d-dimensional integer lattice for every d at least 2. The route is a new conditioned slack hierarchy, a family of conditioned covariance inequalities for increasing functions of a single open cluster whose level-zero case is the Harris inequality, from which a stronger additive gluing inequality is derived. The classical inputs, including Harris, van den Berg, Haggstrom and Kahn, the four functions theorem, Gladkov's decision-tree Harris-Kleitman inequality, the Barsky, Grimmett and Newman half-space theorem and Kesten's critical value for the square lattice, are re-proved from Mathlib, so the statement carries no hypothesis beyond the dimension. The recorded audit reports a clean build, two sorry placeholders confined to the statement file by design, no added axioms, all seven compared theorems depending on exactly the three standard axioms, and a passing comparator run. Lab: Anthropic. Model: Anthropic Claude models, versions not stated. Verification: Claimed. Autonomy: AI-led.

Entropy production of the Boltzmann equation is not always monotone