[ { "id": "2026-07-22-dinitz-garg-goemans", "title": "Counterexample to the Dinitz–Garg–Goemans conjecture", "claim": "An explicit seven-node network whose cheapest single-route (unsplittable) shipping costs more than its fractional cost even under the allowed capacity slack, disproving a conjecture from the late 1990s.", "field": "computer-science", "date": "2026-07-22", "lab": "Independent", "model": "GPT-5.6 Pro", "humans": [ "Dmitry Rybin" ], "year_posed": 1999, "verification": "formal", "autonomy": "ai-led", "detail": "Dinitz, Garg, and Goemans proved that any fractional multicommodity flow can be rounded to an unsplittable (single-path-per-demand) flow while violating each arc's capacity by at most the maximum demand. Goemans conjectured this rounding could also be done without increasing the total cost. The counterexample is a directed graph on seven nodes carrying three demands (15, 10, 15); its fractional solution costs 58, yet every unsplittable routing that stays within the allowed capacity cushion (violation \\(\\le 15\\)) costs at least 60. Rybin reached it with GPT-5.6 Pro in four short prompts and reported that the model returned proof certificates, an exhaustive-enumeration verification program, machine-readable data, and LaTeX source.", "novelty_check": "The Dinitz–Garg–Goemans unsplittable-flow result is from the late 1990s (D. Dinitz, N. Garg, M. Goemans, 'On the single-source unsplittable flow problem'); the cost-preserving strengthening was an open conjecture attributed to Goemans. No prior counterexample or resolution appears in the literature. The construction is a new concrete instance, not a retrieval of an existing example.", "caveats": "Not peer-reviewed. The counterexample was shared on X with a public GPT-5.6 Pro transcript rather than in a paper, and the entry was downgraded from author verified to claimed on 2026-08-02 because no standalone artifact was linked. That gap has since been closed by people other than the author: an Archive of Formal Proofs entry by Arthur Freitas Ramos, David Barros Hulak and Ruy de Queiroz formalizes the seven-vertex instance in Isabelle and proves that no unsplittable routing within the allowed congestion matches the fractional cost, which is the whole content of a finite counterexample, so the grade is formal. The formalization uses its own finite path-flow model of the problem, and its authors disclose AI assistance with the proof engineering. Hensen Juang publicly generalized the instance into an infinite parametric family on the same seven nodes, which is consistent with the claim. Autonomy graded ai-led: the model produced the construction; the human posed the problem and verified.", "independent_checks": [ { "who": "Arthur Freitas Ramos, David Barros Hulak and Ruy J. G. B. de Queiroz (Archive of Formal Proofs)", "url": "https://isa-afp.org/entries/Dinitz_Garg_Goemans_Counterexample.html", "outcome": "Formalized the seven-vertex, nine-arc instance in Isabelle/HOL and proved that its fractional flow of cost 58 cannot be rounded to any unsplittable routing that keeps the cost, even under the non-strict congestion allowance of \\(d_{\\max} = 15\\). AI assistance was used for proof engineering; Isabelle checks the final statements and proofs." } ], "registrations": [ { "registry": "mathdb", "id": "316187", "title": "Dinitz–Garg–Goemans conjecture", "url": "https://mathdb.com/p/316187/dinitz-garg-goemans-conjecture" }, { "registry": "vibemathed", "id": "dinitz-garg-goemans-unsplittable-flow", "url": "https://vibemathed.com/problem/dinitz-garg-goemans-unsplittable-flow" } ], "sources": [ { "label": "Dmitry Rybin announcement (X)", "url": "https://x.com/DmitryRybin1/status/2079907499545919968", "kind": "announcement" }, { "label": "Coverage: officechai", "url": "https://officechai.com/ai/mathematician-says-gpt-5-6-disproved-the-30-year-old-dinitz-garg-goemans-conjecture-with-4-simple-prompts/", "kind": "coverage" }, { "label": "Archive of Formal Proofs: a formal counterexample to the cost-preserving single-source unsplittable flow conjecture", "url": "https://isa-afp.org/entries/Dinitz_Garg_Goemans_Counterexample.html", "kind": "research" } ], "discussion": [ { "label": "Hensen Juang: infinite family generalizing the counterexample (X)", "url": "https://x.com/basedjensen/status/2080020575968543230" } ], "tags": [ "combinatorial-optimization", "network-flow", "counterexample", "unsplittable-flow" ], "added": "2026-07-24", "revisions": [ { "date": "2026-08-02", "kind": "regraded", "note": "Downgraded from author verified to claimed: classifying the sources showed no primary artifact, only a chat transcript." }, { "date": "2026-08-19", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration." }, { "date": "2026-08-21", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. It sits alongside the MathDB record already cited. No grade moved." }, { "date": "2026-09-24", "kind": "regraded", "note": "Upgraded from claimed to formally verified: an Archive of Formal Proofs entry by three authors unconnected to the announcement machine-checks the counterexample in Isabelle, supplying the standalone artifact whose absence caused the August downgrade.", "url": "https://isa-afp.org/entries/Dinitz_Garg_Goemans_Counterexample.html" } ] }, { "id": "2026-07-19-jacobian-conjecture", "title": "Counterexample to the Jacobian conjecture in dimension three", "claim": "An explicit polynomial map in three variables with constant Jacobian determinant \\(-2\\) that is nevertheless not invertible, disproving a conjecture open since 1939.", "field": "mathematics", "date": "2026-07-19", "lab": "Anthropic", "model": "Claude Fable 5", "humans": [ "Levent Alpöge" ], "year_posed": 1939, "verification": "formal", "autonomy": "collaborative", "detail": "The map \\[F(x,y,z) = \\bigl(u^3z + y^2u(4+3xy),\\ y + 3xu^2z + 3xy^2(4+3xy),\\ 2x - 3x^2y - x^3z\\bigr)\\] with \\(u = 1+xy\\) has Jacobian determinant identically \\(-2\\), yet sends the three distinct points \\((0, 0, -1/4)\\), \\((1, -3/2, 13/2)\\) and \\((-1, 3/2, 13/2)\\) all to \\((-1/4, 0, 0)\\). A map with a global inverse cannot be three-to-one. Alpöge, a number theorist at Anthropic, announced it on X the day it was found.", "novelty_check": "The Jacobian conjecture (Keller, 1939) has been a celebrated open problem for 87 years, with many published false proofs in both directions. No prior counterexample in any dimension over characteristic 0 exists in the literature. Ott-Heinrich Keller's original formulation is the one addressed. A separate question was opened in September 2026, about provenance rather than priority of the counterexample: the supplemental statement on page 160 of arXiv:2609.05746 asserts prior systematic use of the \\(xy\\), \\(1+xy\\) and \\(\\delta+xy\\) constructions, including the map to the projective line given by \\((x : 1+xy)\\), in an in-progress monograph circulated publicly in January 2025. That bears on where the construction came from, not on whether a counterexample already existed, and the caveats record what is and is not claimed.", "caveats": "Not yet peer-reviewed. The formula is public and checkable in seconds by computer algebra, which makes conventional peer review less load-bearing than usual, but the official record lists the conjecture as open until the literature catches up. The division of labor between Alpöge and the model has not been fully documented; autonomy graded conservatively pending a transcript. A priority and disclosure question was raised on the record in September 2026 and is unresolved. Adrian Vasiu, the third author of the Borisov, Gabber and Vasiu monograph on endomorphisms of affine spaces, appended a signed supplemental statement to arXiv:2609.05746 recording that an in-progress version of that work, dated 14 January 2025, was posted publicly at IHES two days later without his knowledge, and that in his understanding it subsequently fed AI models. He makes four numbered points: that no prior work aimed to classify etale endomorphisms of affine spaces of geometric degree three; that no prior work systematically used \\(xy\\), \\(1+xy\\) and \\(\\delta+xy\\) in constructing open embeddings, morphisms and etale endomorphisms, including the map \\((x : 1+xy)\\); that between the monograph and the seven-page write-up of this counterexample there is overlap of notation and of intent; and that the prompt history, model version and workflow behind that write-up have not been disclosed publicly. What he does not claim is that the counterexample itself appears in the earlier work, and he notes that one of the overlapping constructions plays no role in the counterexamples. So this is a question about where the construction came from and what was disclosed, not a challenge to whether the map is a counterexample, and the formal grade is unaffected. It is not a generic complaint either: \\((x : 1+xy)\\) is the device the seven-page paper's own abstract is built on.", "independent_checks": [ { "who": "whataifound.org (symbolic recomputation, SymPy)", "outcome": "confirmed, \\(\\det J = -2\\) identically; all three points map to \\((-1/4, 0, 0)\\)" }, { "who": "Multiple mathematicians via public computer-algebra checks", "url": "https://news.ycombinator.com/item?id=48973869", "outcome": "confirmed" } ], "registrations": [ { "registry": "mathdb", "id": "315779", "title": "Jacobian conjecture", "url": "https://mathdb.com/p/315779/jacobian-conjecture" }, { "registry": "vibemathed", "id": "jacobian-conjecture", "url": "https://vibemathed.com/problem/jacobian-conjecture" }, { "registry": "palomar", "id": "PALOMAR-2026-08-21-000006", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-21-000006&version=1", "repository": "Paul-Lez/jacobian-conjecture", "commit": "b244ea482ed086297e379d73e6724cd26b2d301f", "theorems": [ "JacobianConjecture.jacobian_conjecture" ], "checked": "2026-08-21", "note": "A Comparator wrapper of the formalization by Paul Lezeau and Dean Cureton in Google DeepMind's formal-conjectures, whose Lean source names the construction as Alpöge and Fable's counterexample. It records a stronger statement than this entry claims, disproving the generalized conjecture over every nontrivial commutative ring rather than in characteristic zero alone, and omits the two-variable case, which stays open." }, { "registry": "palomar", "id": "PALOMAR-2026-08-27-000013", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-27-000013&version=1", "repository": "Arthur742Ramos/jacobian-conjecture-lean", "commit": "8db55ea3ffa8ba90094d6ef78ee3351ddfce01ca", "theorems": [ "JacobianCounterexample.jacobianDet_F", "JacobianCounterexample.explicit_three_point_fibre", "JacobianCounterexample.not_jacobian_conjecture_complex" ], "checked": "2026-08-27", "note": "A second and independent Lean formalization of the same map, by Ramos, Hulak and de Queiroz, sharing no code with the Lezeau and Cureton development already cited. It records the two-variable case as untouched, as this entry does." } ], "sources": [ { "label": "The Jacobian counterexample, explained", "url": "https://jacobianfun.org/jacobian-explained", "kind": "research" }, { "label": "Ulam: A counterexample to the Jacobian conjecture (PDF), the seven-page algebraic verification of the announced map", "url": "https://www.ulam.ai/research/jacobian.pdf", "kind": "research" }, { "label": "Alexis Gallagher: Jacobian Conjecture Disproved!", "url": "https://alexisgallagher.com/posts/2026/jacobianfun/", "kind": "commentary" }, { "label": "Hacker News discussion", "url": "https://news.ycombinator.com/item?id=48973869", "kind": "commentary" }, { "label": "Borisov, Gabber and Vasiu: on endomorphisms of affine spaces and the Jacobian problem, whose supplemental statement on page 160 raises priority and disclosure questions", "url": "https://arxiv.org/abs/2609.05746", "kind": "challenge" } ], "tags": [ "algebraic-geometry", "counterexample", "polynomial-maps" ], "added": "2026-07-20", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration." }, { "date": "2026-08-21", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. No grade moved." }, { "date": "2026-08-25", "kind": "check", "note": "A third-party Lean formalization of this counterexample was registered at Palomar, mechanically confirming that it typechecks against the stated theorem at a pinned commit. The formalization is by Lezeau and Cureton rather than by the counterexample's author. The grade was already formal and did not move.", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-21-000006&version=1" }, { "date": "2026-08-31", "kind": "check", "note": "A second independent Lean formalization of this counterexample was registered at Palomar, by a different group from the one already cited. Two independent developments of the same construction reinforce the formal grade rather than change it.", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-27-000013&version=1" }, { "date": "2026-09-14", "kind": "challenge", "note": "Cited the supplemental statement on page 160 of arXiv:2609.05746, in which Adrian Vasiu raises a priority and disclosure question about the construction behind this counterexample. Neither grade moved: the statement does not dispute that the map is a counterexample, and does not claim the counterexample appears in the earlier monograph.", "url": "https://arxiv.org/abs/2609.05746" } ] }, { "id": "2026-05-erdos-unit-distance", "title": "Disproof of the Erdős unit-distance conjecture", "claim": "A reasoning model produced the core construction disproving a 1946 conjecture in discrete geometry, finding point configurations with more unit-distance pairs than the conjecture permitted.", "field": "mathematics", "date": "2026-05-20", "lab": "OpenAI", "model": "GPT-5 series reasoning model", "year_posed": 1946, "verification": "independent", "autonomy": "ai-led", "detail": "The model produced a construction beating the conjectured bound, with an inexplicit exponent greater than 1. Nine mathematicians - Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood - published a human-verified version of the argument; Sawin separately made the exponent explicit at \\(n^{1.014}\\). Gowers called it 'the first example of a result produced autonomously by an AI that I find exciting in itself.'", "novelty_check": "The unit-distance problem dates to Erdős (1946). Prior bounds were well documented; the construction is new.", "caveats": "Human mathematicians shaped the problem framing and verified the construction. The explicit \\(n^{1.014}\\) exponent is Sawin's refinement, not the model's output: the model's own bound was inexplicit. The strength of the endorsement from Gowers is notable but is a judgment, not a formal check.", "independent_checks": [ { "who": "Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood", "url": "https://arxiv.org/abs/2605.20695", "outcome": "Nine mathematicians outside OpenAI wrote a digested, human-verified version of the construction and placed it against the work of Ellenberg-Venkatesh, Golod-Shafarevich and Hajir-Maire-Ramakrishna." }, { "who": "Will Sawin (explicit exponent)", "url": "https://arxiv.org/abs/2605.20579", "outcome": "Refined the argument to an explicit exponent of about 1.014, which requires working through it." }, { "who": "Kim Morrison (Palomar comparator registration)", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-08-000001&version=3", "outcome": "Machine-checked the negation of the uniform-constant form of the conjecture, narrower than the entry's claim." } ], "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-08-08-000001", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-08-000001&version=3", "repository": "kim-em/erdos-unit-distance-comparator", "commit": "be6c2ee4c9fb16fd6bed442b4c361fb10369beb5", "theorems": [ "UnitDistance.erdos_unit_distance_uniform_constant_false" ], "checked": "2026-08-14", "note": "Palomar's record describes this as certifying the literal negation of the uniform-constant form of the conjecture, from a Mathlib-only statement of that negation. That is narrower than this entry's claim, which is why the entry stays at independently checked rather than moving to formally verified." }, { "registry": "mathdb", "id": "315785", "title": "Erdős unit-distance conjecture", "url": "https://mathdb.com/p/315785/erdos-unit-distance-conjecture" } ], "sources": [ { "label": "Remarks on the disproof of the unit distance conjecture (human-verified writeup)", "url": "https://arxiv.org/abs/2605.20695", "kind": "research" }, { "label": "Sawin: An explicit lower bound for the unit distance problem", "url": "https://arxiv.org/abs/2605.20579", "kind": "research" }, { "label": "Quanta: The AI Revolution in Math Has Arrived", "url": "https://www.quantamagazine.org/the-ai-revolution-in-math-has-arrived-20260413/", "kind": "coverage" }, { "label": "TechCrunch: OpenAI claims it solved an 80-year-old math problem, for real this time", "url": "https://techcrunch.com/2026/05/20/openai-claims-it-solved-an-80-year-old-math-problem-for-real-this-time/", "kind": "coverage" }, { "label": "Tech Jacks: AI math reasoning milestones", "url": "https://techjacksolutions.com/ai-brief/ai-math-reasoning-milestones-30-days-research-automation/", "kind": "coverage" } ], "tags": [ "discrete-geometry", "combinatorics", "erdos" ], "added": "2026-07-20", "revisions": [ { "date": "2026-08-19", "kind": "check", "note": "Palomar registered Kim Morrison's comparator wrapper as PALOMAR-2026-08-08-000001, a machine-checked formalization by someone outside the group that produced the writeup. The verification grade did not move: what is proved formally is the uniform-constant form, which is narrower than this entry's claim.", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-08-000001&version=3" }, { "date": "2026-08-19", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration." }, { "date": "2026-09-24", "kind": "check", "note": "Recorded the checks the independently checked grade already rested on: the nine-author human-verified write-up, Sawin's explicit refinement and Kim Morrison's Palomar formalization. No grade moved.", "url": "https://arxiv.org/abs/2605.20695" } ] }, { "id": "2026-01-erdos-728", "title": "Erdős problem #728 resolved and formalized in Lean", "claim": "GPT-5.2 produced a proof of a previously open Erdős problem; Harmonic's Aristotle formalized it in Lean, making it the first Erdős problem regarded as fully resolved by AI with machine-checked proof.", "field": "mathematics", "date": "2026-01-13", "lab": "OpenAI / Harmonic", "model": "GPT-5.2 Pro + Aristotle", "verification": "formal", "autonomy": "autonomous", "detail": "The problem concerns the existence of infinitely many integers a, b, n satisfying divisibility and inequality conditions involving factorials. GPT-5.2 generated the argument, Aristotle formalized it in Lean, and a human-readable writeup was posted to arXiv. Modifications of the same argument also resolved problems #729 and #401.", "novelty_check": "Recorded as open in the Erdős problems database prior to resolution. Terence Tao's AI-contributions wiki tracked the resolution and did not find the result in existing literature.", "caveats": "Tao cautioned that the win 'says more about speed than difficulty': the problem was open but not considered deep. Lean formalization makes correctness essentially certain; significance is the contested part, not validity.", "independent_checks": [ { "who": "Lean kernel (machine-checked)", "url": "https://arxiv.org/abs/2601.07421", "outcome": "verified" }, { "who": "Terence Tao", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "outcome": "accepted" } ], "sources": [ { "label": "arXiv 2601.07421: Resolution of Erdős Problem #728", "url": "https://arxiv.org/abs/2601.07421", "kind": "research" }, { "label": "Tao: AI contributions to Erdős problems", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "kind": "research" }, { "label": "The Decoder: Tao on GPT-5.2 Pro", "url": "https://the-decoder.com/terence-tao-says-gpt-5-2-pro-cracked-an-erdos-problem-but-warns-the-win-says-more-about-speed-than-difficulty/", "kind": "coverage" } ], "tags": [ "number-theory", "erdos", "lean", "formalization" ], "added": "2026-07-20", "registrations": [ { "registry": "mathdb", "id": "315787", "title": "Erdős Problem #728", "url": "https://mathdb.com/p/315787/erdos-problem-728" } ], "revisions": [ { "date": "2026-08-21", "kind": "correction", "note": "Recorded the MathDB record for this problem as a registration. No grade moved." } ] }, { "id": "2026-01-alphaevolve-bruhat", "title": "Structure in Bruhat intervals of permutation groups", "claim": "AlphaEvolve identified unexpected special structure in Bruhat intervals for particular permutation groups.", "field": "mathematics", "date": "2026-01-15", "lab": "Google DeepMind", "model": "AlphaEvolve (Gemini-based)", "verification": "independent", "autonomy": "search-scaffold", "detail": "Part of a continuing series of AlphaEvolve results in combinatorics. The system evolves candidate programs under a human-specified evaluation function, so the search is automated but the problem framing is human.", "novelty_check": "Documented as a new observation in the Quanta coverage; the specific structure was not previously catalogued.", "caveats": "The paper is an arXiv preprint, not yet peer reviewed, so this entry was downgraded from 'peer reviewed' once the artifact was located. The five authors are academic mathematicians unaffiliated with the announcing lab, and they turned AlphaEvolve's suggested permutations into proved theorems, which is what the 'independently checked' grade rests on.", "sources": [ { "label": "Ellenberg, Libedinsky, Plaza, Simental, Williamson: Bruhat intervals that are large hypercubes", "url": "https://arxiv.org/abs/2601.01235", "kind": "research" }, { "label": "Quanta: The AI Revolution in Math Has Arrived", "url": "https://www.quantamagazine.org/the-ai-revolution-in-math-has-arrived-20260413/", "kind": "coverage" } ], "tags": [ "combinatorics", "alphaevolve" ], "added": "2026-07-20", "revisions": [ { "date": "2026-08-02", "kind": "regraded", "note": "Downgraded from peer reviewed to independently checked: the linked artifact is an arXiv preprint, not a reviewed paper." } ] }, { "id": "2025-05-alphaevolve-matmul", "title": "4×4 complex matrix multiplication in 48 scalar multiplications", "claim": "AlphaEvolve found a scheme multiplying 4×4 complex matrices with 48 scalar multiplications, improving on Strassen's 49 from 1969.", "field": "computer-science", "date": "2025-05-14", "lab": "Google DeepMind", "model": "AlphaEvolve (Gemini-based)", "year_posed": 1969, "verification": "independent", "autonomy": "search-scaffold", "detail": "Strassen's 1969 algorithm had stood at 49 multiplications for 56 years in this setting. AlphaEvolve was not purpose-built for matrix multiplication; it is a general evolutionary coding agent. The scheme is exactly checkable.", "novelty_check": "Strassen (1969) and subsequent literature exhaustively catalogued; 49 was the standing record for complex-valued 4×4. Confirmed new.", "caveats": "Applies to complex-valued matrices specifically. Practical speedup is limited; the significance is theoretical. Related follow-up work by others has explored rank-23 schemes for 3×3.", "sources": [ { "label": "AlphaEvolve paper (PDF)", "url": "https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf", "kind": "research" }, { "label": "DeepMind: AlphaEvolve", "url": "https://deepmind.google/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/", "kind": "announcement" }, { "label": "IEEE Spectrum", "url": "https://spectrum.ieee.org/deepmind-alphaevolve", "kind": "coverage" }, { "label": "Wikipedia: AlphaEvolve", "url": "https://en.wikipedia.org/wiki/AlphaEvolve", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: AlphaEvolve announcement", "url": "https://news.ycombinator.com/item?id=43985489" }, { "label": "Hacker News: AlphaEvolve independently discovered a novel matrix-multiplication algorithm", "url": "https://news.ycombinator.com/item?id=45803805" } ], "tags": [ "algorithms", "linear-algebra", "alphaevolve" ], "added": "2026-07-20" }, { "id": "2025-05-alphaevolve-kissing", "title": "Improved lower bound for the 11-dimensional kissing number", "claim": "AlphaEvolve improved the best known configuration for the kissing number problem in 11 dimensions.", "field": "mathematics", "date": "2025-05-14", "lab": "Google DeepMind", "model": "AlphaEvolve (Gemini-based)", "year_posed": 1694, "verification": "independent", "autonomy": "search-scaffold", "detail": "Part of a sweep across 50+ open problems in analysis, geometry, combinatorics and number theory. AlphaEvolve rediscovered state-of-the-art solutions in roughly 75% of cases and improved on the best known in about 20%. The kissing number problem asks how many non-overlapping unit spheres can touch a central one.", "novelty_check": "Kissing number bounds are well catalogued per dimension; the 11D improvement was checked against the standing record.", "caveats": "A lower-bound improvement via explicit construction, not a resolution of the problem. The 20% improvement rate means most of the 50+ problems saw no advance.", "registrations": [ { "registry": "mathdb", "id": "315951", "title": "Kissing number problem", "url": "https://mathdb.com/p/315951/kissing-number-problem" } ], "sources": [ { "label": "AlphaEvolve paper (PDF)", "url": "https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf", "kind": "research" }, { "label": "DeepMind: AlphaEvolve", "url": "https://deepmind.google/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/", "kind": "announcement" }, { "label": "IEEE Spectrum: AlphaEvolve Tackles Kissing Problem", "url": "https://spectrum.ieee.org/deepmind-alphaevolve", "kind": "coverage" } ], "discussion": [ { "label": "Hacker News: AlphaEvolve announcement", "url": "https://news.ycombinator.com/item?id=43985489" } ], "tags": [ "geometry", "sphere-packing", "alphaevolve" ], "added": "2026-07-20", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration." } ] }, { "id": "2025-08-gpt5-convex-bound", "title": "Improved step-size bound in smooth convex optimization", "claim": "GPT-5 Pro extended a guaranteed-convexity window for gradient descent from \\(\\eta \\le 1/L\\) to \\(\\eta \\le 1.5/L\\), but the optimal \\(1.75/L\\) bound had already been published months earlier.", "field": "mathematics", "date": "2025-08-01", "lab": "OpenAI", "model": "GPT-5 Pro", "humans": [ "Sébastien Bubeck" ], "verification": "known", "autonomy": "ai-assisted", "detail": "Bubeck posed an open problem from a convex optimization paper. After about 17 minutes of reasoning the model produced an improved bound using Bregman divergence inequalities and cocoercivity. Bubeck verified the proof as correct and described it as new mathematics.", "novelty_check": "Version 2 of the source paper, published 2 April 2025, had already established the optimal 1.75/L bound, strictly stronger than the model's 1.5/L. The model was working from an earlier version and its result was superseded before it was produced.", "caveats": "The proof itself is valid; the novelty claim is not. Analysts noted the argument largely recombined known techniques in different notation. Retained as a cautionary entry: this is the single most common failure mode in this space, and the reason every entry carries a novelty check.", "sources": [ { "label": "Analysis of the claim", "url": "https://hackmd.io/@niloydebbarma/HkGxVPwFgx", "kind": "challenge" }, { "label": "What does GPT-5's 'new math' claim actually mean?", "url": "https://allthings.how/what-does-gpt-5s-new-math-claim-actually-mean/", "kind": "challenge" } ], "discussion": [ { "label": "Hacker News: claim that GPT-5-pro can prove new interesting mathematics", "url": "https://news.ycombinator.com/item?id=44965577" } ], "tags": [ "optimization", "already-known", "cautionary" ], "added": "2026-07-20" }, { "id": "2025-11-gpt5-science-acceleration", "title": "Early science acceleration experiments with GPT-5", "claim": "A multi-domain study documenting cases where GPT-5 contributed to research progress across mathematics, physics, biology and materials science.", "field": "computer-science", "date": "2025-11-20", "lab": "OpenAI", "model": "GPT-5", "verification": "peer-reviewed", "autonomy": "ai-assisted", "detail": "A collection of case studies rather than a single result. Useful as a source of individual entries; each contained claim needs separate grading before it belongs in the registry on its own.", "novelty_check": "Not applicable. This is a meta-report, not a single claim.", "caveats": "Lab-authored evaluation of its own model. Individual case studies vary widely in strength and several have not been independently checked.", "sources": [ { "label": "arXiv 2511.16072", "url": "https://arxiv.org/pdf/2511.16072", "kind": "research" } ], "tags": [ "meta", "multi-domain" ], "added": "2026-07-20" }, { "id": "2023-12-funsearch-capset", "title": "New lower bound constructions for the cap set problem", "claim": "FunSearch discovered larger cap sets than any previously known construction, the first time an LLM-based system produced a genuinely new discovery on an established open problem.", "field": "mathematics", "date": "2023-12-14", "lab": "Google DeepMind", "model": "FunSearch (PaLM 2 / Codey)", "year_posed": 1970, "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "FunSearch pairs an LLM that proposes programs with an automated evaluator that rejects incorrect ones, sidestepping hallucination by construction. Applied to extremal combinatorics, it found new large cap sets in both finite-dimensional and asymptotic cases. Published in Nature. Historically the first entry in this registry's scope.", "novelty_check": "Cap set bounds are a well-tracked literature; the constructions were confirmed to exceed the standing record at publication.", "caveats": "The LLM proposes; a human-designed evaluator and search loop does the selecting. Calling this 'the AI discovered it' overstates the model's role, though the discovery is real.", "sources": [ { "label": "Nature: Mathematical discoveries from program search with LLMs", "url": "https://www.nature.com/articles/s41586-023-06924-6", "kind": "research" }, { "label": "FunSearch code and discovered programs", "url": "https://github.com/google-deepmind/funsearch", "kind": "research" }, { "label": "DeepMind: FunSearch", "url": "https://deepmind.google/blog/funsearch-making-new-discoveries-in-mathematical-sciences-using-large-language-models/", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: FunSearch makes new discoveries in mathematical sciences", "url": "https://news.ycombinator.com/item?id=38643076" } ], "tags": [ "combinatorics", "cap-set", "funsearch", "historic-first" ], "added": "2026-07-20" }, { "id": "2023-12-funsearch-binpacking", "title": "Improved heuristics for online bin packing", "claim": "FunSearch produced bin-packing heuristics outperforming standard baselines on benchmark distributions.", "field": "computer-science", "date": "2023-12-14", "lab": "Google DeepMind", "model": "FunSearch (PaLM 2 / Codey)", "year_posed": 1971, "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "Discovered programs are human-readable, which allowed domain experts to inspect and deploy them. Practical rather than theoretical significance.", "novelty_check": "Compared against best-fit and first-fit families and published heuristics; improvements are empirical on tested distributions.", "caveats": "An empirical improvement on benchmark distributions, not a proved worst-case bound.", "sources": [ { "label": "Nature: Mathematical discoveries from program search with LLMs", "url": "https://www.nature.com/articles/s41586-023-06924-6", "kind": "research" }, { "label": "FunSearch code and discovered programs", "url": "https://github.com/google-deepmind/funsearch", "kind": "research" } ], "discussion": [ { "label": "Hacker News: FunSearch makes new discoveries in mathematical sciences", "url": "https://news.ycombinator.com/item?id=38643076" } ], "tags": [ "algorithms", "heuristics", "funsearch" ], "added": "2026-07-20" }, { "id": "2025-02-ai-coscientist-amr", "title": "AI co-scientist hypotheses on antimicrobial resistance and liver fibrosis", "claim": "A multi-agent system generated hypotheses that were subsequently validated experimentally in the lab.", "field": "biology", "date": "2025-02-19", "lab": "Google", "model": "AI Co-Scientist (Gemini 2.0 multi-agent)", "verification": "peer-reviewed", "autonomy": "ai-assisted", "detail": "The system proposed mechanisms in antimicrobial resistance and liver fibrosis that wet-lab work then supported. Notable as one of the earliest cases of an AI-generated hypothesis surviving experimental test rather than just sounding plausible. The peer-reviewed version, published in Nature in May 2026, covers three biomedical applications together: drug repurposing, novel-target discovery and explaining mechanisms of antimicrobial resistance. It adds drug-repurposing candidates and synergistic combination therapies for acute myeloid leukaemia that were then validated in vitro.", "novelty_check": "The AMR mechanism proposed had reportedly been independently arrived at by a human group whose work was unpublished at the time, which complicates the novelty claim.", "caveats": "Hypothesis generation with human-run validation, not autonomous discovery. Peer review confirms the work, not that every hypothesis in it was unprecedented, and the novelty question above is untouched by publication. Independent replication outside the collaborating labs is limited. Graded conservatively.", "sources": [ { "label": "Nature: Accelerating scientific discovery with Co-Scientist", "url": "https://www.nature.com/articles/s41586-026-10644-y", "kind": "research" }, { "label": "bioRxiv: AI mirrors experimental science to uncover a novel mechanism of gene transfer", "url": "https://www.biorxiv.org/content/10.1101/2025.02.19.639094v1", "kind": "research" }, { "label": "Turing Post: 12 AI Co-Scientists of 2026", "url": "https://www.turingpost.com/p/ai-co-scientists-in-2026", "kind": "coverage" } ], "tags": [ "hypothesis-generation", "microbiology", "multi-agent" ], "revisions": [ { "date": "2026-08-14", "kind": "regraded", "note": "Upgraded from author verified to peer reviewed: the work was published in Nature (655, 487-496) on 19 May 2026, covering drug repurposing, novel-target discovery and antimicrobial resistance mechanisms together.", "url": "https://www.nature.com/articles/s41586-026-10644-y" } ], "added": "2026-07-20" }, { "id": "2025-robin-macular", "title": "Candidate treatment for dry age-related macular degeneration", "claim": "The Robin system automated hypothesis generation, experiment design and data analysis, identifying a novel candidate treatment for dry AMD.", "field": "biology", "date": "2025-05-01", "lab": "FutureHouse", "model": "Robin (multi-agent)", "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "Presented as an end-to-end automated discovery loop with humans executing the physical experiments. Robin proposed enhancing retinal pigment epithelium phagocytosis as a therapeutic strategy, then identified ripasudil and KL001 as candidates and confirmed in vitro efficacy for both. It went on to propose and analyse a follow-up RNA sequencing experiment, which showed upregulation of ABCA1, a lipid efflux pump and a possible novel target in its own right. Published in Nature in May 2026.", "novelty_check": "Ripasudil is a clinically used Rho kinase inhibitor; the paper states that to the authors' knowledge it had never previously been proposed for the treatment of dry age-related macular degeneration, and the peer-reviewed version carries that claim. The ABCA1 mechanism emerged from the system's own follow-up experiment rather than from the prior literature. What is new is the repurposing hypothesis and its in vitro support, not a new compound.", "caveats": "Peer reviewed, which is the right grade for an empirical claim of this kind, but in vitro efficacy is a long way from a treatment: a candidate is many years and several trial phases from the clinic. Verification of the biology is the journal's, not an independent replication. Autonomy stays ai-led rather than autonomous: the paper describes the approach as semi-autonomous, and humans executed every physical experiment, though all hypotheses, experimental directions, data analyses and main-text figures were produced by Robin.", "sources": [ { "label": "Nature: A multi-agent system for automating scientific discovery", "url": "https://www.nature.com/articles/s41586-026-10652-y", "kind": "research" }, { "label": "Turing Post: 12 AI Co-Scientists of 2026", "url": "https://www.turingpost.com/p/ai-co-scientists-in-2026", "kind": "coverage" } ], "tags": [ "drug-discovery", "ophthalmology", "multi-agent" ], "revisions": [ { "date": "2026-08-14", "kind": "regraded", "note": "Upgraded from claimed to peer reviewed: the work was published in Nature (655, 497-505) on 19 May 2026, which also supplies the primary source the entry previously lacked.", "url": "https://www.nature.com/articles/s41586-026-10652-y" } ], "added": "2026-07-20" }, { "id": "2022-10-05-alphatensor", "title": "Faster matrix-multiplication algorithms found by reinforcement learning", "claim": "AlphaTensor discovered matrix-multiplication schemes using fewer scalar multiplications than any previously known, including a way to multiply 4×4 matrices over GF(2) in 47 multiplications versus the 49 of two-level Strassen.", "field": "computer-science", "date": "2022-10-05", "lab": "Google DeepMind", "model": "AlphaTensor (AlphaZero-based)", "year_posed": 1969, "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "AlphaTensor recast the search for matrix-multiplication algorithms as a single-player game of decomposing a 3-D tensor into rank-one terms, then trained an AlphaZero-style agent to play it. It matched or beat the best known rank for many matrix sizes, and over GF(2) found a 4×4 scheme using 47 multiplications, improving on the 49 obtained by applying Strassen's 1969 algorithm twice. Every decomposition it returns is an exact identity checkable by hand. Published in Nature.", "novelty_check": "Fast matrix multiplication has an exhaustively catalogued literature going back to Strassen (1969) and Laderman (1976). AlphaTensor's schemes were checked against the standing per-size records; the 4×4 result over GF(2) was new at publication.", "caveats": "The 4×4 improvement is specific to characteristic-2 arithmetic and the practical speedup is limited. Within about a week, Kauers and Moosbauer used a classical computer search to improve the 5×5 GF(2) case from AlphaTensor's 96 multiplications to 95, showing conventional methods remained competitive. 'Discovered' here means an RL search over decompositions, not an autonomous mathematical insight.", "independent_checks": [ { "who": "Kauers & Moosbauer (computer-algebra search)", "url": "https://www.quantamagazine.org/ai-reveals-new-possibilities-in-matrix-multiplication-20221123/", "outcome": "verified and shortly improved on the 5×5 case" } ], "sources": [ { "label": "Nature: Discovering faster matrix multiplication algorithms with reinforcement learning", "url": "https://www.nature.com/articles/s41586-022-05172-4", "kind": "research" }, { "label": "AlphaTensor code and discovered algorithms", "url": "https://github.com/google-deepmind/alphatensor", "kind": "research" }, { "label": "DeepMind: Discovering novel algorithms with AlphaTensor", "url": "https://deepmind.google/discover/blog/discovering-novel-algorithms-with-alphatensor/", "kind": "announcement" }, { "label": "Quanta: AI Reveals New Possibilities in Matrix Multiplication", "url": "https://www.quantamagazine.org/ai-reveals-new-possibilities-in-matrix-multiplication-20221123/", "kind": "coverage" } ], "discussion": [ { "label": "Hacker News: Discovering novel algorithms with AlphaTensor", "url": "https://news.ycombinator.com/item?id=33096667" }, { "label": "Hacker News: On AlphaTensor's new matrix multiplication algorithms", "url": "https://news.ycombinator.com/item?id=33117192" }, { "label": "Hacker News: human search beats AlphaTensor's 5×5 result days later", "url": "https://news.ycombinator.com/item?id=33180316" }, { "label": "Hacker News: Discovering faster matrix multiplication algorithms with reinforcement learning", "url": "https://news.ycombinator.com/item?id=33096580" } ], "tags": [ "algorithms", "linear-algebra", "reinforcement-learning" ], "added": "2026-07-20", "videos": [ { "label": "How AI Discovered a Faster Matrix Multiplication Algorithm", "channel": "Quanta Magazine", "youtube_id": "fDAPJ7rvcUw" }, { "label": "AlphaTensor by DeepMind explained", "channel": "Yannic Kilcher", "youtube_id": "3N3Bl5AA5QU" } ] }, { "id": "2023-06-07-alphadev", "title": "Faster sorting routines discovered and merged into the LLVM C++ library", "claim": "AlphaDev found shorter branchless routines for sorting small fixed-length inputs; reverse-engineered to C++, they were merged into LLVM's libc++, the first change to those routines in over a decade.", "field": "computer-science", "date": "2023-06-07", "lab": "Google DeepMind", "model": "AlphaDev (AlphaZero-based)", "verification": "independent", "autonomy": "search-scaffold", "detail": "AlphaDev treated the construction of a sorting routine as a single-player game played directly in CPU assembly, rewarding shorter and faster correct programs. For sort-3, sort-4 and sort-5 it removed instructions relative to the human-tuned library code, yielding large speedups on short sequences. The routines were reverse-engineered to C++ and accepted into the LLVM libc++ standard sort, which ships to millions of users. Published in Nature.", "novelty_check": "The libc++ small-sort routines had been hand-optimised by compiler engineers for years. AlphaDev's shorter instruction sequences were reviewed against the standing implementations and confirmed to be improvements before being merged.", "caveats": "The gains are on very short fixed-length inputs; asymptotic sorting complexity is unchanged, and the headline percentage speedups apply only to those small cases. The 'independent' grade rests on LLVM maintainers reviewing and merging the code, not on a formal proof of optimality.", "independent_checks": [ { "who": "LLVM libc++ maintainers (code review and merge)", "url": "https://en.wikipedia.org/wiki/AlphaDev", "outcome": "reviewed and merged into the standard sort library" } ], "sources": [ { "label": "Nature: Faster sorting algorithms discovered using deep reinforcement learning", "url": "https://www.nature.com/articles/s41586-023-06004-9", "kind": "research" }, { "label": "AlphaDev code and discovered sorting routines", "url": "https://github.com/google-deepmind/alphadev", "kind": "research" }, { "label": "DeepMind: AlphaDev discovers faster sorting algorithms", "url": "https://deepmind.google/blog/alphadev-discovers-faster-sorting-algorithms/", "kind": "announcement" }, { "label": "Wikipedia: AlphaDev", "url": "https://en.wikipedia.org/wiki/AlphaDev", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: AlphaDev, faster sorting algorithms via deep RL", "url": "https://news.ycombinator.com/item?id=36228125" } ], "videos": [ { "label": "AlphaDev: Discovering Faster Sorting Algorithms with Reinforcement Learning", "channel": "AI for Good (ITU)", "youtube_id": "_wDCndqGiRE" } ], "tags": [ "algorithms", "sorting", "reinforcement-learning", "deployed" ], "added": "2026-07-20" }, { "id": "2023-11-29-gnome", "title": "Large-scale prediction of new stable crystalline materials (GNoME)", "claim": "GNoME predicted about 2.2 million new inorganic crystal structures, roughly 380,000 of them flagged as thermodynamically stable, presented as an order-of-magnitude expansion of known stable materials.", "field": "materials", "date": "2023-11-29", "lab": "Google DeepMind", "model": "GNoME (graph neural network)", "verification": "disputed", "autonomy": "search-scaffold", "detail": "Graph Networks for Materials Exploration (GNoME) is a graph neural network trained in an active-learning loop against density-functional-theory calculations to predict formation energies and filter candidates by stability. DeepMind released the predicted structures as a public dataset. Published in Nature.", "novelty_check": "Stable-material prediction builds on the Materials Project and OQMD convex-hull databases. The dispute below is precisely about how much of GNoME's set is genuinely new relative to those existing catalogues.", "caveats": "Substantive objections were raised and remain unresolved. Cheetham and Seshadri, reviewing the release, found 'scant evidence for compounds that fulfil the trifecta of novelty, credibility and utility,' noting that many entries are minor substitutional variants, radioactive, or otherwise unlikely to be useful. Stability here means a DFT convex-hull prediction, not experimental synthesis; the headline counts are predicted, not made.", "independent_checks": [ { "who": "Cheetham & Seshadri (critical review)", "url": "https://www.theregister.com/2024/04/11/google_deepmind_material_study/", "outcome": "disputed the novelty and utility of most predicted compounds" } ], "sources": [ { "label": "Nature: Scaling deep learning for materials discovery", "url": "https://www.nature.com/articles/s41586-023-06735-9", "kind": "research" }, { "label": "GNoME models, DFT data and 380k predicted structures", "url": "https://github.com/google-deepmind/materials_discovery", "kind": "research" }, { "label": "DeepMind: Millions of new materials discovered with deep learning", "url": "https://deepmind.google/blog/millions-of-new-materials-discovered-with-deep-learning/", "kind": "announcement" }, { "label": "The Register: Boffins deem DeepMind's material discoveries shallow", "url": "https://www.theregister.com/2024/04/11/google_deepmind_material_study/", "kind": "challenge" } ], "discussion": [ { "label": "The Register: chemists dispute DeepMind's AI materials discoveries", "url": "https://www.theregister.com/2024/01/31/ai_chemistry_research_disputed/" } ], "tags": [ "materials", "crystal-structure", "graph-neural-network", "contested" ], "added": "2026-07-20" }, { "id": "2023-11-29-a-lab-synthesis", "title": "Autonomous laboratory reports solid-state synthesis of new inorganic compounds", "claim": "Berkeley's A-Lab, an AI-driven autonomous laboratory, reported synthesising 41 novel inorganic compounds out of 58 targets over 17 days with minimal human intervention.", "field": "materials", "date": "2023-11-29", "lab": "Lawrence Berkeley National Laboratory", "model": "A-Lab (ML planning + robotics)", "verification": "disputed", "autonomy": "ai-led", "detail": "A-Lab coupled machine-learning synthesis-recipe prediction with robotic sample preparation, heating and X-ray characterisation, drawing targets from stability predictions including GNoME's. It reported making 41 of 58 targeted compounds autonomously. Published in Nature alongside GNoME.", "novelty_check": "Targets were drawn from computed stability databases. The contested question is whether the synthesised phases were correctly identified and genuinely new, rather than known phases or misindexed results.", "caveats": "A detailed critique led by Robert Palgrave argued that many of the 41 claimed compounds were misidentified from the X-ray data, in several cases likely known phases, mixtures, or amorphous products rather than the claimed novel crystals. It is the identification step, not the automation, that is disputed; the autonomy claim itself is not the weak point. A 2026 Author Correction from the team re-analysed the diffraction data and clarified that 'novel' was meant as new to their prediction platform rather than necessarily new to science.", "independent_checks": [ { "who": "Palgrave et al. (analysis of the reported diffraction data)", "url": "https://www.theregister.com/2024/04/11/google_deepmind_material_study/", "outcome": "disputed the identification of most claimed new compounds" }, { "who": "Authors' correction (Nature, 2026)", "url": "https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12872444/", "outcome": "re-analysed diffraction data; clarified 'novel' meant new to the platform, not to science" } ], "sources": [ { "label": "Nature: An autonomous laboratory for the accelerated synthesis of inorganic materials", "url": "https://www.nature.com/articles/s41586-023-06734-w", "kind": "research" }, { "label": "The Register: Boffins deem DeepMind's material discoveries shallow", "url": "https://www.theregister.com/2024/04/11/google_deepmind_material_study/", "kind": "challenge" } ], "discussion": [ { "label": "The Register: chemists dispute DeepMind's AI materials discoveries", "url": "https://www.theregister.com/2024/01/31/ai_chemistry_research_disputed/" } ], "tags": [ "materials", "autonomous-lab", "synthesis", "contested" ], "added": "2026-07-20" }, { "id": "2024-07-25-alphaproof-imo", "title": "Silver-medal standard at the 2024 International Mathematical Olympiad", "claim": "AlphaProof and AlphaGeometry 2 together solved four of the six 2024 IMO problems for 28 of 42 points (one short of the gold threshold), with the algebra and number-theory solutions produced and checked in the Lean proof assistant.", "field": "mathematics", "date": "2024-07-25", "lab": "Google DeepMind", "model": "AlphaProof + AlphaGeometry 2", "verification": "independent", "autonomy": "ai-led", "detail": "AlphaProof, a reinforcement-learning system that works inside Lean, solved two algebra problems and one number-theory problem (including P6, the competition's hardest, which few human contestants solved), while AlphaGeometry 2 solved the geometry problem in seconds. The two combinatorics problems went unsolved. The Lean-based solutions are machine-checked by construction; the full performance was graded by mathematicians Timothy Gowers and Joseph Myers under competition-style marking.", "novelty_check": "These are competition problems with published official solutions, so the achievement is a capability milestone (solving hard, known-answer problems under near-competition conditions) rather than a new mathematical result. The registry records it as such.", "caveats": "Problems were hand-translated into formal Lean statements by people before AlphaProof attempted them, a real human contribution beyond posing the question, so autonomy is graded conservatively. AlphaProof also took far longer than the human time limit on some problems. This is benchmark performance on solved problems, not a discovery.", "independent_checks": [ { "who": "Timothy Gowers & Joseph Myers (competition-style grading)", "url": "https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/", "outcome": "28/42, silver-medal standard" } ], "sources": [ { "label": "DeepMind: AI achieves silver-medal standard solving IMO problems", "url": "https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/", "kind": "announcement" }, { "label": "Nature: Olympiad-level formal mathematical reasoning with reinforcement learning", "url": "https://www.nature.com/articles/s41586-025-09833-y", "kind": "research" }, { "label": "Unite.AI: How AlphaProof and AlphaGeometry 2 achieved silver-medal standard", "url": "https://www.unite.ai/ai-at-the-international-mathematical-olympiad-how-alphaproof-and-alphageometry-2-achieved-silver-medal-standard/", "kind": "coverage" } ], "discussion": [ { "label": "Hacker News: AI solves IMO problems at silver-medal level", "url": "https://news.ycombinator.com/item?id=41069829" } ], "videos": [ { "label": "AlphaProof and AlphaGeometry 2 achieve a silver-medal score at the IMO, explained", "channel": "Elvis Saravia (DAIR.AI)", "youtube_id": "5FMpqA2CELw" } ], "tags": [ "olympiad", "lean", "formalization", "benchmark" ], "added": "2026-07-20" }, { "id": "2025-10-19-gpt5-erdos-retrieval", "title": "GPT-5 \"solved 10 Erdős problems\": it located existing solutions", "claim": "An OpenAI executive announced that GPT-5 had solved 10 previously open Erdős problems and made progress on 11 more; the problems were only 'open' in the sense that one database maintainer was unaware of the already-published solutions the model surfaced.", "field": "mathematics", "date": "2025-10-19", "lab": "OpenAI", "model": "GPT-5", "verification": "known", "autonomy": "retrieval", "detail": "OpenAI VP Kevin Weil posted that 'GPT-5 found solutions to 10 (!) previously unsolved Erdős problems.' Thomas Bloom, who maintains erdosproblems.com, replied that listing a problem as open there only means he personally was unaware of a solution, and that GPT-5 had found existing papers containing the solutions rather than proving anything new. The post was deleted; Demis Hassabis called the episode 'embarrassing' and Yann LeCun mocked it.", "novelty_check": "By definition, that is the whole point of the episode. The solutions existed in the published literature; GPT-5's contribution was locating them, a genuinely useful literature-search result that was misdescribed as original problem-solving.", "caveats": "Retained as a cautionary entry and a direct companion to the GPT-5 convex-optimization case. The underlying literature search was real and valuable; only the 'solved previously unsolved problems' framing was false. This is distinct from the later, genuine GPT-5.2 + Aristotle resolution of Erdős #728.", "independent_checks": [ { "who": "Thomas Bloom (erdosproblems.com maintainer)", "url": "https://techcrunch.com/2025/10/19/openais-embarrassing-math/", "outcome": "the problems were already solved in the literature; no new proofs" } ], "sources": [ { "label": "TechCrunch: OpenAI's 'embarrassing' math", "url": "https://techcrunch.com/2025/10/19/openais-embarrassing-math/", "kind": "challenge" }, { "label": "The Decoder: A GPT-5 math breakthrough that never happened", "url": "https://the-decoder.com/leading-openai-researcher-announced-a-gpt-5-math-breakthrough-that-never-happened/", "kind": "challenge" } ], "discussion": [ { "label": "Hacker News: OpenAI researcher announced a GPT-5 math breakthrough that never happened", "url": "https://news.ycombinator.com/item?id=45633482" } ], "tags": [ "number-theory", "erdos", "already-known", "cautionary", "retrieval" ], "added": "2026-07-20" }, { "id": "2021-07-15-alphafold2", "title": "Accurate protein structure prediction across the known proteome (AlphaFold2)", "claim": "AlphaFold2 predicted three-dimensional structures for nearly all catalogued proteins from amino-acid sequence at accuracy rivalling experiment, work that earned the 2024 Nobel Prize in Chemistry.", "field": "biology", "date": "2021-07-15", "lab": "Google DeepMind", "model": "AlphaFold2", "humans": [ "John Jumper", "Demis Hassabis" ], "year_posed": 1972, "verification": "independent", "autonomy": "ai-led", "detail": "AlphaFold2 pairs an attention-based neural network with evolutionary sequence information to predict how a protein folds from its amino-acid sequence. At the 2020 CASP14 blind assessment it reached accuracy competitive with experimental methods, and DeepMind and EMBL-EBI then released predicted structures for over 200 million proteins, nearly the entire catalogued proteome. Demis Hassabis and John Jumper shared the 2024 Nobel Prize in Chemistry for the work, alongside David Baker for computational protein design.", "novelty_check": "Protein-structure prediction had been an open grand challenge for roughly fifty years, benchmarked every two years by the CASP assessment. AlphaFold2's CASP14 result was a discontinuous jump over all prior methods and is uncontested.", "caveats": "AlphaFold predicts structures rather than determining them experimentally; outputs are computational hypotheses that can be wrong for disordered regions, alternative folds, point mutations, and many complexes. It predicts natural structures, not new biology on its own. The grade reflects the prediction step; the Nobel recognised the human-built method.", "independent_checks": [ { "who": "CASP14 blind assessment (independent assessors)", "outcome": "accuracy competitive with experiment across most targets" }, { "who": "Royal Swedish Academy of Sciences (2024 Nobel in Chemistry)", "url": "https://www.nobelprize.org/prizes/chemistry/2024/press-release/", "outcome": "awarded to Hassabis and Jumper for protein structure prediction" } ], "sources": [ { "label": "Nature: Highly accurate protein structure prediction with AlphaFold", "url": "https://www.nature.com/articles/s41586-021-03819-2", "kind": "research" }, { "label": "AlphaFold code (google-deepmind/alphafold)", "url": "https://github.com/google-deepmind/alphafold", "kind": "research" }, { "label": "DeepMind: AlphaFold", "url": "https://deepmind.google/science/alphafold/", "kind": "announcement" }, { "label": "Nobel Prize: Chemistry 2024 press release", "url": "https://www.nobelprize.org/prizes/chemistry/2024/press-release/", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: AlphaFold source release", "url": "https://news.ycombinator.com/item?id=27848186" } ], "tags": [ "protein-structure", "biology", "alphafold", "nobel-prize" ], "added": "2026-07-21" }, { "id": "2022-02-16-tokamak-plasma-control", "title": "Deep reinforcement learning controls tokamak fusion plasma", "claim": "A reinforcement-learning controller learned to shape and stabilise the magnetic confinement of real fusion plasma inside the TCV tokamak, holding configurations that are hard to sustain by hand.", "field": "physics", "date": "2022-02-16", "lab": "Google DeepMind", "model": "DeepMind RL controller", "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "Working with EPFL's Swiss Plasma Center, DeepMind trained a deep reinforcement-learning agent to command the magnetic control coils of the TCV tokamak. Learning in a simulator against specified target shapes, the controller then ran on the real machine, holding elongated, snowflake and other configurations, including a 'droplet' state with two separate plasmas at once. It is one of the first times a single learned controller replaced the hand-engineered cascade normally required.", "novelty_check": "Magnetic plasma control is normally built from many separately engineered controllers. A single RL-learned controller running on a real tokamak was new; the Nature paper documents the comparison against conventional control.", "caveats": "Demonstrated on the TCV research tokamak, not a power-producing reactor, and the policy was trained in simulation before transfer to hardware. This is a control-engineering advance, not a solution to fusion energy. Independent groups have since extended RL plasma control to other machines.", "sources": [ { "label": "Nature: Magnetic control of tokamak plasmas through deep reinforcement learning", "url": "https://www.nature.com/articles/s41586-021-04301-9", "kind": "research" }, { "label": "DeepMind: Accelerating fusion science through learned plasma control", "url": "https://deepmind.google/discover/blog/accelerating-fusion-science-through-learned-plasma-control/", "kind": "announcement" }, { "label": "EPFL: EPFL and DeepMind use AI to control plasmas for fusion", "url": "https://actu.epfl.ch/news/epfl-and-deepmind-use-ai-to-control-plasmas-for-nu/", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: controlling the fusion plasma in a tokamak with reinforcement learning", "url": "https://news.ycombinator.com/item?id=30379973" } ], "tags": [ "physics", "fusion", "reinforcement-learning", "control" ], "added": "2026-07-21" }, { "id": "2025-05-alphaevolve-minimum-overlap", "title": "Improved bound for the Erdős minimum-overlap problem", "claim": "AlphaEvolve nudged the best known bound for Erdős's minimum-overlap constant, the first improvement since 2016, and sharpened several autocorrelation inequalities.", "field": "mathematics", "date": "2025-05-14", "lab": "Google DeepMind", "model": "AlphaEvolve (Gemini-based)", "year_posed": 1955, "verification": "independent", "autonomy": "search-scaffold", "detail": "As part of a sweep across open problems in mathematical analysis, AlphaEvolve improved the upper bound on the Erdős minimum-overlap constant from about 0.380927 to about 0.380924, the first movement since 2016, and improved constants in autocorrelation inequalities. The gains are numerically small but exceed long-standing records, and the constructions are explicit and checkable.", "novelty_check": "The minimum-overlap constant and the autocorrelation inequalities have well-tracked records. AlphaEvolve's values were compared against the standing bounds and confirmed to improve them.", "caveats": "The improvements are marginal in magnitude, and both were improved again by other automated systems in 2026. As with the other AlphaEvolve results, a human-designed evaluator and search loop did the selecting.", "registrations": [ { "registry": "mathdb", "id": "316064", "title": "Minimum overlap problem", "url": "https://mathdb.com/p/316064/minimum-overlap-problem" } ], "sources": [ { "label": "DeepMind: AlphaEvolve", "url": "https://deepmind.google/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/", "kind": "announcement" }, { "label": "AlphaEvolve paper (PDF)", "url": "https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/AlphaEvolve.pdf", "kind": "research" }, { "label": "Wikipedia: Minimum overlap problem", "url": "https://en.wikipedia.org/wiki/Minimum_overlap_problem", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: AlphaEvolve announcement", "url": "https://news.ycombinator.com/item?id=43985489" } ], "tags": [ "analysis", "erdos", "alphaevolve" ], "added": "2026-07-21", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration." } ] }, { "id": "2026-07-20-gaussian-moments", "title": "Counterexamples to the Gaussian moments conjecture", "claim": "Explicit low-degree polynomials disproving a 2017 conjecture that had been proposed as a route to proving the Jacobian conjecture, found days after that conjecture was itself disproved.", "field": "mathematics", "date": "2026-07-20", "lab": "Independent", "model": "GPT-5.6 Sol Pro + Claude Fable 5", "humans": [ "Christopher D. Long" ], "year_posed": 2017, "verification": "author-verified", "autonomy": "ai-led", "detail": "Derksen, van den Essen and Zhao proposed the Gaussian moments conjecture and showed that the Jacobian conjecture would follow from it. Long reports that a cubic counterexample in four variables was produced by GPT-5.6 Sol Pro with no human intervention after the initial prompt, which simply noted that the Jacobian conjecture had just been disproved and asked whether a small counterexample to GMC might therefore exist; Claude Fable 5 then found a quartic counterexample in three variables and supplied independent algebraic checks. Together these show \\(\\mathrm{GMC}(n)\\) is false for every \\(n \\ge 3\\).", "novelty_check": "The Gaussian moments conjecture is due to Derksen, van den Essen and Zhao (Israel J. Math., 2017), building on their earlier moment-vanishing and integral conjectures. No prior counterexample appears in the literature; the conjecture's interest was precisely that it implied the Jacobian conjecture, which was disproved on 2026-07-19. The two-variable case GMC(2) is explicitly left open by this paper.", "caveats": "arXiv preprint, not peer-reviewed, single author. The constructions are explicit polynomials and checkable by computer algebra, which lowers the cost of verification, and vibemathed has since re-derived both counterexamples in exact arithmetic for the first ten moments, which confirms the objects but not the argument that the identity holds for every m. The result is a corollary of the Jacobian collapse rather than an independent line of attack: the prompt that produced it began by telling the model that the Jacobian conjecture had fallen.", "independent_checks": [ { "who": "vibemathed.com re-derivation", "url": "https://vibemathed.com/problem/gaussian-moments-conjecture", "outcome": "Rebuilt both counterexamples from the paper's stated polynomials and evaluated them in exact rational arithmetic against the Gaussian moment rules, independently of the paper's algebra: for \\(m = 1\\) to \\(10\\) both give \\(E(P^m) = 0\\) and \\(E(QP^m) = m!\\) exactly. That checks instances, not the all-\\(m\\) argument." } ], "sources": [ { "label": "arXiv: Small Counterexamples to the Gaussian Moments Conjecture", "url": "https://arxiv.org/abs/2607.18186", "kind": "research" }, { "label": "Derksen, van den Essen, Zhao: The Gaussian Moments Conjecture and the Jacobian Conjecture", "url": "https://arxiv.org/abs/1506.05192", "kind": "research" } ], "tags": [ "algebra", "polynomial-maps", "counterexample", "jacobian-conjecture" ], "added": "2026-07-27", "registrations": [ { "registry": "vibemathed", "id": "gaussian-moments-conjecture", "url": "https://vibemathed.com/problem/gaussian-moments-conjecture" } ], "revisions": [ { "date": "2026-08-21", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. No grade moved." }, { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's exact-arithmetic re-derivation of both counterexamples for the first ten moments. The all-m argument is unchecked, so no grade moved.", "url": "https://vibemathed.com/problem/gaussian-moments-conjecture" } ] }, { "id": "2026-07-14-zeroth-order-oracle-bound", "title": "Near-quadratic lower bound for derivative-free convex optimization", "claim": "A lower bound of order \\(d^2/\\log d\\) on the number of exact function evaluations needed to minimize a convex Lipschitz function, closing a gap open since 1996 and showing a 1996 algorithm was essentially optimal.", "field": "mathematics", "date": "2026-07-14", "lab": "Independent", "model": "GPT-5.6 Sol Pro", "humans": [ "Phillip Kerger" ], "year_posed": 1996, "verification": "formal", "autonomy": "ai-led", "detail": "In zeroth-order optimization an algorithm may only query a function's value, never its gradient. Protasov's 1996 algorithm needed about \\(d^2\\) evaluations in \\(d\\) dimensions, but the best known lower bound was only about \\(d\\), leaving open whether a much faster method existed. The new bound of \\(\\Omega(d^2/\\log(d+1))\\) matches the upper bound to polylogarithmic factors. Kerger reports that after a ten-page prompt built on his own earlier failed attempts, the model produced the complete proof in a single 2.5-hour session with no intervention; he then reviewed it and formally verified the core bound in Lean 4, which compiles against Mathlib with no `sorry` and no bespoke axioms.", "novelty_check": "The gap between Protasov's \\(O(d^2)\\) upper bound (1996) and the \\(O(d)\\) lower bound is documented in the derivative-free optimization literature and had stood for three decades. No prior matching lower bound appears in the literature. The paper is explicit that the AI produced the argument.", "caveats": "arXiv preprint, not peer-reviewed. The author states plainly that 'it is accurate to say that the AI model used solved the problem, not the author of this paper', which is the basis for the ai-led grade; the human wrote the prompt, reviewed the proof, and did the Lean formalization. Graded `formal` on the strength of the machine-checked Lean artifact for the core bound, not on the preprint as a whole.", "independent_checks": [ { "who": "Lean 4 / Mathlib compiler (author-run, artifact public)", "url": "https://github.com/PhillipKerger/zero-order-bounds-lean-verification", "outcome": "core \\(\\tilde\\Omega(d^2)\\) lower bound compiles with no sorry and no bespoke axioms" } ], "sources": [ { "label": "arXiv: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization", "url": "https://arxiv.org/abs/2607.13335", "kind": "research" }, { "label": "Lean verification, prompt and chat logs", "url": "https://github.com/PhillipKerger/zero-order-bounds-lean-verification", "kind": "research" } ], "tags": [ "optimization", "complexity-theory", "lower-bounds", "lean" ], "added": "2026-07-27" }, { "id": "2026-05-21-alphaproof-nexus", "title": "Nine Erdős problems and 44 OEIS conjectures proved with machine-checked proofs", "claim": "An agent pairing language models with the Lean proof assistant resolved 9 of 353 open Erdős problems and 44 of 492 open OEIS conjectures, with every proof mechanically verified.", "field": "mathematics", "date": "2026-05-21", "lab": "Google DeepMind", "model": "AlphaProof Nexus (LLM + Lean)", "humans": [ "Pushmeet Kohli", "Swarat Chaudhuri" ], "verification": "formal", "autonomy": "search-scaffold", "detail": "The system proposes proofs with a language model and checks every step in Lean, so a hallucinated argument cannot pass. Beyond the Erdős and OEIS totals it settled a 15-year-old question on log-concavity of pure O-sequences in codimension 3 and type 2, and proved an exact \\(O(1/t)\\) convergence rate for anchored gradient descent-ascent via a parameter choice not previously identified. Two of the Erdős problems had been open for 56 years. All formal proofs were released in a public repository.", "novelty_check": "The Erdős problems were listed as open in the erdosproblems.com database and the OEIS conjectures as unproven at the time of the run; the paper documents which. In the course of the work the authors found misformalizations in the stated problems #125 and #741(i), which had to be corrected before resolution, a reminder that 'open' status in a database is itself fallible. Terence Tao's community wiki independently tracks AI contributions to Erdős problems and records these.", "caveats": "The headline is a hit rate, not a sweep: 9 of 353 and 44 of 492. The authors state successes concentrate in combinatorics, convex optimization and number theory where Lean's library is mature, that the agent inherits its LLM's biases and shows high search variance, and that it cannot solve problems requiring substantial new theory. Formal verification guarantees the proofs are correct; it says nothing about whether the problems were deep. Autonomy is graded search-scaffold rather than autonomous: humans built the agent, chose the problem sets, and supplied the formalization targets.", "independent_checks": [ { "who": "Lean compiler via the paper's SafeVerify step", "url": "https://github.com/google-deepmind/alphaproof-nexus-results", "outcome": "all proofs compile with no sorry and no disallowed axioms (sorryAx)" }, { "who": "Terence Tao's AI-contributions wiki", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "outcome": "tracked among AI contributions to Erdős problems" } ], "registrations": [ { "registry": "vibemathed", "id": "erdos-12", "url": "https://vibemathed.com/problem/erdos-12", "title": "Erdős Problem #12" }, { "registry": "vibemathed", "id": "erdos-138", "url": "https://vibemathed.com/problem/erdos-138", "title": "Erdős Problem #138" }, { "registry": "vibemathed", "id": "green-open-problem-57", "url": "https://vibemathed.com/problem/green-open-problem-57", "title": "Ben Green's Open Problem 57" }, { "registry": "vibemathed", "id": "wow-graph-conjecture-2", "url": "https://vibemathed.com/problem/wow-graph-conjecture-2", "title": "Written on the Wall II, Graph Conjecture 2" }, { "registry": "vibemathed", "id": "monochromatic-quantum-graphs-diagonal", "url": "https://vibemathed.com/problem/monochromatic-quantum-graphs-diagonal", "title": "Monochromatic Quantum Graphs in the Diagonal Family" }, { "registry": "vibemathed", "id": "monochromatic-quantum-graphs-four-particles", "url": "https://vibemathed.com/problem/monochromatic-quantum-graphs-four-particles", "title": "Four-Particle Monochromatic Quantum Graphs" }, { "registry": "vibemathed", "id": "pure-o-sequences-log-concavity", "url": "https://vibemathed.com/problem/pure-o-sequences-log-concavity", "title": "Log-Concavity of Codimension-Three Pure O-Sequences" }, { "registry": "vibemathed", "id": "anchored-gda-last-iterate", "url": "https://vibemathed.com/problem/anchored-gda-last-iterate", "title": "Last-Iterate Rate for Anchored Gradient Descent-Ascent", "note": "This record cites the team's April preprint, arXiv:2604.03782, which the AlphaProof Nexus report refers to for this result; the Lean proof sits in the same results repository." } ], "sources": [ { "label": "arXiv: Advancing Mathematics Research with AI-Driven Formal Proof Search", "url": "https://arxiv.org/abs/2605.22763", "kind": "research" }, { "label": "All Lean proofs (DeepMind repository)", "url": "https://github.com/google-deepmind/alphaproof-nexus-results", "kind": "research" }, { "label": "Tao's AI-contributions-to-Erdős-problems wiki", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "kind": "research" } ], "tags": [ "lean", "automated-theorem-proving", "erdos-problems", "combinatorics" ], "added": "2026-07-27", "revisions": [ { "date": "2026-10-01", "kind": "correction", "note": "Recorded the eight vibemathed records that cover parts of this result as registrations. vibemathed describes the work as autonomous, which does not move the search-scaffold grade for the reasons given in the caveats." } ] }, { "id": "2025-11-03-alphaevolve-at-scale", "title": "AlphaEvolve across 67 problems: 20 improvements, 8 regressions", "claim": "A systematic study with Terence Tao of an evolutionary coding agent on 67 open problems in analysis, combinatorics and geometry, reporting where it beat the literature and where it did not.", "field": "mathematics", "date": "2025-11-03", "lab": "Google DeepMind", "model": "AlphaEvolve (Gemini-based)", "humans": [ "Terence Tao", "Javier Gómez-Serrano", "Bogdan Georgiev", "Adam Zsolt Wagner" ], "verification": "independent", "autonomy": "search-scaffold", "detail": "Rather than a single headline result, this is the systematic version: 67 problems attempted, with outcomes recorded either way. Tao reports roughly 20 cases of a new result, most others matching known bounds, and 8 where the tool did worse than the literature. Genuine advances include new three-dimensional constructions for finite-field Nikodym sets, which then prompted a human-refined hybrid construction beating both algebraic and random baselines, and an improved upper bound for Sidon sets, from 1.96365 to about 1.9526. On the well-known conjectures of Sidorenko, Sendov and Crouzeix it found no counterexample, which Tao notes may simply reflect that they are true.", "novelty_check": "Prior bounds are cited per problem in the 81-page paper, with data and prompts released publicly. Tao flags the central novelty risk himself: for well-known problems the tool often proposed the optimal construction immediately, suggesting recall from training data rather than search, and one supposedly new four-dimensional Kakeya construction turned out to essentially match a prior Bukh–Chao paper. Entries in this registry drawn from the earlier AlphaEvolve release (kissing number, minimum overlap, matrix multiplication) are tracked separately.", "caveats": "Tao's own framing is that the contribution is scale and adaptability, not a fundamental breakthrough, and he describes the LLM as supplying 'educated randomness' inside an evolutionary loop rather than doing symbolic reasoning. Substantial human expertise was required to design non-exploitable verifiers: the system reliably finds loopholes, such as placing points at nearly identical locations to satisfy a distance tolerance. Included here because the negative and null results are reported alongside the wins, which is rare and is what makes it citable.", "independent_checks": [ { "who": "Terence Tao (co-author, published assessment)", "url": "https://terrytao.wordpress.com/2025/11/05/mathematical-exploration-and-discovery-at-scale/", "outcome": "reports ~20 new results, 8 worse than literature; cautions on training-data contamination" } ], "sources": [ { "label": "arXiv: Mathematical exploration and discovery at scale", "url": "https://arxiv.org/abs/2511.02864", "kind": "research" }, { "label": "Terence Tao's write-up", "url": "https://terrytao.wordpress.com/2025/11/05/mathematical-exploration-and-discovery-at-scale/", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: Mathematical exploration and discovery at scale", "url": "https://news.ycombinator.com/item?id=45833162" } ], "tags": [ "combinatorics", "geometry", "analysis", "evolutionary-search" ], "added": "2026-07-27" }, { "id": "2025-09-11-gauss-strong-pnt", "title": "Strong prime number theorem formalized in Lean by an autoformalization agent", "claim": "An agent completed a Lean formalization of the strong prime number theorem in about three weeks, finishing a project two expert mathematicians had left blocked after 18 months.", "field": "mathematics", "date": "2025-09-11", "lab": "Math, Inc.", "model": "Gauss (autoformalization agent)", "humans": [ "Terence Tao", "Alex Kontorovich" ], "verification": "author-verified", "autonomy": "ai-assisted", "detail": "Tao and Kontorovich began a Lean formalization of the strong prime number theorem (the version with an explicit error term) in January 2024, and by July 2025 had announced intermediate progress blocked on core difficulties in complex analysis. Math, Inc. reports that Gauss produced over 25,000 lines of Lean and roughly 1,100 theorems and definitions in about three weeks, completing the project. The repository README states that most statements and proofs were produced by the agent, with humans supplying the high-level blueprint, reviewing key lemmas, and adapting prior work.", "novelty_check": "The prime number theorem is not new mathematics; the contribution is formalization. What was open was whether this particular formalization could be completed, and the surrounding claim is about speed. Recorded because formalization at this scale is a distinct kind of contribution and because the AI finished work humans had started and stalled on, rather than producing a new theorem. Prior art is the PrimeNumberTheoremAnd project itself, which this build reuses and which the README credits.", "caveats": "This is a formalization result, not a mathematical discovery: no new theorem was proved. The comparison to '18+ months' is not like-for-like: Tao and Kontorovich worked intermittently, and the repository explicitly reuses definitions and some proofs from their earlier PrimeNumberTheoremAnd project. Math, Inc.'s announcement states the agent 'relies on natural language scaffolding supplied by human mathematicians, and requires high-level expert guidance'; the proportion of human effort is not quantified, and neither Tao nor Kontorovich is quoted in it. Graded author-verified rather than formal: the artifact is public and Lean-checkable, but the README does not itself assert a sorry-free build and no independent audit of that is on the public record. Vendor-announced, so the framing carries the usual caution.", "independent_checks": [ { "who": "Public Lean artifact (not independently audited)", "url": "https://github.com/math-inc/strongpnt", "outcome": "repository public and machine-checkable; README reports the development finished" } ], "sources": [ { "label": "Math, Inc.: Introducing Gauss", "url": "https://www.math.inc/gauss", "kind": "announcement" }, { "label": "strongpnt Lean repository", "url": "https://github.com/math-inc/strongpnt", "kind": "research" } ], "tags": [ "lean", "formalization", "number-theory", "prime-number-theorem" ], "added": "2026-07-27" }, { "id": "2026-07-11-grothendieck-group-schemes", "title": "Counterexample to Grothendieck's question on finite flat group schemes", "claim": "A finite locally free group scheme of order four whose fourth power map is not trivial, answering a question Grothendieck raised in the 1960s; one model produced the construction and another formalised it in Lean.", "field": "mathematics", "date": "2026-07-11", "lab": "Independent", "model": "OpenAI Sol (construction); Claude Fable (Lean formalisation)", "humans": [ "Akhil Mathew", "Kevin Buzzard" ], "verification": "formal", "autonomy": "collaborative", "detail": "Grothendieck asked whether every finite locally free group scheme of order \\(n\\) is killed by \\(n\\). Deligne proved it for commutative group schemes; the non-commutative case stayed open. The counterexample is a Hopf algebra of rank four over the ring \\(\\mathbb{Z}[a,b]/(a^3, b^3, a^2b+2)\\), with coordinate algebra \\(R[U,V]/(U^2 - abU + b^2V,\\ V^2 - a^2V)\\), whose fourth convolution power is not the convolution unit. Kevin Buzzard, told of a 12-page informal write-up, replied that he does not read AI-generated informal mathematics and asked for a Lean proof instead; four hours later a 1,076-line Lean formalisation existed, and it compiles on a laptop in under five minutes. It was submitted to mathlib as pull request #41748.", "novelty_check": "Per Buzzard's account, the question had been settled in special cases by Grothendieck, Deligne and René Schoof, with further partial results published by Emiliano Torti in 2025; no counterexample appeared in the literature. The object here is an explicit new construction rather than a rediscovered example, and the Lean file states and refutes the general claim directly.", "caveats": "The mathlib pull request was open, not merged, when this entry was written, and was filed by a pseudonymous account whose module docstring credits \"Codex (OpenAI) and Claude (Anthropic), under the direction of the author\" rather than naming Sol or Fable; the model attribution here follows Buzzard's blog post. The `formal` grade rests on the Lean proof compiling, which Buzzard reports doing himself; the 12-page informal argument has not been peer reviewed. Autonomy graded collaborative: a human posed the question, directed the work and filed the PR, and the record of who did which step is a blog post rather than a transcript.", "independent_checks": [ { "who": "Kevin Buzzard (compiled the Lean formalisation)", "url": "https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/", "outcome": "confirmed; 1,076-line Lean proof compiles in under five minutes" } ], "registrations": [ { "registry": "mathdb", "id": "315784", "title": "Grothendieck's group-scheme question", "url": "https://mathdb.com/p/315784/grothendieck-s-group-scheme-question" }, { "registry": "vibemathed", "id": "grothendieck-finite-flat-group-schemes", "url": "https://vibemathed.com/problem/grothendieck-finite-flat-group-schemes" } ], "sources": [ { "label": "Kevin Buzzard: Human mathematicians are being outcounterexampled", "url": "https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/", "kind": "announcement" }, { "label": "mathlib4 PR #41748: a finite free group scheme of order four not killed by four", "url": "https://github.com/leanprover-community/mathlib4/pull/41748", "kind": "research" } ], "discussion": [ { "label": "Hacker News: Human mathematicians are being outcounterexampled", "url": "https://news.ycombinator.com/item?id=48983382" } ], "tags": [ "algebraic-geometry", "group-schemes", "counterexample", "lean", "formalization" ], "added": "2026-07-29", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration." }, { "date": "2026-08-31", "kind": "correction", "note": "Recorded the vibemathed record as a second registration, alongside MathDB. It carries the same split of work between the two models that this entry does. No grade moved." } ] }, { "id": "2025-09-17-unstable-singularities", "title": "New families of unstable singularities in fluid equations", "claim": "Physics-informed neural networks found previously unknown unstable self-similar blow-up solutions for three fluid equations, computed to near machine precision, and revealed an unexpected near-linear relation between a solution's instability order and its blow-up rate.", "field": "mathematics", "date": "2025-09-17", "lab": "Google DeepMind (with Brown, NYU and Stanford)", "model": "Physics-informed neural networks with Gauss–Newton optimisation", "humans": [ "Javier Gómez-Serrano", "Tristan Buckmaster", "Ching-Yao Lai", "Yongji Wang" ], "verification": "author-verified", "autonomy": "search-scaffold", "detail": "A physics-informed neural network minimises the residual of the differential equation itself rather than fitting data, which lets it hunt for self-similar blow-up profiles that ordinary simulation cannot hold onto because they are repelling. The team reports a stable and three unstable singularities for the incompressible porous media equation, a stable and three unstable solutions plus a candidate fourth for 2D Boussinesq, and an unstable singularity for 3D Euler with boundary. Residuals were driven near machine precision, the accuracy level a later computer-assisted proof would need. Across equations the blow-up rate depends almost linearly on the order of instability, a pattern nobody had predicted.", "novelty_check": "Stable self-similar blow-up for these equations was already known, including Chen and Hou's computer-assisted proof for 3D Euler with boundary. Unstable singularities had been found only in isolated cases because they are numerically repelling; no systematic families existed in the literature. This is the first paper to produce several unstable families across three different equations by one method.", "caveats": "An arXiv preprint, not peer reviewed at the time of this entry. These are high-precision numerical solutions, not theorems: converting them into proofs requires a computer-assisted verification step the paper leaves to future work. No singularity for 3D Navier–Stokes, the Millennium Prize problem, is claimed, and the equations studied are not that one. Autonomy is search-scaffold: the equations, the self-similar ansatz and the loss function are human-designed and the network is the solver.", "sources": [ { "label": "arXiv: Discovery of Unstable Singularities", "url": "https://arxiv.org/abs/2509.14185", "kind": "research" }, { "label": "DeepMind: Discovering new solutions to century-old problems in fluid dynamics", "url": "https://deepmind.google/blog/discovering-new-solutions-to-century-old-problems-in-fluid-dynamics/", "kind": "announcement" }, { "label": "Physics World: Neural networks discover unstable singularities in fluid systems", "url": "https://physicsworld.com/a/neural-networks-discover-unstable-singularities-in-fluid-systems/", "kind": "coverage" } ], "tags": [ "pde", "fluid-dynamics", "singularity", "physics-informed-neural-networks" ], "added": "2026-07-29" }, { "id": "2025-07-21-gemini-deepthink-imo", "title": "Gold-medal standard at the 2025 International Mathematical Olympiad", "claim": "An advanced version of Gemini Deep Think read the official problem statements and wrote natural-language proofs for five of the six 2025 IMO problems inside the contest time limits, scoring 35 of 42, graded and certified by the official IMO coordinators.", "field": "mathematics", "date": "2025-07-21", "lab": "Google DeepMind", "model": "Gemini Deep Think (advanced version)", "verification": "independent", "autonomy": "ai-led", "detail": "The model worked end to end in natural language within the two 4.5-hour sessions, with no human translating problems into a formal language, and solved problems one through five. Google DeepMind was in the first cohort whose results were officially graded and certified by IMO coordinators against the rubric used for student scripts; 35 points is the gold-medal threshold. OpenAI announced the same 35/42 score two days earlier with an experimental reasoning model, graded internally by three former IMO medallists rather than by the IMO.", "novelty_check": "IMO problems have published official solutions, so this is a capability milestone rather than a new mathematical result, and the registry records it as one. The prior best was AlphaProof and AlphaGeometry 2's 28/42 in 2024, which required humans to hand-translate each problem into Lean first; removing that step is the substantive change.", "caveats": "Benchmark performance on problems with known answers, not a discovery. Google's own account describes training on a curated corpus of high-quality solutions and providing general guidance on how to approach IMO problems, which is human mathematical input beyond posing the problem, hence ai-led rather than autonomous. Problem six went unsolved. The parallel OpenAI claim is self-graded and not IMO-certified, and drew criticism for being published before the closing ceremony, which organisers had asked AI developers to wait for.", "independent_checks": [ { "who": "Official IMO coordinators (certified grading)", "url": "https://deepmind.google/discover/blog/advanced-version-of-gemini-with-deep-think-officially-achieves-gold-medal-standard-at-the-international-mathematical-olympiad/", "outcome": "35/42, gold-medal standard" } ], "sources": [ { "label": "Gemini Deep Think: official IMO 2025 solutions (PDF)", "url": "https://storage.googleapis.com/deepmind-media/gemini/IMO_2025.pdf", "kind": "research" }, { "label": "DeepMind: Gemini with Deep Think officially achieves gold-medal standard at the IMO", "url": "https://deepmind.google/discover/blog/advanced-version-of-gemini-with-deep-think-officially-achieves-gold-medal-standard-at-the-international-mathematical-olympiad/", "kind": "announcement" }, { "label": "Simon Willison: OpenAI's gold medal performance on the IMO", "url": "https://simonwillison.net/2025/Jul/19/openai-gold-medal-math-olympiad/", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: Gemini with Deep Think achieves gold-medal standard at the IMO", "url": "https://news.ycombinator.com/item?id=44637191" } ], "tags": [ "olympiad", "benchmark", "natural-language-proof" ], "added": "2026-07-29" }, { "id": "2025-07-28-aristotle-imo-lean", "title": "Machine-checked Lean proofs for five of six 2025 IMO problems", "claim": "Harmonic's Aristotle produced formally verified Lean 4 proofs for five of the six 2025 IMO problems, so the gold-medal-standard score rests on a compiler rather than on human graders.", "field": "mathematics", "date": "2025-07-28", "lab": "Harmonic", "model": "Aristotle", "verification": "formal", "autonomy": "ai-led", "detail": "Aristotle drafts an informal proof, breaks it into intermediate lemmas, formalises each in Lean 4, and iterates against the compiler's feedback, which lets it use the large natural-language corpus of mathematics while still emitting machine-checked output. Harmonic published the Lean artifacts for the 2025 problems in a public repository. The announcement came a week after the natural-language gold results from Google DeepMind and OpenAI.", "novelty_check": "Formal IMO results are not new (AlphaProof solved three 2024 problems in Lean), but a gold-medal-level score with a formal proof for every claimed solution had not been reported. The problems have published official solutions, so this is a capability milestone rather than new mathematics.", "caveats": "Problems had to be stated formally in Lean before the system could attempt them, the same human step that capped AlphaProof's autonomy grade. Answer-construction problems require the answer to appear in the formal statement, a known weakness of formal olympiad evaluation. No IMO-certified grading and no independently enforced contest time limit. The `formal` grade covers the proofs Aristotle produced, not the claim that this equals a human gold medal.", "sources": [ { "label": "arXiv: Aristotle: IMO-level Automated Theorem Proving", "url": "https://arxiv.org/abs/2510.01346", "kind": "research" }, { "label": "harmonic-ai/IMO2025: Lean proofs and verification artifacts", "url": "https://github.com/harmonic-ai/IMO2025", "kind": "research" } ], "tags": [ "olympiad", "lean", "formalization", "benchmark" ], "added": "2026-07-29" }, { "id": "2024-01-17-alphageometry", "title": "Olympiad geometry solved without human demonstrations", "claim": "A language model trained purely on synthetic data, paired with a symbolic deduction engine, solved 25 of 30 olympiad geometry problems against 10 for the previous best automated system and 25.9 for the average human gold medallist.", "field": "mathematics", "date": "2024-01-17", "lab": "Google DeepMind", "model": "AlphaGeometry", "humans": [ "Trieu Trinh", "Thang Luong" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "Geometry proofs stall when they need an auxiliary construction, an extra point or line that is not in the problem statement. AlphaGeometry pairs a symbolic deduction engine, which exhausts what follows mechanically, with a language model that proposes constructions when deduction runs dry. The model was trained on 100 million synthetic proofs generated from random diagrams, with no human proof data, and its output is a symbolic proof that the deduction engine checks. Published in Nature.", "novelty_check": "Automated geometry provers date to Wu's method in the 1970s and to GEX/Java Geometry Expert; the prior state of the art solved 10 of the 30 benchmark problems. The benchmark is IMO geometry problems from 2000–2022 with known solutions, so the contribution is the method and the score, not a new theorem.", "caveats": "Benchmark performance on solved problems, not a discovery. An independent re-evaluation found the symbolic baseline much stronger than reported: Wu's method alone solves 15 of the 30, not 10, and with classical synthetic methods 21, which narrows the margin the headline comparison implies. The domain is narrow: plane geometry expressible in the system's formal language, which excludes problems involving inequalities or variable numbers of points, and the benchmark set was filtered to what the language can state. Proof steps are machine-checked by the deduction engine, which is what carries the confidence; the peer-reviewed grade reflects Nature review of the system and results.", "independent_checks": [ { "who": "Sinha, Prabhu, Kumaraguru, Bhat and Bethge (re-evaluation of the benchmark)", "url": "https://arxiv.org/abs/2404.06405", "outcome": "Revisited IMO-AG-30 and found Wu's method alone solves 15 problems, not the 10 reported as the baseline, and with classical synthetic methods solves 21 of 30 on a CPU-only laptop, four fewer than AlphaGeometry; combined with AlphaGeometry it reaches 27." } ], "sources": [ { "label": "Nature: Solving olympiad geometry without human demonstrations", "url": "https://www.nature.com/articles/s41586-023-06747-5", "kind": "research" }, { "label": "DeepMind: AlphaGeometry, an olympiad-level AI system for geometry", "url": "https://deepmind.google/discover/blog/alphageometry-an-olympiad-level-ai-system-for-geometry/", "kind": "announcement" }, { "label": "google-deepmind/alphageometry (code)", "url": "https://github.com/google-deepmind/alphageometry", "kind": "research" }, { "label": "arXiv: Wu's method can boost symbolic AI to rival silver medalists and AlphaGeometry to outperform gold medalists", "url": "https://arxiv.org/abs/2404.06405", "kind": "challenge" } ], "discussion": [ { "label": "Hacker News: AlphaGeometry, an Olympiad-level AI system for geometry", "url": "https://news.ycombinator.com/item?id=39029801" } ], "tags": [ "olympiad", "geometry", "neuro-symbolic", "benchmark" ], "added": "2026-07-29", "revisions": [ { "date": "2026-09-24", "kind": "challenge", "note": "Cited an independent re-evaluation of the benchmark finding the symbolic baseline stronger than reported, 21 of 30 rather than 10. No grade moved.", "url": "https://arxiv.org/abs/2404.06405" } ] }, { "id": "2021-12-01-knot-theory-intuition", "title": "Two theorems found by machine pattern-spotting in knot theory and representation theory", "claim": "Networks trained to predict one mathematical invariant from another, then probed for which inputs mattered, pointed mathematicians to a proved theorem linking a knot's signature to a new geometric quantity, and to structure behind the combinatorial invariance conjecture.", "field": "mathematics", "date": "2021-12-01", "lab": "Google DeepMind (with Oxford and Sydney)", "model": "Supervised networks with gradient-based attribution", "humans": [ "Alex Davies", "Marc Lackenby", "Geordie Williamson" ], "verification": "peer-reviewed", "autonomy": "ai-assisted", "detail": "The recipe was to train a network to predict one invariant from another, then use attribution to see which inputs carried the signal, and hand that to a mathematician. In knot theory it led Marc Lackenby to define a new quantity, the natural slope of a knot, and to prove a theorem bounding the signature in terms of it. In representation theory it led Geordie Williamson to structure in Bruhat interval graphs bearing on the combinatorial invariance conjecture for Kazhdan–Lusztig polynomials. Published in Nature.", "novelty_check": "Both theorems are new and neither had a prior form in the literature; both were proved conventionally by the human mathematicians after the models indicated where to look. The general idea of using computation to generate conjectures is old; what was new was the attribution step turning a black-box predictor into a usable hint.", "caveats": "The mathematics was done by humans; the models indicated which relationships were worth studying, which is why autonomy is ai-assisted. Ernest Davis's review argues the \"guiding human intuition\" framing overstates the contribution and that the knot-theory signal was within reach of simpler statistical methods. The theorems themselves are not disputed.", "independent_checks": [ { "who": "Ernest Davis (critical review of the framing)", "url": "https://arxiv.org/abs/2112.04324", "outcome": "theorems accepted; disputes how much the deep learning contributed" } ], "sources": [ { "label": "Nature: Advancing mathematics by guiding human intuition with AI", "url": "https://www.nature.com/articles/s41586-021-04086-x", "kind": "research" }, { "label": "Knot theory and representation theory notebooks", "url": "https://github.com/google-deepmind/mathematics_conjectures", "kind": "research" }, { "label": "University of Oxford: Machine learning helps mathematicians make new connections", "url": "https://www.ox.ac.uk/news/2021-12-01-machine-learning-helps-mathematicians-make-new-connections-0", "kind": "announcement" }, { "label": "Ernest Davis: Deep Learning and Mathematical Intuition (review)", "url": "https://arxiv.org/abs/2112.04324", "kind": "challenge" } ], "tags": [ "knot-theory", "representation-theory", "conjecture-generation" ], "added": "2026-07-29" }, { "id": "2021-04-29-wagner-combinatorics", "title": "Reinforcement learning refutes several conjectures in extremal combinatorics", "claim": "A neural network building graphs edge by edge under a cross-entropy search produced explicit counterexamples to several published conjectures in graph theory and extremal combinatorics.", "field": "mathematics", "date": "2021-04-29", "lab": "Tel Aviv University", "model": "Deep cross-entropy method (custom network)", "humans": [ "Adam Zsolt Wagner" ], "verification": "independent", "autonomy": "search-scaffold", "detail": "Each conjecture was rewritten as a scoring function on graphs. A network proposes graphs one edge at a time, the top-scoring constructions are kept, and the network is retrained on them, so the search drifts toward whatever the conjecture says should be impossible. Among the conjectures refuted are a question of Brualdi and Cao on maximising permanents of pattern-avoiding matrices and several concerning adjacency and distance eigenvalues of graphs. The counterexamples are small finite graphs printed in the paper, so each can be checked by hand or by a few lines of code.", "novelty_check": "Computer search for combinatorial counterexamples predates this work: linear-programming and SAT-based refutations go back years, including Refuting conjectures in extremal combinatorics via linear programming (2019). What was new is the reinforcement-learning formulation and the specific conjectures, which were open at the time. The paper is widely treated as the origin of the \"AI finds counterexamples\" line of work.", "caveats": "An arXiv preprint; no journal publication is listed on the arXiv record five years on. The refuted conjectures are individually modest (none is a named problem of the Erdős or Jacobian class), and several came from open-problem lists rather than headline literature. Autonomy is search-scaffold on the strictest reading: the human chose the conjectures, wrote the scoring function that encodes each one, and designed the search; the network only proposes graphs.", "independent_checks": [ { "who": "Roucairol & Cazenave, Monte Carlo search replication and extension", "url": "https://arxiv.org/abs/2207.03343", "outcome": "reproduced the refutations and found further counterexamples by different search" } ], "sources": [ { "label": "arXiv: Constructions in combinatorics via neural networks", "url": "https://arxiv.org/abs/2104.14516", "kind": "research" }, { "label": "Wagner's cross-entropy code for the counterexamples", "url": "https://github.com/zawagner22/cross-entropy-for-combinatorics", "kind": "research" }, { "label": "Refutation of Spectral Graph Theory Conjectures with Monte Carlo Search", "url": "https://arxiv.org/abs/2207.03343", "kind": "challenge" } ], "tags": [ "combinatorics", "graph-theory", "counterexample", "reinforcement-learning" ], "added": "2026-07-29" }, { "id": "2025-09-25-qma-amplification-limits", "title": "Limits to black-box amplification in QMA, with the key step written by GPT-5", "claim": "A quantum complexity paper proving black-box error amplification for QMA cannot push completeness closer to certainty than doubly exponentially, in which the pivotal technical step was produced by GPT-5 in about half an hour.", "field": "computer-science", "date": "2025-09-25", "lab": "UT Austin / CWI Amsterdam", "model": "GPT-5 Thinking", "humans": [ "Scott Aaronson", "Freek Witteveen" ], "verification": "author-verified", "autonomy": "ai-assisted", "detail": "The authors were stuck on how the largest eigenvalue of a matrix behaved as a parameter varied. GPT-5 proposed reformulating the quantity so that a complex approximation-theory bound applied, and that reformulation became the technical core of the oracle separation. Aaronson wrote that the step would have cost him or a graduate student a couple of weeks and that it was the first substantive AI contribution to a paper of his. The result makes his 2008 oracle separation quantitative and shows recent amplification results are optimal for black-box procedures.", "novelty_check": "The theorem extends Aaronson's 2008 QMA oracle separation and a 2025 result of Jeffery and Witteveen; no prior bound of this form on black-box amplification appears in the literature. The AI contribution is documented in the paper's acknowledgements and on Aaronson's own blog, not inferred from press coverage. That matters, because most \"AI proved a theorem\" stories are not.", "caveats": "An arXiv preprint. The contribution is one lemma inside a human-conceived and human-written paper; Aaronson is explicit that the model did not pose the problem or design the argument and that he could have done the step himself given time. Autonomy is ai-assisted, which stays the right grade even though the step was pivotal.", "sources": [ { "label": "arXiv: Limits to black-box amplification in QMA", "url": "https://arxiv.org/abs/2509.21131", "kind": "research" }, { "label": "Scott Aaronson: The QMA Singularity", "url": "https://scottaaronson.blog/?p=9183", "kind": "announcement" } ], "tags": [ "quantum-complexity", "proof-assistance", "oracle-separation" ], "added": "2026-07-29" }, { "id": "2025-03-12-ai-scientist-workshop-paper", "title": "A fully machine-generated paper passed workshop peer review", "claim": "One of three end-to-end AI-generated manuscripts submitted to an ICLR 2025 workshop averaged 6.33 from reviewers and would have been accepted; the authors withdrew it, and the bar cleared was a workshop track, not the main conference.", "field": "computer-science", "date": "2025-03-12", "lab": "Sakana AI", "model": "The AI Scientist-v2", "verification": "author-verified", "autonomy": "ai-led", "detail": "The AI Scientist-v2 generates a hypothesis, writes and runs the experiments through agentic tree search, and produces the manuscript, figures and related work without human editing. Sakana submitted three such papers to the ICLR 2025 workshop \"I Can't Believe It's Not Better\", with the knowledge and cooperation of the workshop organisers and ICLR leadership, and agreed in advance to withdraw anything accepted. One paper, on whether an explicit compositional regularisation term improves compositional generalisation, scored 6.33 and cleared the bar.", "novelty_check": "Machine-written text had passed review before in the form of nonsense-generator stings (SCIgen, 2005), which tested review rather than produced research. The claim here is different: an autonomous pipeline produced a real experimental paper that reviewers rated acceptable. Sakana's own write-up is the source for the process, and the code and submitted manuscripts are public.", "caveats": "Workshop tracks accept a far higher share of submissions than the main conference, and Sakana says so itself; the company also notes the accepted paper contains a citation error and that its own reviewers judged the work below the main-conference bar. The paper's scientific content is a negative result: the proposed regularisation did not help. Reviewers were not told which submissions were machine-generated, but the experiment ran with organiser consent, so this is not a blind test of peer review at scale. Humans chose the venue and selected which generated papers to submit, which caps autonomy at ai-led.", "sources": [ { "label": "Sakana AI: The AI Scientist generates its first peer-reviewed publication", "url": "https://sakana.ai/ai-scientist-first-publication/", "kind": "announcement" }, { "label": "TechCrunch: Sakana claims its AI paper passed peer review (it's more nuanced than that)", "url": "https://techcrunch.com/2025/03/12/sakana-claims-its-ai-paper-passed-peer-review-but-its-a-bit-more-nuanced-than-that/", "kind": "challenge" }, { "label": "SakanaAI/AI-Scientist-ICLR2025-Workshop-Experiment (manuscripts and reviews)", "url": "https://github.com/SakanaAI/AI-Scientist-ICLR2025-Workshop-Experiment", "kind": "research" } ], "tags": [ "autonomous-research", "peer-review", "agents", "contested" ], "added": "2026-07-29" }, { "id": "2025-01-16-esmgfp", "title": "A working fluorescent protein generated by a language model", "claim": "ESM3 designed esmGFP, a bright green fluorescent protein sharing only 58% of its sequence with the closest known natural fluorescent protein, a distance the authors estimate at over 500 million years of natural evolution.", "field": "biology", "date": "2025-01-16", "lab": "EvolutionaryScale", "model": "ESM3 (98B)", "humans": [ "Alexander Rives", "Tom Sercu" ], "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "ESM3 is a masked language model over protein sequence, structure and function jointly. Prompted with the residues that form the GFP chromophore and little else, and run through iterative generation, it produced candidates that were synthesised and assayed in the lab. esmGFP fluoresces at brightness comparable to natural GFPs while differing at 96 of 229 positions from its nearest natural relative. Published in Science.", "novelty_check": "Engineered GFP variants are routine but sit a handful of mutations from a natural template, and directed evolution moves in small steps. A functional fluorescent protein this far from any known sequence had not been reported, and the paper includes the expression, spectra and brightness measurements rather than a prediction alone.", "caveats": "The 500-million-year figure is an inference from a sequence-divergence-to-time calibration, not a measurement, and has been criticised as a marketing frame. esmGFP was one hit from a generated and screened set; the paper reports the funnel. The design goal and the chromophore constraint were supplied by humans. No independent re-synthesis of esmGFP by an unaffiliated lab was on record when this entry was written.", "sources": [ { "label": "Science: Simulating 500 million years of evolution with a language model", "url": "https://www.science.org/doi/10.1126/science.ads0018", "kind": "research" }, { "label": "EvolutionaryScale: ESM3 release", "url": "https://www.evolutionaryscale.ai/blog/esm3-release", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: ESM3, simulating 500 million years of evolution with a language model", "url": "https://news.ycombinator.com/item?id=40787800" } ], "tags": [ "protein-design", "generative-biology", "fluorescent-protein" ], "added": "2026-07-29" }, { "id": "2025-09-17-evo-phage-genomes", "title": "AI-generated bacteriophage genomes that replicate and kill bacteria", "claim": "Genome language models wrote complete ΦX174-like bacteriophage genomes from scratch; of 285 designs that could be built, 16 produced viable infectious phages, several with faster lysis than the natural virus.", "field": "biology", "date": "2025-09-17", "lab": "Arc Institute / Stanford University", "model": "Evo 1 and Evo 2", "humans": [ "Samuel King", "Brian Hie" ], "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "The team fine-tuned Evo 1 and Evo 2 on a cleaned set of nearly 15,000 Microviridae genomes, generated 302 candidate genomes with ΦX174-like architecture, chemically synthesised the 285 that could be assembled, and tested them for plaque formation in E. coli. Sixteen were viable, with substantial sequence divergence from the template. Several generated phages outcompeted wild-type ΦX174 in growth and lysis kinetics, and a cocktail of them cleared three ΦX174-resistant E. coli strains. The authors frame it as the first generative design of a complete functional genome.", "novelty_check": "Synthetic phage genomes assembled from natural sequence date to Venter's 2003 ΦX174 reconstruction, and generative design of individual proteins was well established. Generating an entire genome that yields a working organism had not been demonstrated. The claim is bounded to bacteriophages, which infect bacteria only.", "caveats": "Peer review does not replicate: no independent group has rebuilt these genomes, and the 16 viable designs remain the authors' own result. The designs are variants within one small, exceptionally well-characterised template (ΦX174, 5,386 bp), not free-form genomes, and 269 of the 285 that were built did not work. On biosafety, the authors excluded eukaryotic and human-infecting viruses from training and confine the work to phages; the result has nonetheless become a reference point in biosecurity debate, which is context this entry records rather than adjudicates. Science published a biosecurity perspective alongside the paper arguing for legally required screening of synthetic DNA orders. Note that some press coverage of the Science paper misattributes the work to OpenAI; the collaboration is Arc Institute and Stanford.", "sources": [ { "label": "Science: Generative design of bacteriophages with genome language models", "url": "https://www.science.org/doi/10.1126/science.aec2657", "kind": "research" }, { "label": "bioRxiv: Generative design of novel bacteriophages with genome language models", "url": "https://www.biorxiv.org/content/10.1101/2025.09.12.675911v1", "kind": "research" }, { "label": "Arc Institute: How we built the first AI-generated genomes", "url": "https://arcinstitute.org/news/hie-king-first-synthetic-phage", "kind": "announcement" }, { "label": "Asimov Press: AI-designed phages", "url": "https://www.asimov.press/p/ai-phages", "kind": "coverage" }, { "label": "Inglesby and Hanke: Guarding against the misuse of AI-designed viruses", "url": "https://www.science.org/doi/10.1126/science.aej8512", "kind": "commentary" } ], "revisions": [ { "date": "2026-08-06", "kind": "regraded", "note": "Upgraded from author verified to peer reviewed: the preprint was published in Science, alongside a biosecurity perspective calling for mandatory screening of synthetic DNA orders.", "url": "https://www.science.org/doi/10.1126/science.aec2657" } ], "tags": [ "genome-design", "bacteriophage", "generative-biology", "biosecurity" ], "added": "2026-07-29" }, { "id": "2024-05-08-alphafold3", "title": "Joint structure prediction for proteins, nucleic acids and ligands (AlphaFold 3)", "claim": "A diffusion-based successor to AlphaFold predicts complexes spanning proteins, DNA, RNA, small molecules and ions in a single model, with large accuracy gains over the specialised docking tools used for each case.", "field": "biology", "date": "2024-05-08", "lab": "Google DeepMind / Isomorphic Labs", "model": "AlphaFold 3", "humans": [ "John Jumper", "Max Jaderberg" ], "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "AlphaFold 2 predicted the fold of a protein chain; drug discovery mostly needs the structure of that protein bound to something else. AlphaFold 3 replaces the structure module with a diffusion model operating directly on atom coordinates, which lets one network cover protein–ligand, protein–nucleic-acid and antibody–antigen complexes. Reported accuracy substantially exceeds specialised docking tools on protein–ligand benchmarks. Published in Nature.", "novelty_check": "Docking and complex prediction had decades of specialised tools (AutoDock, RoseTTAFold All-Atom, and AlphaFold-Multimer for protein assemblies). The contribution is one model spanning molecule types at higher accuracy, not a new biological result; the structures it predicts are natural ones.", "caveats": "At publication neither code nor weights were released, only a rate-limited web server, which drew an open letter from over a thousand researchers about reproducibility; weights for academic use followed in November 2024. Accuracy on antibody–antigen complexes and some nucleic-acid targets is far below the headline figures, and the model can produce confidently wrong hallucinated structure in disordered regions. An independent benchmark of 2,600 protein-ligand systems released after the training cutoff found accuracy falls on complexes unlike the training data and concluded that co-folding methods including this one largely memorise ligand poses. Predictions are hypotheses, not measurements.", "independent_checks": [ { "who": "Škrinjar, Eberhardt, Tauriello, Schwede and Durairaj, University of Basel and SIB (Runs N' Poses benchmark)", "url": "https://www.biorxiv.org/content/10.1101/2025.02.03.636309v3", "outcome": "On 2,600 high-resolution protein-ligand systems released after the training cutoff, accuracy of AlphaFold 3 and three similar co-folding methods declines on complexes unlike the training data; the authors conclude that these methods largely memorise ligand poses." } ], "sources": [ { "label": "Nature: Accurate structure prediction of biomolecular interactions with AlphaFold 3", "url": "https://www.nature.com/articles/s41586-024-07487-w", "kind": "research" }, { "label": "AlphaFold 3 code (google-deepmind/alphafold3)", "url": "https://github.com/google-deepmind/alphafold3", "kind": "research" }, { "label": "Google DeepMind and Isomorphic Labs introduce AlphaFold 3", "url": "https://blog.google/innovation-and-ai/products/google-deepmind-isomorphic-alphafold-3-ai-model/", "kind": "announcement" }, { "label": "bioRxiv: Have protein-ligand co-folding methods moved beyond memorisation?", "url": "https://www.biorxiv.org/content/10.1101/2025.02.03.636309v3", "kind": "challenge" } ], "discussion": [ { "label": "Hacker News: AlphaFold 3 code", "url": "https://news.ycombinator.com/item?id=42106906" } ], "tags": [ "protein-structure", "drug-discovery", "alphafold", "diffusion-model" ], "added": "2026-07-29", "revisions": [ { "date": "2026-09-24", "kind": "challenge", "note": "Cited an independent benchmark finding that protein-ligand accuracy falls on complexes unlike the training data. No grade moved.", "url": "https://www.biorxiv.org/content/10.1101/2025.02.03.636309v3" } ] }, { "id": "2024-10-02-flywire-connectome", "title": "Complete wiring diagram of an adult fruit-fly brain", "claim": "Machine segmentation of an electron-microscopy volume, corrected by a large community of proofreaders, produced the first synapse-level map of an entire adult brain: 139,255 neurons and roughly 50 million synapses.", "field": "neuroscience", "date": "2024-10-02", "lab": "FlyWire Consortium (Princeton, MRC LMB, Cambridge, Vermont)", "model": "Automated electron-microscopy segmentation networks", "humans": [ "Sebastian Seung", "Mala Murthy", "Gregory Jefferis" ], "verification": "peer-reviewed", "autonomy": "ai-assisted", "detail": "A female Drosophila brain was sliced into about 7,000 sections and imaged by electron microscopy, producing a volume no team could trace by hand. Neural networks segmented the neurons automatically; more than 200 labs and a crowd of proofreaders then corrected the segmentation over several years and annotated over 8,000 cell types. Simulations built directly on the resulting wiring diagram predicted which neurons drive feeding and grooming, and the predictions held up experimentally. Published as a Nature package in October 2024.", "novelty_check": "Complete connectomes existed for C. elegans (302 neurons, 1986) and the Drosophila larva (3,016 neurons, 2023). This one is roughly 45 times larger than the previous largest complete connectome, and automated segmentation is what made that scale reachable at all.", "caveats": "Automated segmentation alone was not accurate enough: the result required years of human proofreading, which is why autonomy is graded ai-assisted rather than ai-led. Neurotransmitter identity is predicted rather than measured for most connections, and gap junctions and neuromodulation are not fully captured. It is one individual's brain, and a wiring diagram is not by itself an explanation of behaviour.", "sources": [ { "label": "Nature: Neuronal wiring diagram of an adult brain", "url": "https://www.nature.com/articles/s41586-024-07558-y", "kind": "research" }, { "label": "NIH: Complete wiring map of an adult fruit fly brain", "url": "https://www.nih.gov/news-events/nih-research-matters/complete-wiring-map-adult-fruit-fly-brain", "kind": "announcement" }, { "label": "FlyWire (data and explorer)", "url": "https://flywire.ai/", "kind": "research" } ], "tags": [ "connectome", "electron-microscopy", "segmentation", "drosophila" ], "added": "2026-07-29" }, { "id": "2023-12-20-coscientist", "title": "A language-model agent that planned and ran chemistry experiments on lab robots", "claim": "Coscientist, driven by GPT-4 with web search, documentation search, code execution and lab automation, designed and executed palladium-catalysed cross-coupling reactions on real robotic hardware from a one-line goal.", "field": "chemistry", "date": "2023-12-20", "lab": "Carnegie Mellon University", "model": "GPT-4 (Coscientist agent)", "humans": [ "Daniil Boiko", "Robert MacKnight", "Gabe Gomes" ], "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "Given \"perform Suzuki and Sonogashira reactions\", the agent searched the literature to identify the chemistry, read the hardware documentation for an API it had not seen, wrote the control code, located the reagents on the plate from a UV-Vis reading, and ran the reactions on a liquid handler. The paper reports six tasks in total, including reaction optimisation under Bayesian search. Published in Nature.", "novelty_check": "Robotic synthesis platforms and automated reaction optimisation predate this (Cronin's mobile robotic chemist, Nature 2020, among others). What was new is a general-purpose language model doing the planning, tool selection and code generation across unfamiliar hardware from a natural-language goal, rather than executing a human-written protocol.", "caveats": "The chemistry performed is textbook cross-coupling: the finding is about autonomy, not about a new compound or reaction. The paper's own safety section shows the agent could be steered toward hazardous syntheses and the authors call for safeguards. Humans set up the hardware, reagents and safety envelope, and the tasks were chosen to be within the platform's reach.", "sources": [ { "label": "Nature: Autonomous chemical research with large language models", "url": "https://www.nature.com/articles/s41586-023-06792-0", "kind": "research" }, { "label": "PubMed Central full text", "url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC10733136/", "kind": "research" }, { "label": "Coscientist data and reference implementation", "url": "https://github.com/gomesgroup/coscientist", "kind": "research" } ], "tags": [ "lab-automation", "agents", "cross-coupling", "self-driving-lab" ], "added": "2026-07-29" }, { "id": "2025-02-13-designed-serine-hydrolases", "title": "Enzymes with working catalytic machinery designed from scratch", "claim": "Diffusion-based protein design produced serine hydrolases with catalytic efficiencies up to \\(2.2 \\times 10^5\\ \\mathrm{M^{-1}\\,s^{-1}}\\) on five folds unlike any natural serine hydrolase, with crystal structures matching the design models to under 1 Å.", "field": "chemistry", "date": "2025-02-13", "lab": "Institute for Protein Design, University of Washington", "model": "RFdiffusion with PLACER ensemble scoring", "humans": [ "Anna Lauko", "David Baker" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "An enzyme needs more than the right shape: its active site has to stay organised at every step of the reaction, which is where designed enzymes have historically failed. The team generated backbones with RFdiffusion from a minimal active-site description, then used an ensemble-generation network to check that the catalytic geometry held along the whole reaction coordinate, keeping only designs that passed. The resulting hydrolases approach natural catalytic efficiency on folds not found among natural serine hydrolases, and the crystal structures match the designs closely. Published in Science.", "novelty_check": "De novo enzyme design dates to the 2000s (Kemp eliminases, retro-aldolases), but designed enzymes were typically orders of magnitude slower than natural ones and needed directed evolution to become useful. The advance is reaching this efficiency with multi-step catalytic machinery by design rather than by evolution.", "caveats": "Serine hydrolases are a comparatively tractable target with a well-understood catalytic triad; the result does not generalise automatically to harder chemistry. Efficiencies remain below the best natural hydrolases. The per-round success rate is low and the paper reports the screening funnel. The reaction, the active-site description and the assays were human-specified, which is why autonomy is search-scaffold.", "sources": [ { "label": "Science: Computational design of serine hydrolases", "url": "https://www.science.org/doi/10.1126/science.adu2454", "kind": "research" }, { "label": "Baker Lab: Generating new enzymes with complex active sites", "url": "https://www.bakerlab.org/2025/02/13/ai-enzymes-with-complex-active-sites/", "kind": "announcement" } ], "tags": [ "protein-design", "enzyme-design", "rfdiffusion", "catalysis" ], "added": "2026-07-29" }, { "id": "2023-12-20-antibiotic-structural-class", "title": "A new structural class of antibiotic candidates against MRSA", "claim": "Graph neural networks trained on 39,312 assayed compounds and applied to 12 million molecules surfaced a chemical class active against MRSA in mice, with the substructures driving each prediction made explicit rather than left opaque.", "field": "chemistry", "date": "2023-12-20", "lab": "MIT / Broad Institute / Harvard", "model": "Ensembles of graph neural networks with substructure attribution", "humans": [ "Felix Wong", "Erica Zheng", "James Collins" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "The team measured antibiotic activity and human-cell cytotoxicity for 39,312 compounds, trained network ensembles on that data, and predicted both properties for over 12 million molecules. Rather than reading off top scores, they extracted the chemical substructures the models were keying on, which let them pick a class rather than isolated hits. Two lead compounds cleared MRSA infection in mouse models, topically and systemically, and appear to kill by collapsing the electrochemical gradient across the bacterial membrane. Published in Nature.", "novelty_check": "This is the same group's follow-up to the 2020 halicin work and to abaucin (2023); the new element is selecting a structural class via model interpretation instead of screening for individual hits. \"New structural class\" is a claim about scaffold novelty relative to clinical antibiotics, checked against known antibiotic chemotypes in the paper.", "caveats": "These are candidates, not drugs: mouse models only, with no clinical development at the time of this entry. The mechanism, disrupting membrane potential, is the same one halicin uses, so the novelty is structural rather than mechanistic, and membrane-active compounds carry a known mammalian-toxicity risk. The screening library, assays and interpretation were human-run; the model scored and explained.", "sources": [ { "label": "Nature: Discovery of a structural class of antibiotics with explainable deep learning", "url": "https://www.nature.com/articles/s41586-023-06887-8", "kind": "research" }, { "label": "ScienceDaily: Using AI, researchers identify a new class of antibiotic candidates", "url": "https://www.sciencedaily.com/releases/2023/12/231221012744.htm", "kind": "coverage" } ], "tags": [ "antibiotics", "graph-neural-network", "explainability", "mrsa" ], "added": "2026-07-29" }, { "id": "2020-02-20-halicin", "title": "Halicin, an antibiotic found by a neural network screening a compound library", "claim": "A graph neural network trained on 2,335 molecules picked halicin out of a repurposing library; it killed multidrug-resistant bacteria including Acinetobacter baumannii and Mycobacterium tuberculosis in vitro and cleared infections in mice.", "field": "medicine", "date": "2020-02-20", "lab": "MIT / Broad Institute", "model": "Directed message-passing graph neural network (Chemprop)", "humans": [ "Jonathan Stokes", "Regina Barzilay", "James Collins" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "The model was trained to predict growth inhibition of E. coli from structure alone, then applied to the Drug Repurposing Hub. Halicin, an abandoned diabetes candidate, scored highly despite being structurally unlike known antibiotics. It kills by dissipating the proton-motive force across the bacterial membrane, and the team could not evolve resistant E. coli over 30 days of serial passage. A follow-up screen of about 107 million molecules from ZINC15 produced eight further candidates. Published in Cell.", "novelty_check": "Virtual screening long predates deep learning. What was new was a model surfacing a hit structurally distant from its training set, with a mechanism distinct from existing antibiotic classes, and the follow-through to efficacy in animals. Halicin itself was a known molecule; the discovery is of its antibacterial activity, not of the compound.", "caveats": "Halicin had not entered clinical trials six years on, which is the usual fate of antibiotic leads and a reminder that discovery is not the bottleneck in this field. The model ranked compounds; the mechanism work, the animal studies and the interpretation are human. Membrane-disrupting antibacterials carry a known selectivity risk in mammals.", "independent_checks": [ { "who": "Booq et al., King Abdulaziz City for Science and Technology (Antibiotics, 2021)", "url": "https://doi.org/10.3390/antibiotics10121480", "outcome": "Reproduced broad-spectrum in vitro activity against every Gram-positive and Gram-negative strain tested, including a multidrug-resistant A. baumannii isolate, though at MICs of 128 to 256 micrograms per mL for A. baumannii, and found halicin loses potency after about a week of storage at 4 C." }, { "who": "Higashihira et al., Indiana University School of Medicine (Clinical Orthopaedics and Related Research, 2022)", "url": "https://doi.org/10.1097/corr.0000000000002251", "outcome": "Found halicin effective against S. aureus biofilms in vitro, a setting the original work did not test." } ], "sources": [ { "label": "Cell: A deep learning approach to antibiotic discovery", "url": "https://www.cell.com/cell/fulltext/S0092-8674(20)30102-1", "kind": "research" }, { "label": "MIT News: Artificial intelligence yields new antibiotic", "url": "https://news.mit.edu/2020/artificial-intelligence-identifies-new-antibiotic-0220", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: Artificial intelligence yields a powerful new antibiotic", "url": "https://news.ycombinator.com/item?id=22376178" } ], "tags": [ "antibiotics", "drug-repurposing", "graph-neural-network" ], "added": "2026-07-29", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded two unaffiliated laboratories that reproduced halicin's in vitro antibacterial activity, one with higher MICs for A. baumannii and a storage-stability concern. No grade moved.", "url": "https://doi.org/10.3390/antibiotics10121480" } ] }, { "id": "2023-09-19-alphamissense", "title": "Pathogenicity predictions for 71 million human missense variants", "claim": "An AlphaFold-derived model classified 89% of all 71 million possible human missense variants as likely benign or likely pathogenic, against the roughly 0.1% that carry a clinical classification today.", "field": "medicine", "date": "2023-09-19", "lab": "Google DeepMind", "model": "AlphaMissense", "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "AlphaMissense fine-tunes AlphaFold's structural representation on human and primate variant frequency data, so it learns which substitutions natural selection tolerates without being trained on clinical labels. Applied across 19,233 canonical human proteins it scored every possible single amino-acid change, calling 57% likely benign and 32% likely pathogenic. The full catalogue and the model code were released publicly. Published in Science.", "novelty_check": "Variant-effect predictors are a crowded field (SIFT, PolyPhen-2, EVE, CADD). The contribution is proteome-wide coverage at state-of-the-art benchmark accuracy without training on clinical annotations, which is what makes the predictions usable where no clinical data exists.", "caveats": "These are predictions, not diagnoses, and DeepMind says so: ACMG/AMP guidance treats computational evidence as supporting rather than standalone. Benchmarks have been criticised for circularity, since the model trains on population frequency and is evaluated partly against datasets encoding related signals. Calibration varies by gene, and a \"likely pathogenic\" call on a variant nobody has seen in a patient remains a hypothesis.", "independent_checks": [ { "who": "Ljungdahl et al., Oxford, UCSF and New York Genome Center (bioRxiv, 2023)", "url": "https://doi.org/10.1101/2023.10.24.562294", "outcome": "Against multiplexed-assay data for 17,696 variants in five genes, AlphaMissense was consistently among the top five of 39 predictors and the best-correlated for two genes, a modest improvement, and like the others it overcalls variants as pathogenic." }, { "who": "David Curtis, UCL Genetics Institute (European Journal of Human Genetics, 2024)", "url": "https://doi.org/10.1038/s41431-024-01675-y", "outcome": "Across 18 genes with rare-variant associations to hyperlipidaemia, hypertension or type 2 diabetes, AlphaMissense gave the strongest evidence on average but the best result for only 4 genes, with marked inconsistency between genes." } ], "sources": [ { "label": "Science: Accurate proteome-wide missense variant effect prediction with AlphaMissense", "url": "https://www.science.org/doi/10.1126/science.adg7492", "kind": "research" }, { "label": "DeepMind: A catalogue of genetic mutations to help pinpoint the cause of diseases", "url": "https://deepmind.google/blog/a-catalogue-of-genetic-mutations-to-help-pinpoint-the-cause-of-diseases/", "kind": "announcement" }, { "label": "Nature Biotechnology: Advancing missense variant pathogenicity prediction", "url": "https://www.nature.com/articles/s41587-023-01999-y", "kind": "research" }, { "label": "AlphaMissense predictions and code", "url": "https://github.com/google-deepmind/alphamissense", "kind": "research" } ], "tags": [ "genomics", "variant-effect", "clinical-genetics" ], "added": "2026-07-29", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded two independent evaluations: one finds AlphaMissense among the best predictors against functional assays with a modest margin, the other finds inconsistent performance for common disease genes. No grade moved.", "url": "https://doi.org/10.1101/2023.10.24.562294" } ] }, { "id": "2025-06-03-rentosertib-phase2a", "title": "Phase 2a results for a drug whose target and molecule both came from AI", "claim": "Rentosertib, a TNIK inhibitor whose target was picked by one AI system and whose molecule was generated by another, improved forced vital capacity by 98.4 mL at the top dose against a 20.3 mL decline on placebo in a 71-patient randomised trial.", "field": "medicine", "date": "2025-06-03", "lab": "Insilico Medicine", "model": "PandaOmics (target) + Chemistry42 (molecule)", "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "TNIK was prioritised as a fibrosis target by PandaOmics from omics and literature data; the inhibitor was generated and optimised by Chemistry42. GENESIS-IPF was a 12-week double-blind placebo-controlled Phase 2a across 22 sites in China, randomising 71 idiopathic pulmonary fibrosis patients across placebo and three dose arms. The 60 mg once-daily arm showed the largest lung-function improvement, and the drug was generally well tolerated. Published in Nature Medicine.", "novelty_check": "AI-generated molecules had reached the clinic before, and AI-nominated targets had been published before, but this is the first randomised placebo-controlled efficacy signal for a compound where both the target and the molecule came from generative systems. TNIK's role in fibrosis was not an established target hypothesis when it was nominated.", "caveats": "Phase 2a is small, short and exploratory: 71 patients over 12 weeks, single-country sites, not powered for efficacy, with FVC change over 12 weeks not a registrational endpoint. The sponsor designed, ran, analysed and published the trial. \"AI-discovered\" covers target selection and generative chemistry; the medicinal chemistry, preclinical work and trial conduct were conventional. A Phase 3 was subsequently initiated, which does not retrospectively strengthen this readout.", "sources": [ { "label": "Nature Medicine: A generative AI-discovered TNIK inhibitor for idiopathic pulmonary fibrosis", "url": "https://www.nature.com/articles/s41591-025-03743-2", "kind": "research" }, { "label": "Insilico Medicine: positive topline results of ISM001-055", "url": "https://insilico.com/news/tnik-ipf-phase2a", "kind": "announcement" } ], "tags": [ "drug-discovery", "clinical-trial", "target-discovery", "fibrosis" ], "added": "2026-07-29" }, { "id": "2025-01-16-mattergen", "title": "Generative model designs crystals to order; its flagship synthesis turned out to be a known compound", "claim": "MatterGen generates crystal structures conditioned on target properties, and its headline experimental validation was later shown to match a compound reported in 1972 that sits in the model's own training data.", "field": "materials", "date": "2025-01-16", "lab": "Microsoft Research", "model": "MatterGen", "verification": "disputed", "autonomy": "search-scaffold", "detail": "MatterGen is a diffusion model over crystal structures (atom types, coordinates and lattice jointly), trained on about 608,000 stable materials from the Materials Project and Alexandria, and fine-tunable to condition on a target property. Conditioned on a bulk modulus of 200 GPa it proposed a structure that collaborators at the Shenzhen Institutes of Advanced Technology synthesised, measuring 169 GPa, within 20% of the specification. Published in Nature.", "novelty_check": "Generative models for crystals existed (CDVAE, and screening pipelines like GNoME); the contribution claimed is property-conditioned generation validated experimentally. The novelty question is precisely what is disputed below: whether the synthesised compound was new.", "caveats": "A published critique in Materials Horizons argues the synthesised disordered phase Ta₁⁄₃Cr₂⁄₃O₂ is the same material as Ta₁⁄₂Cr₁⁄₂O₂ reported in 1972, that this compound is present in MatterGen's training set, and that its composition differs from the reported TaCr₂O₆, i.e. the flagship validation recovered a known material rather than a new one. The generative method itself is not refuted; the disputed part is the experimental novelty claim, and the entry is graded on the strength of that published objection. This registry keeps entries like this one rather than removing them.", "independent_checks": [ { "who": "Materials Horizons: Continued challenges in high-throughput materials predictions", "url": "https://pubs.rsc.org/en/content/articlelanding/2026/mh/d6mh00268d", "outcome": "disputed the novelty of the synthesised compound; identified it as present in the training data" } ], "sources": [ { "label": "Nature: A generative model for inorganic materials design", "url": "https://www.nature.com/articles/s41586-025-08628-5", "kind": "research" }, { "label": "MatterGen code and model weights", "url": "https://github.com/microsoft/mattergen", "kind": "research" }, { "label": "Microsoft Research: MatterGen, a new paradigm of materials design with generative AI", "url": "https://www.microsoft.com/en-us/research/blog/mattergen-a-new-paradigm-of-materials-design-with-generative-ai/", "kind": "announcement" }, { "label": "Materials Horizons: MatterGen predicts compounds from the training dataset", "url": "https://pubs.rsc.org/en/content/articlelanding/2026/mh/d6mh00268d", "kind": "challenge" } ], "discussion": [ { "label": "Hacker News: A generative model for inorganic materials design", "url": "https://news.ycombinator.com/item?id=42727329" } ], "tags": [ "materials", "crystal-structure", "diffusion-model", "contested" ], "added": "2026-07-29" }, { "id": "2024-02-21-tearing-mode-avoidance", "title": "Reinforcement learning steers a tokamak away from tearing instabilities", "claim": "A controller trained on past DIII-D shots forecast tearing-mode instabilities up to 300 ms ahead and adjusted the plasma in real time to avoid them, holding high-performance conditions that would otherwise have collapsed.", "field": "physics", "date": "2024-02-21", "lab": "Princeton University / PPPL / DIII-D National Fusion Facility", "model": "Deep reinforcement learning controller over a learned plasma model", "humans": [ "Jaemin Seo", "Egemen Kolemen" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "Tearing modes break up the magnetic surfaces confining a fusion plasma and can end a discharge. Existing approaches suppress them once they appear; this one predicts them. A network trained on past DIII-D discharges estimates the instability probability, and a reinforcement-learning policy trained against a learned plasma model adjusts beam power and plasma shape to keep that probability low while holding pressure high. It was tested live on DIII-D, not only in simulation. Published in Nature.", "novelty_check": "Distinct from the 2022 DeepMind–TCV work already in this registry, which learned magnetic shape control rather than instability avoidance. Tearing-mode prediction from machine learning had been published before; closing the loop so the controller acts on the prediction during a live shot is the new part.", "caveats": "Demonstrated on one machine in specific scenarios. A data-driven controller trained on DIII-D discharges has no guarantee of transferring to another tokamak, still less to ITER-scale burning plasma where the training data does not exist. Tearing modes are one disruption pathway among several. The physics model, actuators and safety limits are human-designed; the policy operates inside them.", "sources": [ { "label": "Nature: Avoiding fusion plasma tearing instability with deep reinforcement learning", "url": "https://www.nature.com/articles/s41586-024-07024-9", "kind": "research" }, { "label": "Princeton Engineering: Engineers use AI to wrangle fusion power for the grid", "url": "https://engineering.princeton.edu/news/2024/02/21/engineers-use-ai-wrangle-fusion-power-grid", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: Avoiding fusion plasma tearing instability with deep reinforcement learning", "url": "https://news.ycombinator.com/item?id=39572871" } ], "tags": [ "fusion", "plasma-control", "reinforcement-learning", "tokamak" ], "added": "2026-07-29" }, { "id": "2024-11-20-alphaqubit", "title": "A neural decoder that identifies quantum errors more accurately than hand-designed methods", "claim": "A transformer trained on Google's Sycamore surface-code data cut decoding errors by 6% against tensor-network decoding and 30% against correlated matching, the best reported accuracy at the time.", "field": "physics", "date": "2024-11-20", "lab": "Google DeepMind / Google Quantum AI", "model": "AlphaQubit", "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "A quantum error-correcting code produces a stream of syndrome measurements; a decoder has to infer from them which errors actually occurred. Hand-designed decoders assume a simplified noise model, whereas AlphaQubit is a transformer trained first on simulated data and then fine-tuned on hundreds of millions of real syndrome samples from Sycamore, so it can learn the device's actual correlated noise, leakage and crosstalk. It beat both the accuracy-oriented tensor-network decoder and the fast matching decoder, and held up in simulation to distance-11 codes. Published in Nature.", "novelty_check": "Machine-learning decoders had been proposed for years but had not beaten the best classical decoders on real hardware data at scale. The result is a decoder that does, on a specific processor.", "caveats": "Accuracy, not speed: AlphaQubit is far slower than matching decoders and does not run inside the real-time budget a fault-tolerant machine needs, which the paper states. It is trained per device on that device's data, so it does not transfer for free. Scaling behaviour beyond the tested code distances is open. This improves a component of error correction; it does not itself demonstrate a fault-tolerant computation.", "sources": [ { "label": "Nature: Learning high-accuracy error decoding for quantum processors", "url": "https://www.nature.com/articles/s41586-024-08148-8", "kind": "research" }, { "label": "Google: AlphaQubit, research on quantum error correction", "url": "https://blog.google/innovation-and-ai/models-and-research/google-deepmind/alphaqubit-quantum-error-correction/", "kind": "announcement" } ], "tags": [ "quantum-computing", "error-correction", "transformer" ], "added": "2026-07-29" }, { "id": "2023-11-14-graphcast", "title": "Medium-range weather forecasts from a graph neural network beat the operational physics model", "claim": "GraphCast produced more accurate 10-day forecasts than ECMWF's HRES on roughly 90% of the evaluated variable and lead-time combinations, in under a minute on a single TPU against hours on a supercomputer.", "field": "climate", "date": "2023-11-14", "lab": "Google DeepMind", "model": "GraphCast", "humans": [ "Remi Lam", "Peter Battaglia" ], "verification": "peer-reviewed", "autonomy": "ai-led", "detail": "GraphCast takes the two most recent global atmospheric states and predicts the next state six hours ahead on a roughly 0.25° grid, applied repeatedly to reach ten days. It is trained on ERA5 reanalysis rather than on the equations of motion, and it beat HRES, the operational gold standard, on the large majority of evaluated variables, rising to almost all of them within the troposphere. It also flagged severe weather events, including cyclone tracks, earlier than the physics model despite never being trained to look for them. Published in Science, with code and weights released.", "novelty_check": "Machine-learning weather models existed before (FourCastNet, Pangu-Weather), and Pangu had already reported beating HRES on some measures. GraphCast's contribution is the breadth of the win across variables and lead times, at operational resolution, with public code. That is what moved national weather services to adopt learned models.", "caveats": "GraphCast is trained on ECMWF reanalysis and initialised from ECMWF analyses, so it depends on the physics-based system it is measured against rather than replacing it. Deterministic models trained on a mean-squared-error objective blur fine structure and underestimate extremes such as peak cyclone intensity. It has no conservation guarantees and cannot forecast anything outside its trained variables. Later ensemble models such as GenCast address part of this.", "independent_checks": [ { "who": "Wolfgang R. Rowell and Lucas S. Kupssinskü (regional evaluation over Brazil)", "url": "https://arxiv.org/abs/2606.06348", "outcome": "Against ECMWF HRES over four Brazilian sub-regions, found a regime-dependent profile: GraphCast underperforms for 500 hPa geopotential at days 2 to 7 over southern Brazil in austral winter, regains an edge at longer range, and captures large-scale moisture transport well in the summer wet season." } ], "sources": [ { "label": "Science: Learning skillful medium-range global weather forecasting", "url": "https://www.science.org/doi/10.1126/science.adi2336", "kind": "research" }, { "label": "DeepMind: GraphCast, AI model for faster and more accurate global weather forecasting", "url": "https://deepmind.google/blog/graphcast-ai-model-for-faster-and-more-accurate-global-weather-forecasting/", "kind": "announcement" }, { "label": "GraphCast code and weights (google-deepmind/weathernext)", "url": "https://github.com/google-deepmind/weathernext", "kind": "research" } ], "tags": [ "weather", "forecasting", "graph-neural-network" ], "added": "2026-07-29", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded an independent regional evaluation over Brazil that finds GraphCast's skill against HRES depends on regime and season. No grade moved.", "url": "https://arxiv.org/abs/2606.06348" } ] }, { "id": "2017-12-14-kepler-90i", "title": "An eighth planet around Kepler-90 found by a neural network", "claim": "A convolutional network re-examining Kepler light curves recovered transit signals the standard pipeline had ranked below threshold, yielding Kepler-90i and making Kepler-90 the first star besides the Sun known to host eight planets.", "field": "astronomy", "date": "2017-12-14", "lab": "Google Brain / University of Texas at Austin", "model": "Convolutional neural network (AstroNet)", "humans": [ "Christopher Shallue", "Andrew Vanderburg" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "The network was trained on 15,000 previously vetted Kepler signals to separate genuine transits from false positives, then pointed at 670 stars already known to host multiple planets, searching the weak signals the automated pipeline had discarded. It surfaced Kepler-90i (about 30% larger than Earth, orbiting every 14.4 days, with a surface hot enough to rival Mercury) and Kepler-80g, which completes a five-planet resonant chain. Published in The Astronomical Journal.", "novelty_check": "Machine classification of Kepler transit candidates already existed in the pipeline (Autovetter, Robovetter). What was new was training a network to vet signals below the pipeline's detection threshold and getting confirmable planets out of the discard pile.", "caveats": "The network vetted candidate signals; the transit search, the follow-up and the statistical validation were conventional. \"First system with as many planets as ours\" is a statement about what has been detected, not what exists. Detection is heavily biased toward close-in planets, and all of Kepler-90's known planets orbit within roughly Earth's distance from its star. The 670-star search produced two new planets in total.", "sources": [ { "label": "The Astronomical Journal: Identifying Exoplanets with Deep Learning", "url": "https://iopscience.iop.org/article/10.3847/1538-3881/aa9e09", "kind": "research" }, { "label": "AstroNet: Kepler light-curve models (exoplanet-ml)", "url": "https://github.com/google-research/exoplanet-ml", "kind": "research" }, { "label": "NASA: Artificial intelligence, NASA data used to discover eighth planet circling distant star", "url": "https://www.nasa.gov/news-release/artificial-intelligence-nasa-data-used-to-discover-eighth-planet-circling-distant-star/", "kind": "announcement" } ], "tags": [ "exoplanets", "kepler", "transit-detection" ], "added": "2026-07-29" }, { "id": "2026-06-25-herculaneum-scroll-read", "title": "First Herculaneum scroll read end to end without unrolling it", "claim": "PHerc. 1667 was virtually unwrapped and read in full from X-ray scans using learned ink detection: about 1.4 metres of papyrus across roughly 22 columns of Greek, identified as Philodemus, On Gods, Book 8.", "field": "archaeology", "date": "2026-06-25", "lab": "Vesuvius Challenge", "model": "Community-developed ink-detection neural networks", "humans": [ "Brent Seales", "Luke Farritor", "Youssef Nader", "Julian Schilliger" ], "verification": "author-verified", "autonomy": "ai-assisted", "detail": "The Herculaneum scrolls were carbonised by the AD 79 eruption of Vesuvius and disintegrate if physically unrolled. The pipeline scans a scroll by X-ray tomography, traces the rolled sheet as a three-dimensional surface, flattens it, and runs a network that detects near-invisible carbon ink from sub-millimetre surface texture. Luke Farritor's model recovered the first word, πορφύρας (\"purple\"), in October 2023, and the 2023 Grand Prize for four passages of 140 characters was claimed in February 2024. In June 2026, PHerc. 1667 became the first scroll recovered continuously from beginning to end, and a title read in a related scroll established that Philodemus' On Gods ran to at least eight books.", "novelty_check": "Virtual unwrapping itself is older (Seales read the En-Gedi scroll in 2015), but that worked because its ink contained metal and showed up in X-ray. Herculaneum ink is carbon-based and nearly invisible to tomography, which is why the ink signal had to be learned from texture instead of density. No Herculaneum scroll had been read by any method before this programme.", "caveats": "Readings are probabilistic reconstructions from a learned ink signal, not photographs of text: characters come with varying confidence and papyrologists supply conventional editorial restoration on top. The project's papyrology team reviews prize submissions, but that team is part of the challenge rather than an outside body, and no peer-reviewed critical edition of PHerc. 1667 had appeared when this entry was written, hence author-verified rather than independent. The attribution to Philodemus rests on a title read in a different scroll.", "sources": [ { "label": "Complete virtual unwrapping and reading of a rolled Herculaneum papyrus", "url": "https://arxiv.org/abs/2606.29085", "kind": "research" }, { "label": "Vesuvius Challenge: An entire Herculaneum scroll has been read for the first time", "url": "https://scrollprize.org/firstscroll", "kind": "announcement" }, { "label": "Vesuvius Challenge", "url": "https://scrollprize.org/", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: An entire Herculaneum scroll has been read for the first time", "url": "https://news.ycombinator.com/item?id=48675179" } ], "tags": [ "papyrology", "virtual-unwrapping", "ink-detection", "ancient-texts" ], "added": "2026-07-29" }, { "id": "2026-07-20-kourovka-notebook", "title": "Eight problems from the Kourovka Notebook solved and formalized in Lean", "claim": "A formal reasoning agent developed the proof strategies for eight open problems in group theory without human guidance on which mathematical steps to take, and machine-checked every one in Lean 4.", "field": "mathematics", "date": "2026-07-20", "lab": "Harmonic", "model": "Aristotle", "humans": [ "Wouter van Doorn", "Elias Judin", "Pietro Monticone", "Daniel Morrison" ], "year_posed": 1969, "verification": "formal", "autonomy": "autonomous", "detail": "The Kourovka Notebook has collected open problems in group theory since 1965, with a new issue every two to four years. Aristotle solved eight of them: 3.46 (a group with exactly two maximal locally soluble normal subgroups), 18.50, 19.25, 20.125 (a surjective non-injective Rota-Baxter operator on a non-abelian group), 21.8, 21.24, 21.147 and 21.150. The oldest first appeared in the third issue in 1969. Solutions take the form of proofs, counterexamples and constructions, each formalized in Lean 4 and published in a companion repository. The authors then informalized the Lean proofs into conventional prose and submitted both to the Notebook editors and the original problem proposers, who accepted them before publication.", "novelty_check": "The Kourovka Notebook is itself the authoritative open-problems register for group theory, and each of the eight is recorded there as open with a named proposer, so novelty is established by the source of record rather than by a literature search. The authors submitted solutions to the Notebook editors and the problem proposers for acceptance before publishing, and the initial submissions remain in the Notebook's own preprint repository. Problem 3.46 traces to Plotkin's survey question on products of locally soluble normal subgroups, which Baumslag, Kovács and Neumann had partly addressed in the negative.", "caveats": "Not peer reviewed as a journal article, though the individual solutions were accepted by the Kourovka Notebook editors and the problem proposers, which is the relevant gatekeeping for this venue. The autonomy grade rests on the authors' own disclosure: they state Aristotle developed the proof strategies without human guidance about which steps to take, while they monitored its work, intervened when clarifications were needed, and built a library of relevant concepts with it. That library work and the final canonisation for Mathlib upstreaming were done partly by hand. The Lean certificates are the load-bearing evidence and anyone can rerun them; the claim about how the arguments were found rests on the authors' account.", "independent_checks": [ { "who": "Kourovka Notebook editors (E. I. Khukhro, V. D. Mazurov)", "url": "https://arxiv.org/abs/1401.0300v46", "outcome": "The 2026-09-01 update of the Notebook marks all eight problems (3.46, 18.50, 19.25, 20.125, 21.8, 21.24, 21.147, 21.150) as solved, citing the authors' preprints hosted on the Notebook's site and this paper." } ], "sources": [ { "label": "arXiv: On Some Problems from the Kourovka Notebook", "url": "https://arxiv.org/abs/2607.17477", "kind": "research" }, { "label": "Lean 4 formalizations of all eight solutions", "url": "https://github.com/pitmonticone/Kourovka", "kind": "research" } ], "tags": [ "group-theory", "lean", "formalization", "kourovka-notebook", "autonomous-discovery" ], "added": "2026-08-06", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded the Kourovka Notebook editors' 2026-09-01 update, which marks all eight problems solved. No grade moved.", "url": "https://arxiv.org/abs/1401.0300v46" } ] }, { "id": "2026-08-01-astra-ten-advances", "title": "Ten results in mathematics and theoretical computer science with Lean certificates", "claim": "An unreleased OpenAI model produced ten results including the first explicit non-sofic group and a disproof of Connes's rigidity conjecture, each published with a machine-checkable Lean 4 proof.", "field": "mathematics", "date": "2026-08-01", "lab": "OpenAI", "model": "Astra", "year_posed": 1999, "verification": "formal", "autonomy": "ai-led", "detail": "The ten cover high-dimensional sphere packing, binary and spherical codes, non-sofic groups, Connes's rigidity conjecture, arithmetic circuit complexity, quantum parallel repetition, the closest vector problem, Ehrhart's volume conjecture, multicolor Ramsey numbers and extremal number conjectures. Soficity was introduced by Gromov in 1999 and whether a non-sofic group exists had stood open since; Connes posed his rigidity conjecture in 1980, and the disproof constructs infinitely many non-isomorphic property (T) groups sharing a von Neumann algebra. Humans used the same model to prepare the manuscripts and then to formalize each argument in Lean 4.32.0. OpenAI put the token cost of finding all ten at roughly $2,000 at Sol API rates.", "novelty_check": "Each of the ten is a named open problem with a documented origin: soficity from Gromov 1999, Connes rigidity from 1980, and the sphere-packing exponent improvement is the first on the general bound since 1978. The Lean repository states which prior bound each result improves. Thomas Bloom, who maintains erdosproblems.com and publicly corrected OpenAI's 2025 Erdős retrieval episode, called these results big news, which is meaningful given that he is the person who caught the previous overclaim.", "caveats": "The certificates are checkable and the claim that the theorems hold is as solid as Lean makes it. The claim about provenance is not checkable: Astra is unreleased, so no outside party can test whether the model produces these results, and the account of how they were found rests entirely on OpenAI. Grading verification as formal reflects the proofs, not the provenance; a reader who cares only about who found it should read this as claimed. Specialist review of the mathematical significance is still pending in several cases. An audit of 18 chapter-specific reviews by Mikołaj and Krzysztof Sienicki found no confirmed substantive error in a principal result, but records that a specialist review asks for major revision of Chapter 8's compressed analytic arguments. The Connes rigidity conclusion has independent support: Shuoxing Zhou reached a counterexample concurrently, and Kun and Thom and Fournier-Facio have built further non-sofic groups on the same mechanism. Noam Brown noted there are no Millennium Prize problems here. Press summaries of the ten disagree with the repository's own list, some substituting specific Erdős problem numbers that do not appear in it; the list above follows the repository. Andreas Thom disputed the framing of the non-sofic-group result in a guest post on Terence Tao's blog on 2026-09-11. He objects that the announcement described a decade without progress while the proof leaned on his and Gabor Kun's 2019 centralizer-rigidity paper, and he asked whether his own private ChatGPT conversations on the same techniques were available during training or solving. OpenAI's reply, that this did not happen, did not address de-identified training data. The dispute is about attribution and disclosure rather than whether the ten proofs check, so neither grade moved.", "independent_checks": [ { "who": "Shuoxing Zhou (independent concurrent counterexample)", "url": "https://arxiv.org/abs/2608.02327", "outcome": "Constructed two non-isomorphic ICC property (T) groups with isomorphic group von Neumann algebras, a counterexample to Connes' rigidity conjecture, with GPT-5.6 Sol assistance and independently of OpenAI's work. Corroborates the conclusion, not OpenAI's stronger infinite-family result." }, { "who": "Gabor Kun and Andreas Thom", "url": "https://arxiv.org/abs/2608.06222", "outcome": "Analyzed the proof mechanism behind the non-sofic group and extended it to new families of non-sofic generalized wreath products and group doubles." }, { "who": "Mikołaj Sienicki and Krzysztof Sienicki (audit of the review record)", "url": "https://arxiv.org/abs/2608.14673", "outcome": "Audited 18 chapter-specific reviews of the ten results with the Lean formalizations and follow-up work: no confirmed substantive error in a principal result remains; a specialist review requests major revision of Chapter 8; subsequent work reuses the Chapter 3 mechanism and confirms that Connes' rigidity conjecture is false." } ], "sources": [ { "label": "Lean 4 certificates for all ten results", "url": "https://github.com/openai/ten-proofs", "kind": "research" }, { "label": "OpenAI: Ten advances in mathematics and theoretical computer science", "url": "https://openai.com/index/ten-advances-in-mathematics/", "kind": "announcement" }, { "label": "Andreas Thom on the non-sofic-group result, guest post on Terence Tao's blog", "url": "https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/", "kind": "challenge" }, { "label": "arXiv: Zhou, ICC property (T) groups without W*-superrigidity", "url": "https://arxiv.org/abs/2608.02327", "kind": "commentary" }, { "label": "arXiv: Kun and Thom, Nonsofic wreath products of residually finite groups", "url": "https://arxiv.org/abs/2608.06222", "kind": "commentary" }, { "label": "arXiv: Fournier-Facio, A torsion-free non-sofic group", "url": "https://arxiv.org/abs/2608.02025", "kind": "commentary" }, { "label": "arXiv: Sienicki and Sienicki, A human audit of OpenAI's AI-generated mathematical proofs", "url": "https://arxiv.org/abs/2608.14673", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: Ten advances in mathematics and theoretical computer science", "url": "https://news.ycombinator.com/item?id=49157930" } ], "tags": [ "group-theory", "lean", "formalization", "sphere-packing", "quantum-complexity" ], "added": "2026-08-06", "revisions": [ { "date": "2026-09-15", "kind": "challenge", "note": "Cited Andreas Thom's guest post on Terence Tao's blog, an attribution and data-provenance dispute over the non-sofic-group result. Neither grade moved: the objection is to the framing and to what was disclosed, not to the proofs.", "url": "https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/" }, { "date": "2026-09-24", "kind": "check", "note": "Recorded Shuoxing Zhou's concurrent independent counterexample to Connes' rigidity conjecture, Kun and Thom's extension of the non-sofic mechanism, and an audit of the review record that finds no confirmed error in a principal result but a request for major revision of Chapter 8. No grade moved.", "url": "https://arxiv.org/abs/2608.14673" } ] }, { "id": "2026-05-18-pevac-ps-sarbecovirus", "title": "Phase 1 trial of a computationally designed pan-sarbecovirus vaccine", "claim": "A vaccine whose antigen was designed entirely in silico from sarbecovirus sequence data was safe in 39 healthy volunteers and raised immune responses against SARS-CoV-2, SARS, and bat viruses that have never infected humans.", "field": "medicine", "date": "2026-05-18", "lab": "University of Cambridge / DIOSynVax", "model": "Unknown", "humans": [ "Alasdair P. S. Munro", "Jonathan L. Heeney", "Saul N. Faust" ], "verification": "peer-reviewed", "autonomy": "ai-assisted", "detail": "Rather than target one strain, the team analysed sarbecovirus genetic data to compute a single synthetic antigen capturing structural features shared across the subgenus, on the theory that a vaccine aimed at what the family has in common survives the evolution of any one member. The candidate, pEVAC-PS, is a DNA vaccine delivered needle-free by micro fluid jet. The phase 1 dose-escalation trial ran in 39 healthy volunteers aged 18 to 50, sponsored by University Hospital Southampton NHS Foundation Trust, and reported no significant side effects.", "novelty_check": "Computationally designed and consensus or mosaic antigens have a long history in HIV and influenza vaccine work, and broad sarbecovirus vaccine candidates were pursued by several groups after 2020. What is new here is a first-in-human safety and immunogenicity readout for an antigen with no natural sequence, generated computationally rather than selected from circulating strains. Searched PubMed for prior phase 1 results on pan-sarbecovirus immunogens; earlier candidates in this class were preclinical at the time of publication.", "caveats": "Phase 1: safety, tolerability and immunogenicity only. No efficacy claim, and immune response is not protection. n=39, healthy adults 18 to 50, so nothing follows about older or immunocompromised populations. Approval is years away and a phase 2 is planned. The autonomy grade is deliberately conservative because the public record is thin on what the computational method actually was: the reporting says artificial intelligence and machine learning without naming a model, architecture or developer, so model is recorded as Unknown rather than inferred. The bat-virus immunogenicity result is a laboratory assay against viruses that have not infected humans, not evidence of protection against a future outbreak.", "sources": [ { "label": "Journal of Infection: A phase I, needle free, dose escalation clinical trial of pEVAC-PS, a candidate pan-Sarbecovirus vaccine", "url": "https://www.journalofinfection.com/article/S0163-4453(26)00084-8/fulltext", "kind": "research" }, { "label": "University of Cambridge: New universal vaccine technology could protect us from future virus outbreaks", "url": "https://www.cam.ac.uk/research/news/new-universal-vaccine-technology-could-protect-us-from-future-virus-outbreaks", "kind": "announcement" }, { "label": "ScienceDaily: AI-designed universal coronavirus vaccine passes first human trial", "url": "https://www.sciencedaily.com/releases/2026/06/260605023357.htm", "kind": "coverage" } ], "tags": [ "vaccine-design", "clinical-trial", "coronavirus", "antigen-design", "phase-1" ], "added": "2026-08-06" }, { "id": "2026-06-17-kagome-superconductors", "title": "Two kagome superconductors predicted by machine learning and confirmed in the lab", "claim": "A screening pipeline that narrowed over 1.3 million candidate structures picked out YRu3B2 and LuRu3B2, which were then synthesized and measured to superconduct at 0.81 K and 0.95 K.", "field": "materials", "date": "2026-06-17", "lab": "Aalto University / Rice University", "model": "Unknown", "humans": [ "Rose Albu Mustaf", "Päivi Törmä", "Emilia Morosan", "B. Andrei Bernevig", "Miguel A. L. Marques" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "The SuperC consortium used machine-learning pre-screening to cut a very large chemical space down to a tractable candidate list, then ran first-principles calculations on what survived, arriving at 741 dynamically and thermodynamically stable compounds with DFT-predicted Tc above 5 K. Morosan's group at Rice synthesized two of them and confirmed bulk superconductivity by magnetization and specific heat measurement. Both compounds carry a kagome lattice, in which electrons form flat bands.", "novelty_check": "Machine-learning screening of superconductor candidates is an established and crowded field, and the JHU APL work on novel superconductors predates this. What is claimed as new is the end-to-end path: candidates identified from scratch by a machine-learning-guided pipeline, then synthesized and experimentally confirmed, rather than screening validated only against already-known superconductors. YRu3B2 and LuRu3B2 were not previously reported as superconductors. Whether this is the first such end-to-end confirmation is the consortium's framing and is not independently adjudicated here.", "caveats": "The gap between prediction and measurement is the story the coverage mostly buries: the screen selected for DFT-predicted Tc above 5 K, and the two compounds actually measured at 0.81 K and 0.95 K, roughly a factor of five to six low. These are sub-1-kelvin superconductors, nowhere near room temperature, and the consortium's stated goal of a room-temperature superconductor by 2033 is a research programme, not a result. Two confirmed compounds out of 741 stable candidates is also not a hit rate the paper establishes as generalizable. The specific machine-learning method is not named in the coverage consulted, so model is recorded as Unknown.", "sources": [ { "label": "arXiv: Machine Learning-Guided Discovery of Kagome Superconductors YRu3B2 and LuRu3B2", "url": "https://arxiv.org/abs/2512.16945", "kind": "research" }, { "label": "ScienceDaily: AI just supercharged the race to find room temperature superconductors", "url": "https://www.sciencedaily.com/releases/2026/07/260701205006.htm", "kind": "coverage" } ], "tags": [ "superconductivity", "kagome", "high-throughput-screening", "dft", "experimental-confirmation" ], "added": "2026-08-06" }, { "id": "2026-08-10-zeta-zeros-critical-line", "title": "Proportion of zeta zeros on the critical line raised to 67.25%", "claim": "An unconditional proof that at least 67.25% of the nontrivial zeros of the Riemann zeta function are simple and lie on the critical line, up from a previous record of about 41.6%, with a machine-checked Lean formalization.", "field": "mathematics", "date": "2026-08-10", "lab": "Anthropic", "model": "Claude (unreleased research version)", "humans": [ "Jarred Sumner", "Levent Alpöge", "Ralph Furman", "Eric Easley" ], "verification": "formal", "autonomy": "ai-led", "detail": "The Riemann hypothesis asserts every nontrivial zero lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg established a positive proportion, Levinson reached a third in 1974, Conrey two fifths in 1989, and the record had crept to roughly 41.6%. This work proves at least \\(\\tfrac32 - \\tfrac{1}{\\sqrt2}\\cot\\tfrac{1}{\\sqrt2} = 67.25\\ldots\\%\\) of zeros are simple and on the line, and that at least 0.83625 of zeros are distinct. The decisive step combined the unconditional pair-correlation work of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh with a 2000 paper of Bombieri, treating the whole function space at once with the quadratic form allowed to be non-diagonal rather than splitting it. The process is itself the record worth noting: the model was asked to attempt the Riemann hypothesis outright, with the mathematical choices left to it, and the bound improvement fell out of failing at that. It generated and discarded roughly 650 ideas in a first session; in a second it coordinated about 60 subagents which ran some 2,400 shell commands, wrote hundreds of scripts, checked numerically against known zeros and refereed one another. Two subagents developed the key ideas, thirteen fed them, thirty tried and failed, thirteen validated, two drafted the paper, across roughly 31 million output tokens in two Claude Code sessions. The model also proposed writing the result up, checked 54 arXiv papers for prior art, and recommended a human number theorist validate it.", "novelty_check": "The proportion ladder is a documented literature running from Hardy (1914) and Selberg (1942) through Levinson's one third (1974) and Conrey's two fifths (1989), with subsequent refinements reaching about 41.6%; the paper states the prior record and cites it. The model itself searched 54 arXiv papers for prior art before write-up. No earlier unconditional bound above 42% appears in the literature. The result is a new bound, not a retrieval of a known one.", "caveats": "Not peer-reviewed. The Riemann hypothesis itself is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The Lean formalization is sorry-free and anchors its statements on Mathlib's own riemannZeta and analyticOrderAt with the corresponding counting functions rather than an assumed form of the result, which is real statement anchoring; but the assembly of those primitives into the proportion claim is the paper's own, and that assembly is exactly what nobody independent has audited. The bound itself has since been reproved by Youness Lamzouri by a different route, a single Hilbert space inequality in place of the paper's finite-dimensional matrix framework, which corroborates the result without auditing the formalization. The same organisation produced both the proof and its formalization. Two Anthropic mathematicians studied and validated the work, which is author-side review, not independent replication. Autonomy graded ai-led rather than autonomous: humans posed the problem, supplied the environment and validated the output.", "independent_checks": [ { "who": "Youness Lamzouri (independent new proof)", "url": "https://arxiv.org/abs/2609.02882", "outcome": "Gave a new proof that more than two thirds of the zeros are simple and on the critical line, replacing the paper's matrix framework with a single Hilbert space inequality, and proved two further estimates the paper does not state. Reports no error in the original." }, { "who": "vibemathed.com source audit", "url": "https://vibemathed.com/problem/more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line", "outcome": "At tag v1.0 found no sorry outside the comparator challenge files, no native_decide and no axiom declarations, and found the headline theorems unconditional: the explicit formula and other inputs are proved in Lean rather than assumed. Statement fidelity against the paper not audited; kernel not re-run." } ], "registrations": [ { "registry": "vibemathed", "id": "more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line", "url": "https://vibemathed.com/problem/more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line", "title": "The Proportion of Zeta Zeros on the Critical Line" } ], "sources": [ { "label": "Anthropic: paper (PDF)", "url": "https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf", "kind": "research" }, { "label": "Lean 4 formalization (anthropics/zeta-23-lean)", "url": "https://github.com/anthropics/zeta-23-lean", "kind": "research" }, { "label": "Session transcript (PDF)", "url": "https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e9548c3f44f.pdf", "kind": "announcement" }, { "label": "arXiv: Lamzouri, A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line", "url": "https://arxiv.org/abs/2609.02882", "kind": "commentary" } ], "tags": [ "lean", "number-theory", "riemann-hypothesis", "multi-agent" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." }, { "date": "2026-09-24", "kind": "check", "note": "Recorded Youness Lamzouri's independent new proof of the two-thirds bound and vibemathed's source audit, which finds the Lean theorems unconditional. No grade moved.", "url": "https://arxiv.org/abs/2609.02882" } ] }, { "id": "2026-07-27-feige-conjecture", "title": "Feige's conjecture on sums of nonnegative random variables settled", "claim": "For independent nonnegative random variables with mean at most 1 and sum S, the probability that S is below its mean plus one is at least 1/e, the sharp constant Feige conjectured in 2004; three independent proofs appeared within days.", "field": "mathematics", "date": "2026-07-27", "lab": "Independent", "model": "ChatGPT 5.6 Pro", "humans": [ "Weibo Fu", "Yanjun Han", "Guanyang Wang", "Jun Yan", "Peng Zhang", "Zhengqing Zhou" ], "year_posed": 2004, "verification": "formal", "autonomy": "ai-led", "detail": "Feige proved in 2004 that for independent nonnegative random variables \\(X_1, \\dots, X_n\\) with \\(\\mathbb{E}[X_i] \\le 1\\) and \\(S\\) their sum, \\(\\Pr(S < \\mathbb{E}[S] + 1)\\) is bounded below by a positive constant, established \\(1/13\\), and conjectured the sharp value \\(1/e\\). The primary paper states plainly in its abstract that the proof was found by ChatGPT 5.6 Pro, which combined the Vlassis-Thomas Dirichlet calibration theorem with Grunbaum-type convex geometry; the authors then checked, revised and rewrote the argument. The sharper form determines the optimal small-deviation bound. An end-to-end Lean formalization accompanies the paper, covering the Vlassis-Thomas theorem, Grunbaum's centroid theorem and the combining argument, developed with Codex.", "novelty_check": "Feige's conjecture is documented as open from his 2004 paper on sums of independent random variables, where he proved the weaker constant 1/13. The sharp 1/e constant is recorded as an open problem in the subsequent small-deviation literature and had stood for 22 years. No prior proof of the sharp constant appears; the near-simultaneous arrival of three independent proofs within days confirms it was live and open, not settled and forgotten.", "caveats": "Not peer-reviewed; all three proofs are preprints. The Lean formalization accompanies the primary paper and is author-produced. Autonomy is graded ai-led rather than autonomous because the human authors posed the problem, then checked, revised and rewrote the argument for publication. The three independent proofs are genuine corroboration of the result, though two of the three were themselves AI-assisted, so they are not fully independent of the method.", "independent_checks": [ { "who": "Zipei Nie and Jiaye Wei (independent second proof)", "url": "https://arxiv.org/abs/2607.24528", "outcome": "independently proved the same sharp 1/e bound, obtained with assistance from GPT-5.6 Sol" }, { "who": "Mark Stander (independent third proof)", "url": "https://doi.org/10.5281/zenodo.21622950", "outcome": "further independent proof of the conjecture" } ], "registrations": [ { "registry": "vibemathed", "id": "feiges-conjecture", "url": "https://vibemathed.com/problem/feiges-conjecture", "title": "Feige's Conjecture" }, { "registry": "mathdb", "id": "315661", "title": "Feige's 1/e conjecture", "url": "https://mathdb.com/p/315661/feige-s-1-e-conjecture" } ], "sources": [ { "label": "arXiv: Sharp small-deviation inequalities for sums of independent nonnegative random variables", "url": "https://arxiv.org/abs/2607.23980", "kind": "research" }, { "label": "Lean formalization (pengzhang91/Feige)", "url": "https://github.com/pengzhang91/Feige", "kind": "research" }, { "label": "arXiv: On Feige's conjecture (Nie and Wei)", "url": "https://arxiv.org/abs/2607.24528", "kind": "research" } ], "tags": [ "lean", "probability", "concentration-inequalities" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-08-04-moore-bound", "title": "Asymptotic degree-diameter problem resolved for fixed diameter", "claim": "The maximum order of a graph with maximum degree \\(d\\) and diameter \\(k\\) satisfies \\(n_k(d)/d^k\\) tending to 1 as \\(d\\) grows, for every fixed \\(k\\), resolving the asymptotic form of a question open since 1978, with a Lean 4 formalization.", "field": "mathematics", "date": "2026-08-04", "lab": "Independent", "model": "GPT-5.6 Pro", "humans": [ "Wouter Cames van Batenburg", "Samuel Korsky" ], "year_posed": 1978, "verification": "formal", "autonomy": "ai-assisted", "detail": "The Moore bound caps the order of a graph of maximum degree \\(d\\) and diameter \\(k\\) at roughly \\(d^k\\). Whether that cap is asymptotically attainable for fixed diameter was open: Bannai and Ito had settled which exact Moore graphs exist, and constructions had approached the bound only for small diameters. This proves the limit of \\(n_k(d)/d^k\\) is 1 for every fixed \\(k\\), with a companion lower bound on the edge variant and a tight asymptotic for the bipartite version. The paper carries an explicit tool and computational resource disclosure: GPT-5.6 Pro was used in exploratory brainstorming directed by the authors, who formulated the required parameter scale and proposed higher-rank spherical buildings as the geometric setting. The model suggested splitting complete flags into their odd- and even-rank subflags, initially in connection with the edge problem; that suggestion became the halved-flag construction carrying the main theorem, and the graph is named for it.", "novelty_check": "The degree-diameter problem is a named and long-tracked question in extremal graph theory, surveyed by Bermond and Bollobas (1981) and with Bollobas's 1978 Extremal Graph Theory as the reference point for the asymptotic question; Bachraty, Siagiova and Siran had approached the bound for diameter three in 2019. The authors note that after completing the work they learned the projective-plane construction of an earlier paper shares the rank-2 incidence graph underlying the \\(k = 1\\) case of theirs, and state the higher-rank construction and its odd-even routing argument are independent and different. No prior proof of the general asymptotic result appears.", "caveats": "Not peer-reviewed. Autonomy is graded ai-assisted, the weaker defensible reading, because the disclosure is explicit that the authors developed the suggestion into the construction, formulated and verified all mathematical arguments, and take full responsibility for the content; the model's contribution was one structural idea during brainstorming. Generative tools also assisted with the Lean formalization. The Lean repository is author-produced and had no independent audit at announcement.", "independent_checks": [ { "who": "vibemathed.com curator check of the Lean repository", "url": "https://github.com/woutercvb/wewantmoore", "outcome": "checked at commit 32beb227: DegreeDiameter.theorem_1_1 states Theorem 1.1 itself and corollary_1_2 states Corollary 1.2, neither a weakened lemma, with a second independent route proved alongside each; no sorry or admit in the sources" } ], "registrations": [ { "registry": "vibemathed", "id": "asymptotically-attaining-the-moore-bound", "url": "https://vibemathed.com/problem/asymptotically-attaining-the-moore-bound", "title": "Asymptotically attaining the Moore bound" }, { "registry": "mathdb", "id": "316015", "title": "Degree diameter problem", "url": "https://mathdb.com/p/316015/degree-diameter-problem" } ], "sources": [ { "label": "arXiv: Asymptotically attaining the Moore bound", "url": "https://arxiv.org/abs/2608.03965", "kind": "research" }, { "label": "Lean 4 formalization (woutercvb/wewantmoore)", "url": "https://github.com/woutercvb/wewantmoore", "kind": "research" } ], "tags": [ "lean", "graph-theory", "combinatorics", "extremal-combinatorics" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-07-20-gaussian-product-inequality", "title": "Gaussian product inequality conjecture proved", "claim": "For any centered Gaussian vector and positive exponents, the expectation of the product of absolute powers is at least the product of the individual expectations, settling a conjecture posed in 2007.", "field": "mathematics", "date": "2026-07-20", "lab": "Independent", "model": "ChatGPT 5.6 Sol", "humans": [ "Frédéric Ouimet", "Dylan Greaves" ], "year_posed": 2007, "verification": "formal", "autonomy": "ai-led", "detail": "The conjecture states that for a centered Gaussian vector \\(X = (X_1, \\dots, X_n)\\), not necessarily nondegenerate, and any positive exponents \\(a_1, \\dots, a_n\\), the expectation of the product of the \\(|X_i|^{a_i}\\) is at least the product of the expectations of the \\(|X_i|^{a_i}\\). The stronger form adds that when every variance is positive, equality holds exactly when the coordinates are independent. The model returned a complete and correct solution from Greaves's prompt. It did not include the equality characterization, but only because that case was not in the prompt as posed. The proof was formalized in Lean using Codex, and the full prompt and output are public as a shared chat transcript.", "novelty_check": "The Gaussian product inequality is a named conjecture dating to 2007, with a substantial partial-results literature covering special cases such as even integer exponents and small dimensions; it was recorded as open in the general case. The proof settles the general statement. No prior general proof appears in the literature.", "caveats": "Not peer-reviewed; released as a preprint with a public chat transcript rather than through a journal. The Lean formalization is author-produced. The AI output omitted the equality-characterization half of the strong form, which was supplied around it. Checks by other models (ChatGPT 5.6 Sol Pro, Gemini 3.1 Pro, Grok 4.5) are not independent human verification and do not raise the grade; the human check by Ouimet is author-side.", "registrations": [ { "registry": "vibemathed", "id": "gaussian-product-inequality-conjecture", "url": "https://vibemathed.com/problem/gaussian-product-inequality-conjecture", "title": "Gaussian product inequality conjecture" }, { "registry": "mathdb", "id": "330065", "title": "The Gaussian Product Conjecture", "url": "https://mathdb.com/p/330065/the-gaussian-product-conjecture" } ], "sources": [ { "label": "A proof of the strong Gaussian product inequality conjecture", "url": "https://doi.org/10.13140/RG.2.2.17569.77923/1", "kind": "research" }, { "label": "Lean formalization (dylgre/gaussian-product-inequality)", "url": "https://github.com/dylgre/gaussian-product-inequality", "kind": "research" }, { "label": "Public chat transcript", "url": "https://chatgpt.com/share/6a5ea69b-1648-83e8-80b1-014ae0b1003c", "kind": "announcement" } ], "tags": [ "lean", "probability", "gaussian-inequalities" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-07-29-sumset-difference-exponent", "title": "Optimal exponent relating sumsets and difference sets determined", "claim": "The least universal exponent \\(c\\) with \\(|A+A|/|A|\\) bounded by \\((|A-A|/|A|)^c\\) for every finite integer set \\(A\\) is determined, settling the extremal question with a Lean 4 formalization.", "field": "mathematics", "date": "2026-07-29", "lab": "Tencent Hunyuan", "model": "Hy3 (Hyra research agent)", "humans": [ "Haowei Lin", "Shanda Li" ], "verification": "formal", "autonomy": "ai-led", "detail": "For a finite set \\(A\\) of integers with at least two elements, the ratio of \\(\\log(|A+A|/|A|)\\) to \\(\\log(|A-A|/|A|)\\) measures how far sumset growth can outrun difference-set growth. Determining its largest possible value fixes the least universal exponent \\(c\\) such that \\(|A+A|/|A| \\le (|A-A|/|A|)^c\\) for every such \\(A\\). Tencent Hunyuan's Hyra research agent, powered by the Hy3 model, explored and optimized finite-set constructions over roughly a 24-hour run, moving from finite numerical searches to natural-language proposals of general constructions and supporting arguments, and produced the construction underlying the paper. The human authors independently checked the construction, corrected and rewrote the exposition, and prepared the final proof manually.", "novelty_check": "The relationship between sumset and difference-set growth is a standard question in additive combinatorics with a long record of partial bounds on the exponent; the extremal value was recorded as undetermined. The paper states the prior bounds it improves on. The construction is new, not a retrieval of a known extremal family.", "caveats": "A newly released arXiv v1 preprint, not peer-reviewed. The Lean 4 and Mathlib formalization is author-provided; the repository reports lake build completing with no sorry declarations or warnings, but the principal asymptotic and supremum results rely on three native_decide certificates for elementary finite computations over a 12-element base-39 digit block, so those steps are trusted to the Lean kernel's decision procedure rather than proved term by term. Autonomy graded ai-led rather than autonomous: the agent produced the construction, but humans posed the problem, checked it and wrote the final proof.", "registrations": [ { "registry": "vibemathed", "id": "optimal-exponent-relating-sumsets-and-difference-sets", "url": "https://vibemathed.com/problem/optimal-exponent-relating-sumsets-and-difference-sets", "title": "Optimal Exponent Relating Sumsets and Difference Sets" }, { "registry": "mathdb", "id": "315713", "title": "Optimal exponent relating sumsets and difference sets", "url": "https://mathdb.com/p/315713/optimal-exponent-relating-sumsets-and-difference-sets" } ], "sources": [ { "label": "arXiv: Settling the Optimal Exponent Relating Sumsets and Difference Sets", "url": "https://arxiv.org/abs/2607.27199", "kind": "research" } ], "tags": [ "lean", "additive-combinatorics", "number-theory" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-07-28-kemeny-three-voters", "title": "Kemeny rank aggregation shown NP-hard for three voters", "claim": "Computing a Kemeny-optimal aggregate ranking is NP-hard when the input is exactly three complete rankings, closing the minimal open case left by hardness results for even voter counts.", "field": "computer-science", "date": "2026-07-28", "lab": "Independent", "model": "GPT-5.6 Sol Ultra, Claude Fable 5", "humans": [ "Dominik Peters" ], "year_posed": 2001, "verification": "formal", "autonomy": "ai-led", "detail": "Kemeny aggregation asks for the ranking minimizing total disagreement with a set of input rankings. Hardness was known for every even number of voters \\(n\\) at least 4, while \\(n = 2\\) is solvable in polynomial time, leaving three voters as the minimal open case since 2001. GPT-5.6 Sol Ultra found the reduction from MAX CUT; Claude Fable 5 helped simplify parts of it. Combined with the earlier results, every fixed number of voters \\(n\\) at least 3 is now known to be hard. The reduction is Lean-checked alongside an author-written preprint.", "novelty_check": "The complexity of Kemeny aggregation for three voters is a specifically tracked open case in the computational social choice literature, following Dwork, Kumar, Naor and Sivakumar's 2001 work establishing hardness for four voters and the subsequent extension to all even counts. The odd cases, and three in particular, were repeatedly noted as open. No prior hardness proof for exactly three voters appears. Péter Madarasi posted an independent proof of the same hardness result on 2026-07-30, two days after this one.", "caveats": "Not peer-reviewed. The Lean check covers the reduction; it is author-produced and had no independent audit at announcement. Autonomy graded ai-led rather than autonomous: the model found the reduction, but the human author posed the problem, reviewed the argument and wrote the paper.", "independent_checks": [ { "who": "Péter Madarasi (independent proof)", "url": "https://arxiv.org/abs/2607.28588", "outcome": "Proved independently, two days later, that Kemeny Score is NP-complete for exactly three unweighted rankings, even when every candidate pair is split 2-to-1, which corroborates the result by a separate argument." } ], "registrations": [ { "registry": "vibemathed", "id": "kemeny-three-voters", "url": "https://vibemathed.com/problem/kemeny-three-voters", "title": "Kemeny Rank Aggregation for Three Voters" }, { "registry": "mathdb", "id": "315674", "title": "Kemeny rank aggregation for three voters", "url": "https://mathdb.com/p/315674/kemeny-rank-aggregation-for-three-voters" } ], "sources": [ { "label": "arXiv: Kemeny Rank Aggregation is NP-Hard for Three Voters", "url": "https://arxiv.org/abs/2607.25540", "kind": "research" }, { "label": "arXiv: Madarasi, The complexity of Kemeny aggregation with three rankings", "url": "https://arxiv.org/abs/2607.28588", "kind": "commentary" } ], "tags": [ "lean", "complexity-theory", "computational-social-choice" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." }, { "date": "2026-09-24", "kind": "check", "note": "Recorded Péter Madarasi's independent proof that Kemeny aggregation is NP-complete for three rankings, posted two days after this result.", "url": "https://arxiv.org/abs/2607.28588" } ] }, { "id": "2026-07-14-sabidussi-compatibility", "title": "Sabidussi's compatibility conjecture proved", "claim": "The edges of a finite connected multigraph carrying a closed eulerian trail can be partitioned into circuits so that no circuit contains two edges used consecutively in the trail, with a Lean 4 formalization.", "field": "mathematics", "date": "2026-07-14", "lab": "Independent", "model": "GPT-5.6 Pro, GPT-5.6 Sol", "humans": [ "Nikolay Ulyanov" ], "verification": "formal", "autonomy": "collaborative", "detail": "Sabidussi's conjecture asks whether, given a finite connected multigraph together with a closed eulerian trail, the edge set can always be partitioned into circuits none of which contains two edges that the trail uses consecutively. The proof establishes this and in fact does more: it four-colours the edges so as to satisfy the constraints. The work was developed with GPT-5.6 Pro and GPT-5.6 Sol, with the author reviewing the proof, and a Lean 4 formalization accompanies the preprint in the author's repository.", "novelty_check": "Sabidussi's compatibility conjecture is a named open problem in graph theory concerning eulerian trails and compatible circuit decompositions, with a partial-results literature covering restricted degree conditions. It was recorded as open in the general case. No prior general proof appears.", "caveats": "Not peer-reviewed. The Lean formalization is in the author's own repository and had no independent audit at announcement. Autonomy graded collaborative rather than ai-led: the source describes the proof as developed with the models and reviewed by the author, without attributing the key idea specifically to the model, so the weaker defensible reading applies.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-08-17-000003", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-17-000003&version=1", "repository": "gexahedron/sabidussi-lean", "commit": "58307d74da99ec6e4fe28e012c30c83e369c54c3", "theorems": [ "sabidussi_compatibility_ordinary" ], "checked": "2026-08-17", "note": "Palomar's record does not restate the theorem in prose, so what the registration certifies is that formal statement rather than the preprint's phrasing of it." }, { "registry": "vibemathed", "id": "sabidussi-compatibility", "url": "https://vibemathed.com/problem/sabidussi-compatibility", "title": "Sabidussi's Compatibility Conjecture" }, { "registry": "mathdb", "id": "388102", "title": "Sabidussi's compatibility conjecture", "url": "https://mathdb.com/p/388102/sabidussi-s-compatibility-conjecture", "note": "A bare problem record: MathDB has no progress summary and no solution posted for it, so this places the problem in that catalogue rather than corroborating the resolution." } ], "sources": [ { "label": "arXiv: Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture", "url": "https://arxiv.org/abs/2607.13225", "kind": "research" }, { "label": "Lean 4 formalization (gexahedron/sabidussi-lean)", "url": "https://github.com/gexahedron/sabidussi-lean", "kind": "research" } ], "tags": [ "lean", "graph-theory", "combinatorics" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "check", "note": "Palomar registered the author's Lean development as PALOMAR-2026-08-17-000003, so the formalization now carries the mechanical audit it did not have at announcement. The repository was also added as a source: the entry had linked no artifact at all.", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-17-000003&version=1" }, { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." }, { "date": "2026-08-31", "kind": "correction", "note": "Recorded the MathDB problem record as a third registration, alongside Palomar and vibemathed. The page carries nothing about the resolution, so no grade moved." } ] }, { "id": "2026-06-02-jamming-exponent-identity", "title": "Identity for the critical exponents of jamming derived analytically", "claim": "The relation \\(a + b = 1\\) between critical exponents at the jamming transition, previously seen only numerically in the full replica-symmetry-breaking solution of hard spheres, is derived from the scaling equations and published after peer review.", "field": "physics", "date": "2026-06-02", "lab": "Independent", "model": "Claude Sonnet 4.6, Claude Opus 4.7", "humans": [ "Giorgio Parisi", "Francesco Zamponi" ], "year_posed": 2014, "verification": "peer-reviewed", "autonomy": "collaborative", "detail": "In the full replica-symmetry-breaking solution for hard spheres, the critical exponents \\(a\\) and \\(b\\) at the jamming transition had been observed numerically to high precision to satisfy \\(a + b = 1\\), without an analytic derivation from the scaling equations. Parisi and Zamponi asked Claude for help; the model quickly proposed what proved to be essentially the correct idea, combining integration-by-parts identities with a maximum principle. Its first formal write-up contained errors, which the authors then fixed and verified before publication in the Journal of Statistical Mechanics.", "novelty_check": "The numerical observation that \\(a + b = 1\\) in the FullRSB jamming solution dates to the 2014 work of Charbonneau, Kurchan, Parisi, Urbani and Zamponi on the exact mean-field theory of hard-sphere glasses, where the relation was reported as observed but unexplained. It remained without analytic derivation in the subsequent literature. The derivation is new.", "caveats": "Peer-reviewed and published, which is the appropriate grade for a physics result of this kind, but this is an analytic derivation within an established mean-field framework rather than an independently replicated calculation. Autonomy graded collaborative: the model supplied the key idea, but its first formal write-up contained errors that the human authors fixed and verified, and the authors framed the exchange as asking for help on a problem they posed.", "registrations": [ { "registry": "vibemathed", "id": "fullrsb-jamming-identity", "url": "https://vibemathed.com/problem/fullrsb-jamming-identity", "title": "FullRSB Jamming Identity a + b = 1" } ], "sources": [ { "label": "arXiv: A proof of an identity for the critical exponents of jamming", "url": "https://arxiv.org/abs/2606.03300", "kind": "research" } ], "tags": [ "statistical-physics", "jamming", "critical-exponents" ], "added": "2026-08-11", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. The vibemathed link it replaces pointed at that site's home page rather than at this result, and was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-07-10-cycle-double-cover", "title": "Cycle double cover conjecture proved for all bridgeless multigraphs", "claim": "Every finite bridgeless loopless multigraph has a multiset of cycles covering each edge exactly twice, resolving a conjecture open since 1973, with a machine-checked Lean proof of the unconditional statement.", "field": "mathematics", "date": "2026-07-10", "lab": "OpenAI", "model": "GPT-5.6 Sol Ultra", "year_posed": 1973, "verification": "formal", "autonomy": "ai-led", "detail": "The cycle double cover conjecture, posed by Szekeres in 1973 and independently by Seymour in 1979, asks whether every bridgeless graph admits a multiset of cycles in which every edge appears exactly twice. It is a textbook open problem in graph theory and sits at the centre of a cluster of neighbouring conjectures. OpenAI announced on 10 July 2026 that GPT-5.6 Sol Ultra, running 64 concurrent subagents, produced a three-page proof of the full statement in under an hour, and published the proof and the prompt, with a Lean 4 formalization following. The formalized route goes through the Jaeger–Kilpatrick eight-flow theorem: it constructs a nowhere-zero flow valued in three copies of the integers mod two, then converts that flow into a cycle double cover. Sang-il Oum published a ten-page exposition a week later, presenting the argument with slight modifications intended to make it more accessible.", "novelty_check": "The conjecture is recorded as open in the standard references, including Wolfram MathWorld, and has a documented history of proof claims posted to arXiv that were later withdrawn or found to have gaps, so the prior state was unambiguously unresolved. One thing complicates the picture rather than the novelty: Shiva Kintali posted an independent Codex-assisted proof claim of the same statement three days later (arXiv:2607.14140), also routed through nowhere-zero three-bit flows and likewise unreviewed. The result is a new proof, not a retrieval of a known one.", "caveats": "Not peer-reviewed. The conjecture's history of gapped proof claims is itself a reason for caution, and mathematicians are treating this one as unconfirmed pending expert review. Two graph theorists outside OpenAI have since written the proof up themselves, Sang-il Oum as an exposition with slight modifications and Jim Geelen as notes clarifying it to himself; neither reports a gap, and neither presents the write-up as a certification. The Lean development does check the unconditional statement for finite loopless bridgeless multigraphs, sorry-free and admit-free with no project-specific axioms, but it is first-party, produced by the same organisation as the proof, and had no independent audit at entry time. Autonomy graded ai-led rather than autonomous: the published prompt supplies more than the problem. It instructs the model to assume for purposes of the task that a complete affirmative proof exists, enumerates the approach families to explore including the flow formulations the proof in fact used, and specifies the 64-agent search strategy in detail.", "independent_checks": [ { "who": "Sang-il Oum (exposition)", "url": "https://arxiv.org/abs/2607.16356", "outcome": "Presented the proof with slight modifications intended to make it more accessible, later adding consequences for the small cycle double cover conjecture and for matroids. Reports no gap; does not claim to certify correctness." }, { "who": "Jim Geelen (notes)", "url": "https://arxiv.org/abs/2607.15399", "outcome": "Wrote three pages of notes clarifying the proof to himself, stating they offer no new insight. Reports no gap; does not claim to certify correctness." } ], "registrations": [ { "registry": "vibemathed", "id": "cycle-double-cover-conjecture", "url": "https://vibemathed.com/problem/cycle-double-cover-conjecture" }, { "registry": "mathdb", "id": "316019", "title": "Cycle double cover conjecture", "url": "https://mathdb.com/p/316019/cycle-double-cover-conjecture" } ], "sources": [ { "label": "OpenAI: A proof of the cycle double cover conjecture (PDF)", "url": "https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf", "kind": "research" }, { "label": "Lean 4 formalization (openai/cdc-lean)", "url": "https://github.com/openai/cdc-lean", "kind": "research" }, { "label": "OpenAI: the prompt used for the proof (PDF)", "url": "https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf", "kind": "announcement" }, { "label": "Ethan Knight announcement (X)", "url": "https://x.com/__eknight__/status/2075643450196971805", "kind": "announcement" }, { "label": "arXiv: A proof of the cycle double cover conjecture by OpenAI: An exposition", "url": "https://arxiv.org/abs/2607.16356", "kind": "commentary" }, { "label": "arXiv: Three-Bit Flows and Cycle Covers. Part I", "url": "https://arxiv.org/abs/2607.14140", "kind": "commentary" }, { "label": "Scientific American: ChatGPT just proved another 50-year-old math conjecture", "url": "https://www.scientificamerican.com/article/chatgpt-just-proved-another-50-year-old-math-conjecture/", "kind": "coverage" }, { "label": "arXiv: Geelen, OpenAI's proof of the Cycle Double Cover Theorem", "url": "https://arxiv.org/abs/2607.15399", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture", "url": "https://news.ycombinator.com/item?id=48863490" } ], "tags": [ "lean", "graph-theory", "combinatorics", "multi-agent" ], "added": "2026-08-14", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces was classified as commentary, which a registry listing is not." }, { "date": "2026-09-24", "kind": "check", "note": "Recorded independent write-ups of the proof by Sang-il Oum and Jim Geelen, neither reporting a gap. They are expositions rather than referee reports, so no grade moved.", "url": "https://arxiv.org/abs/2607.16356" } ] }, { "id": "2026-08-05-sendov-conjecture", "title": "Sendov's conjecture proved for every degree", "claim": "For every complex polynomial of degree at least two whose zeros all lie in the closed unit disk, each zero has a critical point of the polynomial within distance one, closing a question open since 1959.", "field": "mathematics", "date": "2026-08-05", "lab": "Independent", "model": "GPT-5.6 Pro", "humans": [ "Lech Mazur" ], "year_posed": 1959, "verification": "formal", "autonomy": "collaborative", "detail": "Sendov's conjecture states that if every zero of a complex polynomial of degree n at least two lies in the closed unit disk, then every zero has a critical point within distance one. Degrees up to eight were settled piecemeal between 1969 and 1999, and Tao proved the conjecture for all sufficiently large degree in 2020 without specifying the threshold, which left the middle range open. Lech Mazur, working with GPT-5.6 Pro, produced an argument covering every degree along with a Lean development of roughly 90,000 lines. Terence Tao then digested the proof, describing it as remarkably elementary with no complex analysis used beyond the fundamental theorem of algebra, and formalized the entire argument himself in about 15,000 lines. He reports that the same argument resolves the Phelps–Rodriguez conjecture in full generality.", "novelty_check": "Sendov's conjecture is a named 1959 problem with its own Wikipedia article in four languages and a partial-results literature spanning 67 years. Tao's 2020 paper on the sufficiently-high-degree case (arXiv:2012.04125) states the general case as open, which fixes the prior state of the art precisely. No earlier proof covering all degrees appears in the literature, and the Phelps–Rodriguez corollary is new with it. The result is a new proof, not a retrieval. On priority between the two formalizations: Palomar's record for Tao's development notes that it shares no code with Mazur's, was written from the blog-post digestion rather than from that proof, and that priority for the first machine-checked proof of Sendov's conjecture belongs to Mazur's ProofAtlas work.", "caveats": "Not peer-reviewed. Mazur's own Lean package cannot be recompiled as distributed: the published bundle ships no lakefile or manifest and excludes Mathlib, so the formal grade rests on Tao's independent formalization rather than on the original artifact. Autonomy graded collaborative rather than ai-led: the disclosure credits GPT-5.6 Pro with substantial contribution to the discovery and derivation of the proof, including exploration, proof development, exact computational testing and adversarial auditing, but it also has Mazur directing the research workflow and selecting and reconciling the model's outputs, which is mathematical judgment rather than cleanup. The weaker defensible reading applies.", "independent_checks": [ { "who": "Terence Tao", "url": "https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/", "outcome": "digested the argument and formalized the whole of it in Lean independently, at about 15,000 lines against the original's roughly 90,000, and states that it resolves both the Sendov and Phelps–Rodriguez conjectures in full generality" } ], "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-08-13-000001", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-13-000001&version=1", "repository": "teorth/sendov", "commit": "e356ef6d706cc8687369f3b21bb009587e88cbe9", "theorems": [ "SendovConjecture.sendov", "SendovConjecture.phelps_rodriguez" ], "checked": "2026-08-13", "note": "Permitted axioms are exactly propext, Quot.sound and Classical.choice, so the development adds none of its own. This is the artifact the formal grade already rested on, now checked by someone other than its author." }, { "registry": "vibemathed", "id": "sendov-s-conjecture", "url": "https://vibemathed.com/problem/sendov-s-conjecture" }, { "registry": "mathdb", "id": "315904", "title": "Sendov's conjecture", "url": "https://mathdb.com/p/315904/sendov-s-conjecture" }, { "registry": "proofatlas", "id": "sendov-conjecture", "url": "https://proofatlas.ai/formalizations/sendov-conjecture/", "title": "Sendov's conjecture", "note": "ProofAtlas records that the build passed with no unfinished proof steps, and that the record has not reached its accepted-result status, which needs four independent reviews. Its recorded build is not the same claim as the published bundle being rebuildable; this entry's caveat that the distributed package ships no lakefile or manifest still stands." } ], "sources": [ { "label": "ProofAtlas: A Computer-Assisted Proof of Sendov's Conjecture (PDF)", "url": "https://www.proofatlas.ai/papers/sendov-conjecture/SENDOV_CONJECTURE_PROOF_AUGUST_5_2026.pdf", "kind": "research" }, { "label": "Terence Tao: A digestion of the proof of Sendov's conjecture", "url": "https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/", "kind": "commentary" } ], "tags": [ "lean", "complex-analysis", "polynomials", "formalization" ], "added": "2026-08-14", "revisions": [ { "date": "2026-08-19", "kind": "check", "note": "Palomar registered Tao's Lean development as PALOMAR-2026-08-13-000001, mechanically confirming that it typechecks against both stated theorems at a pinned commit under the three standard axioms.", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-13-000001&version=1" }, { "date": "2026-08-19", "kind": "correction", "note": "Recorded priority for the first machine-checked proof, which belongs to Mazur's ProofAtlas formalization rather than to Tao's independent one. Neither grade moved." }, { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces was classified as commentary, which a registry listing is not." }, { "date": "2026-08-21", "kind": "correction", "note": "Recorded the ProofAtlas record for Mazur's formalization as a registration. It was previously carried as a research source, which a registry listing is not. The paper it accompanies stays a source, and no grade moved.", "url": "https://proofatlas.ai/formalizations/sendov-conjecture/" } ] }, { "id": "2026-08-05-schiffer-conjecture", "title": "Counterexamples to Schiffer's conjecture and the Pompeiu problem", "claim": "Infinitely many planar domains that are not balls admit a Neumann eigenfunction of the Laplacian that is constant on the boundary, disproving Schiffer's conjecture and, through a classical equivalence, the 1929 Pompeiu problem.", "field": "mathematics", "date": "2026-08-05", "lab": "Independent", "model": "GPT-5.6, Claude Opus 4.8, Claude Fable 5", "humans": [ "Gonzalo Cao-Labora", "Jaume de Dios Pont" ], "year_posed": 1929, "verification": "formal", "autonomy": "ai-assisted", "detail": "Pompeiu asked in 1929 whether a bounded domain over which some nonzero function integrates to zero under every rigid motion must be a ball. Schiffer's 1957 reformulation asks the same thing through Neumann eigenfunctions of the Laplacian that are constant on the boundary, and appears as Problem 80 on Yau's 1982 list; Williams proved the two formulations equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many N-fold symmetric planar domains, with N large, that are not balls and carry such an eigenfunction. The strategy is to relax the problem so that N may be any real number, which corresponds to the Schiffer problem only at integers, apply bifurcation theory there, and show the size of the local bifurcation branch can be taken independently of N. Their Corollary 1.2 applies Williams' equivalence to the same domains, so a single construction settles both problems.", "novelty_check": "Schiffer's conjecture is Problem 80 on Yau's list, with a partial-results literature running since the 1970s, and the Pompeiu problem has stood since 1929. Both are recorded as open in the standard references and in DeepMind's formal-conjectures repository, which carries a Lean statement of the Pompeiu problem as an unsolved challenge that this work's formalization closes. It is not the first counterexample. On 2026-08-03, two days before this paper, Matthew Colbrook and George Stepaniants posted an independent computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures, a single ten-fold symmetric domain; this paper acknowledges it and takes a different route, bifurcation theory, to infinitely many domains. The construction is new; the disproof is not the first.", "caveats": "A preprint, not peer-reviewed. The Lean 4 verification was written by GPT-5.6 from an early draft of the paper and lives in an author's own repository; it had no independent audit at entry time, and the same group produced both the proof and its formalization. Autonomy graded ai-assisted: the paper is explicit that the novel construction strategy is the authors' own, and that the models were used to verify the asymptotic estimates numerically, to produce first drafts of the proofs of the Bessel function estimates, and to help with exposition.", "independent_checks": [ { "who": "Matthew J. Colbrook and George Stepaniants (independent earlier counterexample)", "url": "https://arxiv.org/abs/2608.01579", "outcome": "Constructed, two days earlier and by a different method, a ten-fold symmetric planar domain with real-analytic boundary that is a counterexample to both Schiffer's conjecture and the planar Pompeiu conjecture, established by a computer-assisted contraction argument. Corroborates that planar counterexamples exist; does not check this paper's construction." } ], "registrations": [ { "registry": "vibemathed", "id": "schiffer-conjecture", "url": "https://vibemathed.com/problem/schiffer-conjecture" }, { "registry": "mathdb", "id": "315903", "title": "Pompeiu problem", "url": "https://mathdb.com/p/315903/pompeiu-problem" } ], "sources": [ { "label": "arXiv: Counterexamples to Schiffer's Conjecture", "url": "https://arxiv.org/abs/2608.05114", "kind": "research" }, { "label": "Lean 4 formalization (jaumededios/Schiffer)", "url": "https://github.com/jaumededios/Schiffer", "kind": "research" }, { "label": "Wikipedia: Pompeiu problem", "url": "https://en.wikipedia.org/wiki/Pompeiu_problem", "kind": "commentary" }, { "label": "arXiv: Colbrook and Stepaniants, A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures", "url": "https://arxiv.org/abs/2608.01579", "kind": "research" } ], "tags": [ "lean", "spectral-geometry", "counterexample", "analysis" ], "added": "2026-08-14", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. The vibemathed link it replaces was classified as commentary, which a registry listing is not." }, { "date": "2026-08-21", "kind": "correction", "note": "Recorded the MathDB record for the Pompeiu problem, the second of the two questions this entry settles, as a registration. No grade moved." }, { "date": "2026-09-24", "kind": "correction", "note": "Corrected the novelty check, which said no prior counterexample appears: Colbrook and Stepaniants posted an independent one two days earlier, which this paper itself acknowledges. No grade moved.", "url": "https://arxiv.org/abs/2608.01579" } ] }, { "id": "2026-08-08-petersen-coloring", "title": "A 112-vertex counterexample to the Petersen coloring conjecture", "claim": "An explicit simple bridgeless cubic graph on 112 vertices admits no Petersen coloring, and hence no normal 5-edge-coloring, refuting a conjecture of Jaeger from 1985 with machine-checked UNSAT certificates.", "field": "mathematics", "date": "2026-08-08", "lab": "Independent", "model": "Unnamed OpenAI model", "humans": [ "Bryce Putman" ], "year_posed": 1985, "verification": "formal", "autonomy": "ai-assisted", "detail": "Jaeger conjectured in 1985 that every bridgeless cubic graph admits a Petersen coloring: a map from its edges into the edges of the Petersen graph sending the three edges at each vertex to three edges meeting at a common vertex of the Petersen graph. Equivalently, every bridgeless cubic graph has a normal 5-edge-coloring. The conjecture implies both the Berge–Fulkerson conjecture and the 5-cycle-double-cover conjecture. This paper gives an explicit simple connected bridgeless cubic graph on 112 vertices, of girth five and edge and vertex connectivity three, with no such coloring; the graph is pinned by a SHA-256 digest in the theorem statement itself. Non-existence is established by direct SAT formulations, with CaDiCaL 3.0.1 returning UNSAT and drat-trim verifying the resulting DRAT proofs. A second, non-isomorphic counterexample of the same order is supplied. The implication runs one way, so Berge–Fulkerson and the 5-cycle-double-cover conjecture stay open; combined with a theorem of Ma, Mattiolo, Steffen and Wolf, one counterexample yields infinitely many.", "novelty_check": "The Petersen coloring conjecture is tracked as open in the Open Problem Garden, whose entry calls it an extraordinary conjecture, and it carries a substantial literature on normal edge-colorings and sublinear approximations built on the assumption that it holds. No prior counterexample appears anywhere in that literature. The construction is a new concrete graph, not a retrieval of an existing example.", "caveats": "A preprint, not peer-reviewed. The paper's computational provenance section states only that OpenAI language-model systems were used extensively in the discovery, computational search, verification and preparation of the work, and names no product, version or division of labour, so the model field records an unnamed OpenAI system and autonomy is graded ai-assisted, the weakest reading the disclosure permits. The paper does not claim 112 is the minimum order for a counterexample.", "independent_checks": [ { "who": "vibemathed.com curator reproduction", "url": "https://vibemathed.com/problem/petersen-coloring-conjecture", "outcome": "rebuilt the 112-vertex graph from the appendix edge table with a SHA-256 matching Theorem 1.1, rederived every property claimed there, and re-proved non-existence with an independently written CNF encoder solved by CaDiCaL rather than by replaying the shipped certificates; six control graphs including the Petersen graph itself came back satisfiable through the same encoder" } ], "registrations": [ { "registry": "vibemathed", "id": "petersen-coloring-conjecture", "url": "https://vibemathed.com/problem/petersen-coloring-conjecture" }, { "registry": "mathdb", "id": "316003", "title": "Petersen coloring conjecture", "url": "https://mathdb.com/p/316003/petersen-coloring-conjecture" } ], "sources": [ { "label": "arXiv: A 112-Vertex Counterexample to the Petersen Coloring Conjecture", "url": "https://arxiv.org/abs/2608.10012", "kind": "research" }, { "label": "Zenodo: reproducibility artifacts and checked DRAT certificates", "url": "https://doi.org/10.5281/zenodo.21845291", "kind": "research" }, { "label": "Open Problem Garden: Petersen coloring conjecture", "url": "https://www.openproblemgarden.org/op/petersen_coloring_conjecture", "kind": "commentary" } ], "tags": [ "graph-theory", "counterexample", "sat-solver", "combinatorics" ], "added": "2026-08-14", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-08-12-liquid-drop-minimizers", "title": "Complete minimizer picture for Gamow's liquid drop model", "claim": "Balls uniquely minimize the Gamow liquid drop energy at every volume up to a sharp threshold of about 3.51, and above that threshold no minimizer exists, closing a gap that partial results had narrowed from both sides without meeting.", "field": "mathematics", "date": "2026-08-12", "lab": "Independent", "model": "ChatGPT 5.6 Pro", "humans": [ "Otis Chodosh", "Matilde Gianocca" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "Gamow's liquid drop functional adds De Giorgi perimeter to a Coulomb repulsion term, and the central question in its literature is the fixed-volume minimization picture: for which volumes a minimizer exists, and whether it is the ball. This proves that balls uniquely minimize, up to translation and null sets, at every volume up to a threshold of \\(5(2 - 2^{2/3})/(2^{2/3} - 1)\\), roughly 3.51, and that above that threshold no minimizer exists at all. A corollary settles the minimal binding energy question of Frank and Lieb, identifying the infimum of energy per unit volume and the ball of volume 5/2 as the unique minimizer. The mechanism is a capacitary estimate sharpening an Agostiniani–Mazzieri monotonicity formula through Gauss–Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.", "novelty_check": "The sharp threshold is the tracked open problem of the liquid drop literature, followed across calculus of variations, mathematical physics and geometric analysis, with a 2017 Notices of the AMS survey and partial results carrying Lieb, Otto, Figalli and Maggi. The prior state of the art is stated in the paper and checks out: minimality of the ball was known only up to volume 1 (Chodosh–Ruohoniemi, CPAM 2025) and nonexistence only from volume 7.5 upward (Schulz, arXiv:2608.09000, posted two days earlier), leaving the interval from 1 to 7.5 open. Frank and Nam had proved existence up to the threshold in 2021, and the new proof uses that result rather than superseding it. Uniqueness across the whole range and nonexistence immediately above the threshold are new.", "caveats": "A preprint, not peer-reviewed, with no formalization. Verification is author-side: the authors state they checked and reworked the proof, which is not independent replication. Autonomy graded ai-led on the paper's own disclosure, which says the results were obtained by ChatGPT 5.6 Pro over a series of chats without significant assistance from the authors and that the fundamental strategy remains close to the model's original output; the authors checked the proof, reworked it and wrote the manuscript, and state the article contains no AI-written text.", "registrations": [ { "registry": "vibemathed", "id": "gamow-liquid-drop-minimizer-conjecture", "url": "https://vibemathed.com/problem/gamow-liquid-drop-minimizer-conjecture" }, { "registry": "mathdb", "id": "355211", "title": "Liquid-drop minimizer conjecture", "url": "https://mathdb.com/p/355211/liquid-drop-minimizer-conjecture" } ], "sources": [ { "label": "arXiv: No compromise in the liquid drop model", "url": "https://arxiv.org/abs/2608.11517", "kind": "research" }, { "label": "arXiv: An improved nonexistence bound for the liquid drop model", "url": "https://arxiv.org/abs/2608.09000", "kind": "commentary" } ], "tags": [ "calculus-of-variations", "geometric-analysis", "mathematical-physics" ], "added": "2026-08-14", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed and MathDB records for this result as registrations. The vibemathed link it replaces was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-08-13-sop2-sop3", "title": "SOP₂ and SOP₃ theories shown to coincide", "claim": "The classes of \\(\\mathrm{SOP}_2\\) and \\(\\mathrm{SOP}_3\\) first-order theories are the same, answering a 2004 question of Džamonja and Shelah and collapsing the bottom of the SOP hierarchy to a single class.", "field": "mathematics", "date": "2026-08-13", "lab": "Independent", "model": "ChatGPT 5.6", "humans": [ "Artem Chernikov" ], "year_posed": 2004, "verification": "author-verified", "autonomy": "collaborative", "detail": "Džamonja and Shelah introduced the tree configurations \\(\\mathrm{SOP}_1\\) and \\(\\mathrm{SOP}_2\\) in 2004 and asked whether either implication in the chain from \\(\\mathrm{SOP}_3\\) to \\(\\mathrm{SOP}_2\\) to \\(\\mathrm{SOP}_1\\) reverses. Mutchnik answered the second, proving \\(\\mathrm{SOP}_1\\) equals \\(\\mathrm{SOP}_2\\), and the surviving half organised much of the subsequent \\(\\mathrm{NSOP}_1\\) literature, with partial results by Conant, by Kaplan, Ramsey and Simon, and by Mutchnik. This five-page paper proves that \\(\\mathrm{SOP}_2\\) implies \\(\\mathrm{SOP}_3\\), the converse having been known from the start, which collapses the bottom of the hierarchy to a single class and carries consequences for Keisler-order maximality. The \\(\\mathrm{SOP}_n\\) hierarchy for \\(n\\) at least three is untouched, as is everything above it.", "novelty_check": "The question is one of the two 2004 Džamonja–Shelah questions that organised two decades of classification theory, and the paper's own introduction traces it as open through Mutchnik's celebrated resolution of the other half and the partial results that followed. It is repeatedly highlighted in the \\(\\mathrm{NSOP}_1\\) literature as the surviving open case. No prior proof appears. The result is new.", "caveats": "A preprint one day old at entry, not peer-reviewed and not formalized. The acknowledgements record comments from Itay Kaplan and Scott Mutchnik on a preliminary version, which is expert attention on a draft rather than independent verification, so the entry stays at author verified. Autonomy graded collaborative: the entire disclosure is one sentence, that the proof was found using ChatGPT 5.6 and simplified and streamlined by the author, which is ambiguous between the model supplying the key insight and the author using it as a tool. The weaker defensible reading applies.", "registrations": [ { "registry": "vibemathed", "id": "sop-2-sop-3", "url": "https://vibemathed.com/problem/sop-2-sop-3" } ], "sources": [ { "label": "arXiv: SOP₂ = SOP₃", "url": "https://arxiv.org/abs/2608.13291", "kind": "research" } ], "tags": [ "model-theory", "logic", "classification-theory" ], "added": "2026-08-14", "revisions": [ { "date": "2026-08-19", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. The vibemathed link it replaces was classified as commentary, which a registry listing is not." } ] }, { "id": "2026-04-23-synthecin", "title": "An antibiotic designed by reinforcement learning clears an MRSA infection in mice", "claim": "A reinforcement-learning generator searching a 46-billion-compound synthesizable space produced synthecin, a structurally novel antibacterial that cleared a methicillin-resistant Staphylococcus aureus wound infection in a mouse model.", "field": "medicine", "date": "2026-04-23", "lab": "McMaster University / Stanford University", "model": "SyntheMol-RL", "humans": [ "Kyle Swanson", "Gary Liu", "Denise B. Catacutan", "Eric D. Brown", "James Zou", "Jonathan M. Stokes" ], "verification": "peer-reviewed", "autonomy": "search-scaffold", "detail": "SyntheMol-RL replaces the Monte Carlo tree search of the earlier SyntheMol with reinforcement learning over roughly 150,000 commercial building blocks and about 50 reaction templates, which lets it generalize across chemically similar blocks and optimize several properties at once. Run against S. aureus with joint objectives of antibacterial activity and aqueous solubility, it proposed candidates that the team then made: 79 compounds unique relative to the training set were synthesized, 13 were potently active in vitro, and 7 of those passed the authors' structural-novelty filters against known antibiotics. One hit, synthecin, was formulated as a topical and cleared a murine MRSA wound infection. The design step is a search harness with a learned policy, not a language model reasoning about chemistry.", "novelty_check": "The compound is new by construction: the generator is restricted to a combinatorial space of unsynthesized products, and the paper reports the 79 tested molecules as unique relative to the training set, with structural-novelty filters applied against known antibiotics leaving seven. The prior art in this registry is the 2020 halicin screen and the 2023 MRSA structural class, both of which selected from existing libraries; the distinguishing claim here is generation of a molecule nobody had made, followed by synthesis and an in vivo test. Searched PubMed and Europe PMC for synthecin: the only records are this paper and its 2025 bioRxiv preprint. The framework itself is an increment on the authors' own SyntheMol, which they state.", "caveats": "Preclinical. One compound, one pathogen, one mouse wound model, and topical administration; the paper describes systemic use as a possibility to be optimized toward, not a result. Structural novelty is measured by the authors' own filters rather than by an outside assessment. The 13 active compounds out of 79 synthesized is a hit rate on a set the model itself chose, so it is not an unbiased estimate of the generator's precision. Autonomy graded search-scaffold rather than ai-led: humans set the objectives, the scoring functions and the building-block space, and selected which candidates to synthesize, which is the same shape as the halicin and MRSA entries.", "sources": [ { "label": "Molecular Systems Biology: SyntheMol-RL", "url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC13230741/", "kind": "research" }, { "label": "bioRxiv preprint (2025)", "url": "https://www.biorxiv.org/content/10.1101/2025.05.17.654017v1", "kind": "research" }, { "label": "McMaster University announcement", "url": "https://news.mcmaster.ca/mcmaster-built-ai-model-speeds-up-drug-discovery-designs-new-antibiotic/", "kind": "announcement" } ], "tags": [ "antibiotics", "drug-discovery", "reinforcement-learning", "generative-model", "mrsa", "in-vivo" ], "added": "2026-08-21" }, { "id": "2026-08-13-banach-isometric", "title": "Banach's isometric conjecture settled in the remaining odd dimensions", "claim": "A real Banach space whose n-dimensional subspaces are all isometric to one another must be a Hilbert space for every odd n, which together with Gromov's even-dimensional theorem closes a question Banach asked in 1932.", "field": "mathematics", "date": "2026-08-13", "lab": "Independent", "model": "ChatGPT 5.5 Pro, ChatGPT 5.6 Pro and GPT-5.6 Sol", "humans": [ "Xinbao Lu", "Kaiwen Yang" ], "year_posed": 1932, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "Banach asked in 1932 whether a real Banach space X all of whose n-dimensional subspaces are isometric, for some fixed n strictly between 1 and the dimension of X, has to be a Hilbert space. Gromov settled every even n, and later work closed several odd cases, leaving a gap. This 21-page paper proves the remaining odd n, combining bundle topology with Brouwer degree theory, which completes the conjecture in the real case. The complex case is not addressed.", "novelty_check": "The conjecture is a named 1932 problem of Banach with a documented partial-results ladder: Gromov's theorem covers even n, and subsequent work settled particular odd dimensions, which is the prior state of the art the paper states and cites. Searched arXiv and MathSciNet for a proof covering all odd n; none appears before this preprint, and the surveys of the problem still list the general odd case as open. The result is a new proof rather than a retrieval.", "caveats": "A preprint, unrefereed and not formalized, and only days old at entry. Autonomy graded ai-assisted rather than collaborative: the declaration states that the authors had already reduced the main problem to proving Theorem 3.10 before using generative AI, that an approach to that theorem then emerged through extensive interactions with ChatGPT 5.5 Pro and 5.6 Pro, that GPT-5.6 Sol generated the initial draft of Section 3 and corresponding material in Section 2, and that the authors checked and rewrote it and take full responsibility. The reduction, which is the framing step, is human, and the machine contribution is one theorem inside it. The weaker defensible reading applies.", "registrations": [ { "registry": "vibemathed", "id": "banach-s-isometric-conjecture", "url": "https://vibemathed.com/problem/banach-s-isometric-conjecture" }, { "registry": "mathdb", "id": "383555", "title": "Banach's isometric conjecture", "url": "https://mathdb.com/p/383555/banach-s-isometric-conjecture" } ], "sources": [ { "label": "arXiv: A solution to Banach's isometric conjecture", "url": "https://arxiv.org/abs/2608.13536", "kind": "research" } ], "tags": [ "functional-analysis", "banach-spaces", "argument", "geometry" ], "added": "2026-08-21" }, { "id": "2026-08-16-talagrand-convolution", "title": "Talagrand's convolution conjecture proved on the Boolean hypercube", "claim": "Convolution by a measure on the Boolean hypercube obeys the dimension-free weak-type decay Talagrand conjectured in 1989, removing the iterated-logarithm factor carried by the best previous bound.", "field": "mathematics", "date": "2026-08-16", "lab": "Independent", "model": "Odin Automatic AI Research Agent", "humans": [ "Junwei Lu", "Shengtao Guo", "Ethan X. Fang" ], "year_posed": 1989, "verification": "author-verified", "autonomy": "ai-led", "detail": "For the Boolean hypercube with uniform measure and convolution by a finite positive measure, Talagrand conjectured that the anti-concentration profile decays like \\(1/\\sqrt{\\log u}\\), with a constant free of the dimension. Yuansi Chen reached that decay in November 2025 up to an additional \\(\\log\\log\\) factor, using a perturbed reverse heat flow. This paper reworks the endpoint comparison so that the exponent has a fixed numerator, which removes the need for a growing stopping buffer and with it the \\(\\log\\log\\) factor, giving the conjectured bound.", "novelty_check": "Talagrand posed the hypercube conjecture in 1989 and it remained on O'Donnell's open-problem list; even the Gaussian special case was open as of 2012. The prior state of the art is fixed precisely by Chen, arXiv:2511.19374, November 2025, which proves the bound up to a \\(\\log\\log\\) correction and says so in its title. This paper's stated contribution is the removal of exactly that factor, and the mechanism it credits, a fixed-numerator frozen exponent that dispenses with the growing stopping buffer, is specific to the gap. The result is one step in a chain rather than a proof from scratch, and it is new.", "caveats": "A preprint days old at entry, unrefereed and not formalized. Alexander Shaposhnikov has since given a self-contained martingale proof of the Boolean case with explicit dimension-free constants, which corroborates the theorem by a different method but is not a check of this paper's argument. The AI disclosure is two sentences, in the abstract and again under a heading 'The role of AI in this proof': the Odin Automatic AI Research Agent was used to discover the proof, and the final proofs were reorganized by the authors. Autonomy graded ai-led on that statement taken at face value, but the claim cannot be audited: Odin is not a publicly documented system, no vendor is named, no transcript or artifact is published, and there is no way to separate discovery from reorganization. Read the model field as the authors' attribution rather than as a verified fact.", "independent_checks": [ { "who": "Alexander Shaposhnikov (independent second proof)", "url": "https://arxiv.org/abs/2609.11290", "outcome": "Gave self-contained martingale proofs of the convolution conjecture in the Gaussian and Boolean settings with explicit dimension-free constants, by a stopped density martingale rather than the reverse-heat approach, and credits Lu, Guo and Fang with settling the Boolean case." } ], "registrations": [ { "registry": "vibemathed", "id": "talagrand-s-convolution-conjecture", "url": "https://vibemathed.com/problem/talagrand-s-convolution-conjecture" }, { "registry": "mathdb", "id": "371652", "title": "Talagrand's convolution conjecture for the Boolean hypercube", "url": "https://mathdb.com/p/371652/talagrand-s-convolution-conjecture-for-the-boolean-hypercube" } ], "sources": [ { "label": "arXiv: Weak-Type Bounds for Convolution on the Boolean Hypercube", "url": "https://arxiv.org/abs/2608.15515", "kind": "research" }, { "label": "arXiv: Chen, Talagrand's convolution conjecture up to loglog (prior bound)", "url": "https://arxiv.org/abs/2511.19374", "kind": "research" }, { "label": "arXiv: Shaposhnikov, Some remarks on Talagrand's convolution conjecture", "url": "https://arxiv.org/abs/2609.11290", "kind": "commentary" } ], "tags": [ "probability", "boolean-analysis", "concentration", "argument" ], "added": "2026-08-21", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded Alexander Shaposhnikov's independent martingale proof of the Boolean case. It confirms the theorem by another route rather than checking this proof, so no grade moved.", "url": "https://arxiv.org/abs/2609.11290" } ] }, { "id": "2026-08-17-matmul-exponent", "title": "Matrix multiplication exponent lowered to below 2.371177", "claim": "A reformulated optimization inside the laser method, with AlphaEvolve applied as the final refinement, lowers the best known upper bound on the matrix multiplication exponent from 2.371339 to 2.371177.", "field": "computer-science", "date": "2026-08-17", "lab": "Google DeepMind", "model": "AlphaEvolve", "humans": [ "Emilien Dupont", "Marvin Eisenberger", "Borislav Kozlovskii", "Abbas Mehrabian", "Francisco J. R. Ruiz", "Abigail See", "Renfei Zhou", "Josh Alman", "Virginia Vassilevska Williams", "Matej Balog" ], "year_posed": 1969, "verification": "author-verified", "autonomy": "search-scaffold", "detail": "The best bounds on the matrix multiplication exponent come from combination loss analysis, a refinement of the laser method. The paper makes three changes to the optimization at its core, and AlphaEvolve is only the third: first the problem is reformulated so it can be solved over a larger space than was previously tractable, then a new optimization algorithm is designed using recent machine-learning methods, and then that algorithm is refined with AlphaEvolve. The combination gives an upper bound of 2.371177 against a previous best of 2.371339. The paper is a note reporting a record, not a structural advance: whether the exponent equals 2 is untouched, and nothing here suggests the laser method can reach it.", "novelty_check": "The exponent has a well-tracked ladder of published upper bounds running through Duan, Wu and Zhou (2022), Vassilevska Williams, Xu, Xu and Zhou (2024) and Alman, Duan, Vassilevska Williams, Xu, Xu and Zhou (2025), and the paper states the prior best as 2.371339 and cites it. No lower published upper bound appears before this note. The result is a new bound rather than a retrieval, and it is an improvement of roughly 1.6 times ten to the minus four, which the authors themselves describe as a small step.", "caveats": "A preprint, unrefereed at entry. A bound of this kind is not the sort of claim a reader can spot-check: the number falls out of a large optimization over laser-method parameters, so reproducing it means re-running the optimization rather than checking a certificate, and no such independent rerun is recorded. Autonomy graded search-scaffold, consistent with the other AlphaEvolve entries here: AlphaEvolve is a human-built evolutionary harness with a language model inside, and in this paper it is the last stage of a pipeline whose first two stages are ordinary human applied mathematics. The author list mixes the DeepMind team with the complexity theorists who set the previous record, which is where the reformulation comes from.", "sources": [ { "label": "arXiv: Improving the matrix multiplication exponent with modern optimization and AlphaEvolve", "url": "https://arxiv.org/abs/2608.16884", "kind": "research" } ], "tags": [ "alphaevolve", "algorithms", "matrix-multiplication", "complexity", "computation" ], "added": "2026-08-21", "registrations": [ { "registry": "vibemathed", "id": "matrix-multiplication-exponent-2371177", "url": "https://vibemathed.com/problem/matrix-multiplication-exponent-2371177" }, { "registry": "mathdb", "id": "388199", "title": "Is the exponent of matrix multiplication 2?", "url": "https://mathdb.com/p/388199/is-the-exponent-of-matrix-multiplication-2", "note": "Bare problem page for the open question of whether the exponent is 2. It records the problem's standing, not this result: nothing is posted against it." } ], "revisions": [ { "date": "2026-08-31", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. It carries the same bound and the same AlphaEvolve attribution this entry does. No grade moved." } ] }, { "id": "2026-08-18-prime-gaps-246", "title": "Bounded prime gaps of 246 formalized in Lean from Bombieri-Vinogradov", "claim": "A machine-checked Lean development proves that infinitely many pairs of primes differ by at most 246, taking the Bombieri-Vinogradov theorem as a hypothesis rather than proving it.", "field": "mathematics", "date": "2026-08-18", "lab": "Axiom Math", "model": "AxiomProver", "humans": [ "Evan Chen", "Sidharth Hariharan", "Kenny Lau", "Bhavik Mehta", "Ken Ono", "Ashvin Swaminathan", "Jesse Thorner", "Yunzhou Xie" ], "verification": "formal", "autonomy": "collaborative", "detail": "The 246 bound is the closest mathematics has come to the twin prime conjecture: Zhang broke the bounded-gaps barrier in 2013, Maynard's multidimensional sieve brought it to 600, and the Polymath8b collaboration tightened that to 246 in 2014. This project formalizes the majority of Maynard's paper together with the Polymath8b extension, in Lean 4 on top of Mathlib and the PrimeNumberTheoremAnd development led by Kontorovich and Tao. AxiomProver generated machine-checkable proofs which human formalisers at Axiom Math then curated and librarized into PrimeGapsLib, released as reusable infrastructure. The mathematics is not new; the machine-checked proof of it is.", "novelty_check": "The theorem is not new and the project does not claim it is. H at most 246 is Polymath8b, 2014, building on Maynard 2013 and Zhang 2013, and all three are cited by the project as the sources being formalized. What did not previously exist is a machine-checked proof: no formalization of the Maynard sieve or the Polymath8b bound appears in Mathlib or in the Lean community's project index before this, and the related PrimeNumberTheoremAnd project it builds on formalizes a different theorem. This entry records a formalization milestone, in the same sense as the strong prime number theorem entry, not a new mathematical result.", "caveats": "The formalization is not self-contained. Palomar registered the development under the theorem names bombieriVinogradov_implies_prime_gap_le_246 and bombieriVinogradov_implies_nth_prime_gap_le_246, and Palomar's own record describes the flagship result as conditional on the Bombieri-Vinogradov theorem, which is assumed as a hypothesis because it is not yet in Mathlib. Bombieri-Vinogradov has been a theorem since 1965, so the mathematics is unconditional in the literature and only the Lean development is conditional, but the project's public page states the 246 result without that qualification, and the two readings should not be conflated. Beyond the mechanical Palomar check there has been no mathematical peer review of the formalization choices. Autonomy graded collaborative: the disclosure is that the proof was generated collaboratively between human formalisers and AxiomProver, with humans curating and librarizing, and roughly 41 named contributors are credited.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-08-18-000002", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-18-000002&version=1", "repository": "AxiomMath/PrimeGapsLib", "commit": "1faa7b14e82ddebc2772dfb9153922f01b106477", "theorems": [ "bombieriVinogradov_implies_prime_gap_le_246", "bombieriVinogradov_implies_nth_prime_gap_le_246" ], "checked": "2026-08-18", "note": "Registered the day Palomar opened for submissions. The theorem names are where the Bombieri-Vinogradov hypothesis is visible; the project's own page does not carry that qualification." } ], "sources": [ { "label": "Axiom Math: Lean formalization of bounded gaps between primes", "url": "https://primegaps.axiommath.ai/", "kind": "research" }, { "label": "PrimeGapsLib (Lean 4 source)", "url": "https://github.com/AxiomMath/PrimeGapsLib", "kind": "research" }, { "label": "Axiom announcement (X)", "url": "https://x.com/axiommathai/status/2089732764279132449", "kind": "announcement" }, { "label": "IEEE Spectrum coverage", "url": "https://spectrum.ieee.org/axiom-math-246-theorem-formalization", "kind": "coverage" } ], "tags": [ "lean", "formalization", "number-theory", "prime-gaps", "sieve-methods" ], "added": "2026-08-21" }, { "id": "2026-08-19-yau-tian-donaldson", "title": "Counterexample to the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics", "claim": "A smooth polarized fivefold that is K-polystable but admits no constant scalar curvature Kähler metric, disproving the form of the Yau–Tian–Donaldson conjecture that Donaldson stated in 2002.", "field": "mathematics", "date": "2026-08-19", "lab": "Independent", "model": "Claude Fable 5, GPT-5.6-sol and Danus", "humans": [ "Jihao Liu", "Bin Dong", "Guoxiong Gao" ], "year_posed": 2002, "verification": "author-verified", "autonomy": "collaborative", "detail": "Donaldson conjectured in 2002 that a smooth polarized variety carries a constant scalar curvature Kähler metric exactly when it is K-polystable, the general-polarization successor to the Kähler-Einstein expectation going back to Yau and to Tian's K-stability. This 79-page paper constructs a polarized smooth projective fivefold, proves it K-polystable against all normal ample algebraic test configurations of every positive exponent, and shows it carries no such metric. Its appendix draws a distinction worth keeping: some counterexamples reduce to a finite certificate checkable once the object is written down, as the Jacobian conjecture counterexample does, while others are themselves theorems quantified over everything, and this is the second kind. Candidate manifolds of this shape have been available since 2008; the mathematical content is the classification showing every test configuration has nonnegative Donaldson-Futaki invariant with equality only for products.", "novelty_check": "The conjecture is named, celebrated and traceable: the paper states it as Donaldson's 2002 conjecture, records the normality repair of the original scheme-theoretic formulation due to Ross and Thomas, and cites the Matsushima and Futaki obstructions and the Chen-Cheng and Berman-Darvas-Sjöström Dinew lines of work as the surrounding literature. No prior counterexample in any dimension appears. The paper is explicit about what was and was not already available: candidate manifolds of the present shape date to 2008, and what was missing is the proof that the mechanism works. The result is new.", "caveats": "A preprint days old at entry, 79 pages, unrefereed and not formalized, and a counterexample of the kind that cannot be checked by evaluating an object, so verification means reading the classification argument. The theorem actually proved is the extremal-metric statement, which is stronger than the cscK wording of the abstract. Autonomy graded collaborative rather than ai-led: the appendix says the author found initial signs of a breakthrough while exploring open problems with Claude Code and then had Claude Code, Codex and Danus work on it in collaboration, with the three systems together producing the counterexample and its proof, but it also records that human input was crucial at one point, when the author recognized the example had to refute either the Codogni-Stoppa conjecture or cscK Yau–Tian–Donaldson and directed the agents to settle which. The appendix further reports that an improved version of Danus, given only the original problem and none of the earlier findings, later settled it alone in 5 hours and 29 minutes without that human input, which would be autonomous; that system is not public and the report is by the same author, so it is not gradeable evidence. The weaker defensible reading applies.", "registrations": [ { "registry": "mathdb", "id": "383718", "title": "Yau–Tian–Donaldson conjecture for constant scalar curvature metrics", "url": "https://mathdb.com/p/383718/yau-tian-donaldson-conjecture-for-constant-scalar-curvature", "note": "MathDB carries three records for this conjecture under permuted name orders (321390, 331106, 383718). This is the one whose title matches the usual attribution order." } ], "sources": [ { "label": "arXiv: Disproof of the Yau–Tian–Donaldson conjecture", "url": "https://arxiv.org/abs/2608.19301", "kind": "research" } ], "tags": [ "algebraic-geometry", "differential-geometry", "k-stability", "counterexample", "construction" ], "added": "2026-08-21", "revisions": [ { "date": "2026-09-07", "kind": "correction", "note": "Removed the vibemathed registration: the cited record now returns 404, and the result appears under no other slug in vibemathed's dataset. The arXiv preprint and the MathDB record are unaffected, and no grade moved." } ] }, { "id": "2026-08-20-marton-inner-bound", "title": "Marton's inner bound shown not to reach the broadcast channel capacity region", "claim": "There are discrete memoryless broadcast channels whose capacity region is strictly larger than Marton's inner bound, settling in the negative a question open since 1979.", "field": "computer-science", "date": "2026-08-20", "lab": "Independent", "model": "GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5", "humans": [ "Mian Huang", "Yanxiao Liu", "Yi Liu" ], "year_posed": 1979, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "Marton's inner bound has been the best known achievable region for the general two-receiver discrete memoryless broadcast channel since 1979, and whether it is tight has been the central open question of the area. The paper exhibits channels whose capacity strictly exceeds it. The route ran through a related conjecture: the authors first found counterexamples to the Markovity conjecture of Gohari, Liu and Nair, then built a local tensorization test for a ternary-component broadcast channel, developed that into a counterexample to the additivity conjecture, and finally into an unconstrained counterexample to Marton's region. The violation gaps are small, of order ten to the minus six, and are certified with interval arithmetic.", "novelty_check": "Marton posed the inner bound in 1979 and its tightness is the standing open problem of broadcast-channel capacity, restated in the network information theory literature and in the Gohari-Liu-Nair line of work the paper builds on and partly refutes. The paper records that the Markovity conjecture had survived an experiment over more than ten thousand random channels in earlier work, which fixes the prior state of the art: no counterexample to either conjecture had been found. No prior demonstration of sub-optimality appears. The result is a new construction rather than a retrieval.", "caveats": "A preprint one day old at entry, unrefereed and not formalized. The gaps are of order ten to the minus six, so the claim rests on the authors' interval-arithmetic certification rather than on anything visible by inspection, and no independent rerun is recorded. Autonomy graded ai-assisted rather than collaborative or ai-led: the acknowledgement credits each contribution individually and the recurring verb is assisted. Chandra Nair asked Yanxiao Liu to search using AI; GPT-5.6 Sol assisted Liu to a Markovity counterexample and, independently, assisted Mian Huang via the human-devised method of elimination geometry; Claude Fable 5 assisted the tensorization test; Claude Fable 5 and Opus 5 assisted a fixed-input-distribution example after Amin Gohari and Nair suggested that constraint; and Yanxiao Liu then modified that example into the unconstrained counterexample. Humans chose the method, set each search target and did the final construction. The weaker defensible reading applies.", "registrations": [ { "registry": "vibemathed", "id": "marton-inner-bound-capacity-region", "url": "https://vibemathed.com/problem/marton-inner-bound-capacity-region" }, { "registry": "mathdb", "id": "390544", "title": "Marton's conjecture on broadcast-channel capacity", "url": "https://mathdb.com/p/390544/marton-s-conjecture-on-broadcast-channel-capacity", "note": "MathDB records the same three models and describes the result as remaining unverified as a theorem, which matches the caveat this entry already carries." } ], "sources": [ { "label": "arXiv: Sub-optimality of Marton's Inner Bound for the Two-Receiver Broadcast Channel", "url": "https://arxiv.org/abs/2608.19869", "kind": "research" } ], "tags": [ "information-theory", "broadcast-channel", "counterexample", "construction" ], "added": "2026-08-21", "revisions": [ { "date": "2026-08-31", "kind": "correction", "note": "Recorded the MathDB record as a second registration, alongside vibemathed. Its own status text agrees with this entry's author-verified grade and its caveat, so no grade moved." } ] }, { "id": "2026-08-23-s6-complex-structure", "title": "A complex structure on the six-sphere", "claim": "An explicit compact complex threefold, built as a family of 2-tori over the \\((3, 4, \\infty)\\) orbifold, is diffeomorphic to the six-sphere, which settles the question Hopf raised in 1948 of whether the six-sphere admits a complex structure.", "field": "mathematics", "date": "2026-08-23", "lab": "Anthropic", "model": "Claude, version not stated", "humans": [ "Levent Alpöge" ], "year_posed": 1948, "verification": "independent", "autonomy": "ai-assisted", "detail": "Whether the six-sphere admits a complex structure is the surviving case of a question going back to Hopf: the two-sphere and the six-sphere are the only spheres carrying an almost complex structure, and only the six-dimensional case is open. The manuscript builds a threefold \\(X\\) fibred over the projective line by complex 2-tori, using the period functions of the \\((3, 4, \\infty)\\) triangle group, degenerating to a degree-six del Pezzo fibre at one special point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. It argues that \\(X\\) is simply connected with the integral homology of the six-sphere, hence diffeomorphic to it, and that its algebraic dimension is exactly 1. That last number is where the paper collides with the literature, and it says so itself rather than passing over it.", "novelty_check": "The problem is named, ancient and heavily attempted: Hopf posed it in 1948, and the modern literature runs through Chern, Bryant, LeBrun and the Campana-Demailly-Peternell papers the manuscript cites and disputes. Its 60-item bibliography is real and checkable. The specific construction, a family of 2-tori over the \\((3, 4, \\infty)\\) orbifold completed at its three special points, does not appear in the prior literature, and no accepted complex structure on the six-sphere exists. Atiyah's 2016 attempt is the best-known recent failure and is not this argument. The construction is new. Its correctness is what the independent checks recorded here address.", "caveats": "Entered at claimed and upgraded to independently checked on 2026-09-24, on two checks by people outside Anthropic who used different means. Philip Engel wrote a self-contained proof of the construction by a geometric route and states that he verified the mathematical content independently. Boris Alexeev formalized the full statement in Lean against a statement adapted from Formal Conjectures, which now lists the Hopf problem as solved and links his proof. It is not graded formal because no third-party rebuild of that 248,818-line development is on record, and Mohammed Abouzaid, while saying the formalization is making the community confident, called the original manuscript so poorly written that checking it will take a while. Not peer-reviewed. The AI attribution still rests on the author's post on X: the 108-page manuscript was read here in full and has no acknowledgements section, no methods note and no mention of Claude, Anthropic or any paraphrase, and Engel records that the precise nature of the interaction between Alpöge and Claude is not documented. Autonomy is graded ai-assisted because that is all the disclosure can carry, not because the contribution is known to be small. The model version is not stated. The result contradicts published work: the paper carries a section on why the argument of the Campana-Demailly-Peternell corrigendum does not apply, and a remark that its theorem contradicts that paper's Corollary 2.3 as published, locating the divergence in a non-vanishing higher direct image and in a lemma that assumes trivial monodromy where here the monodromy is non-trivial. Engel's write-up flags the same collision with the 1998 result that such a structure must have algebraic dimension zero. With two independent checks on record, the burden has shifted to that earlier literature.", "independent_checks": [ { "who": "Philip Engel, University of Illinois Chicago", "url": "https://philip-engel.github.io/S6.pdf", "outcome": "Wrote a self-contained proof of the construction by a different, geometric route (a rational elliptic surface, logarithmic transforms, then van Kampen and Leray computations giving a simply connected integral homology six-sphere) and states that he verified the mathematical content independently. He discloses using ChatGPT to explore the original and draft arguments." }, { "who": "Boris Alexeev (OpenAI), Lean formalization", "url": "https://github.com/plby/HopfProblem", "outcome": "Formalized that the unit six-sphere carries a complex manifold structure compatible with its standard topology, against a statement adapted from Formal Conjectures (Mathoverflow 1973). At commit 9ac8a456 the proof file has no sorry and no axiom declarations and the comparator permits only the three standard axioms; Formal Conjectures now marks the problem solved and links this proof. Counted here from a clone, not rebuilt." } ], "registrations": [ { "registry": "vibemathed", "id": "modular-family-of-2-tori-as-a-complex-structure-on-s6", "url": "https://vibemathed.com/problem/modular-family-of-2-tori-as-a-complex-structure-on-s6" } ], "sources": [ { "label": "Alpöge: The (3,4,infinity) modular family of 2-tori is a complex structure on S⁶ (PDF)", "url": "https://alpo.ge/s6.pdf", "kind": "research" }, { "label": "Coverage: officechai", "url": "https://officechai.com/ai/anthropic-researcher-says-claude-helped-build-a-complex-structure-on-s%E2%81%B6-taking-aim-at-the-unsolved-hopf-problem/", "kind": "coverage" }, { "label": "Levent Alpöge, announcement (X)", "url": "https://x.com/__alpoge__/status/2091639597193368014", "kind": "announcement" }, { "label": "GitHub: plby/HopfProblem, Boris Alexeev's Lean formalization", "url": "https://github.com/plby/HopfProblem", "kind": "research" }, { "label": "Philip Engel: Complex structures on S6, a self-contained proof (PDF)", "url": "https://philip-engel.github.io/S6.pdf", "kind": "commentary" }, { "label": "Quanta Magazine: update on the six-sphere, with Engel and Abouzaid on checking it", "url": "https://www.quantamagazine.org/updates/transformation/#167201", "kind": "coverage" }, { "label": "Scientific American: AI solves 79-year-old math mystery of six-dimensional spheres", "url": "https://www.scientificamerican.com/article/ai-solves-79-year-old-math-mystery-of-six-dimensional-spheres/", "kind": "coverage" } ], "discussion": [ { "label": "Hacker News: A complex structure on S⁶", "url": "https://news.ycombinator.com/item?id=49412947" }, { "label": "Hacker News: Formalization of the Solution to the Hopf Problem", "url": "https://news.ycombinator.com/item?id=49467053" } ], "tags": [ "complex-geometry", "differential-geometry", "hopf-problem", "construction", "contested" ], "added": "2026-08-25", "revisions": [ { "date": "2026-09-24", "kind": "regraded", "note": "Upgraded from claimed to independently checked: Philip Engel published a self-contained proof he verified independently, and Boris Alexeev formalized the full statement in Lean against a Formal Conjectures statement. Autonomy did not move.", "url": "https://philip-engel.github.io/S6.pdf" } ] }, { "id": "2026-08-21-dubickas-square-roots", "title": "Dubickas's question on integral parts of powers of square roots settled", "claim": "The square root of 3 admits a real multiplier making every integral part of its powers even, and among integers only the square root of 2 fails to, answering a question Dubickas posed in 2006 and classifying the whole square-root case.", "field": "mathematics", "date": "2026-08-21", "lab": "Independent", "model": "Claude Fable 5 and Claude Opus 5", "humans": [ "Ralf Stephan" ], "year_posed": 2006, "verification": "formal", "autonomy": "ai-led", "detail": "Dubickas splits the reals above 1 into two classes: \\(\\alpha\\) lies in \\(Z\\) when some nonzero real \\(\\xi\\) makes every integral part \\(\\lfloor \\xi\\alpha^n \\rfloor\\) even, and in \\(S\\) otherwise. At \\(\\alpha = 3/2\\) this is Mahler's question and still open; his Problem 3 asks which class contains \\(\\sqrt{3}\\). The answer is that it lies in \\(Z\\), by an explicit multiplier near 1.3416, and the argument classifies the entire square-root slice: for an integer \\(m\\) at least 2, \\(\\sqrt{m}\\) lies in \\(S\\) exactly when \\(m\\) is 2. A further theorem in the paper replaces parity by divisibility by any integer \\(p\\) at least 2.", "novelty_check": "Dubickas posed the question as Problem 3 of his 2006 Glasgow Mathematical Journal paper (48, 331-336), and it is stated as open there and in the subsequent \\(Z\\)-number literature descending from Mahler. The neighbouring case \\(\\alpha = 3/2\\) is Mahler's problem and remains open, which fixes the boundary of what is claimed here. No prior resolution of Problem 3 appears. The result is new, and the paper also generalizes past the original question rather than only answering it.", "caveats": "The Lean covers less than the paper. The author states that the case m equal to 3 is what is machine-verified, and the repository flags the thickness computation of section 4.1 and the whole of section 8 as not formalized, so the formal grade attaches to the central claim rather than to every theorem in the write-up. Not peer-reviewed. Autonomy graded ai-led on an unusually explicit disclosure: the author records that the mathematical discovery is the Fable 5 agent's, that the formalization is the Fable 5 and Opus 5 agents', that the agents drafted the prose which the author revised, and that the direction, review and responsibility are the author's. The Lean development carries the same framing in its copyright line.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-08-23-000002", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-23-000002&version=1", "repository": "rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "theorems": [ "SZ.sqrtThree_mem_MahlerZ", "SZ.sqrtThree_notMem_S", "SZ.sqrt_natCast_mem_S_iff" ], "checked": "2026-08-23", "note": "Eleven modules, no unfinished proof steps and no declared axioms beyond the three standard ones." } ], "sources": [ { "label": "Even integral parts of powers of square roots (ResearchGate)", "url": "https://www.researchgate.net/publication/413520035_Even_integral_parts_of_powers_of_square_roots", "kind": "research" }, { "label": "Lean development: rwst/Square-Roots", "url": "https://github.com/rwst/Square-Roots", "kind": "research" } ], "tags": [ "lean", "formalization", "number-theory", "z-numbers", "argument" ], "added": "2026-08-25" }, { "id": "2026-08-11-prescribed-cycle-recovery", "title": "A prescribed Hamiltonian cycle that a book-embedding algorithm cannot produce", "claim": "A 16-vertex graph carries a Hamiltonian cycle that the Alam et al. book-embedding algorithm cannot return for any choice of dual spanning tree, answering a question of Bekos, Kaufmann and Pfister in the negative.", "field": "computer-science", "date": "2026-08-11", "lab": "Independent", "model": "OpenAI Codex, with Claude for adversarial review", "humans": [ "Lennart Rudolph" ], "verification": "formal", "autonomy": "ai-assisted", "detail": "Bekos, Kaufmann and Pfister asked whether every prescribed Hamiltonian cycle of a Barnette graph can be recovered from the book-embedding algorithm of Alam and co-authors by choosing the dual spanning tree appropriately. The answer is no. The paper exhibits an explicit 16-vertex Barnette graph whose Hamiltonian cycle has a complementary perfect matching meeting all three edge-colour classes of the simultaneous edge and face colouring the algorithm uses. Every output of the algorithm contains each edge designated green, whatever spanning tree is chosen, and the prescribed cycle omits a green edge under every global colour labelling, so no run can produce it. Note what is not being claimed: the graph satisfies Barnette's conjecture and is Hamiltonian. What fails is the algorithm's ability to reach a particular cycle, not the conjecture.", "novelty_check": "The question is stated by Bekos, Kaufmann and Pfister about the earlier book-embedding algorithm of Alam and co-authors, and the paper answers it directly rather than a nearby question. Searched for a prior counterexample to prescribed-cycle recovery and none appears; the surrounding literature treats the recovery question as open. The construction is a new explicit instance, not a retrieval. Barnette's conjecture itself is untouched and remains open, so nothing in the novelty claim depends on it.", "caveats": "The formalization is partial by its own account: the Lean covers the finite combinatorial core, and the topological realization of the incidence data as a sphere embedding, together with the published algorithm's Property 1, remain external inputs. So the machine checking establishes that the displayed graph has the stated combinatorial properties and that the obstruction fires, not the whole argument end to end. Not peer-reviewed. Autonomy graded ai-assisted on the paper's own disclosure, which is the authoritative one: Codex assisted with literature and novelty searches, certificate verification, code testing, proof development, and manuscript drafting and editing, while Claude was used for independent adversarial review, and the author states that neither system qualifies as an author and takes full responsibility. Worth recording a discrepancy: the Palomar record lists the authors as the human plus two model names, which the paper explicitly declines to do. The paper's statement is the one graded here.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-08-21-000002", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-21-000002&version=1", "repository": "lennrt/palomar-formalizations", "commit": "090b3b300ba8ec76fdba09a17f086953d13f2653", "theorems": [ "BarnettePalomar.concrete_graph_certificate", "BarnettePalomar.face_coloring_certificate", "BarnettePalomar.property_one_excludes_prescribed_cycle" ], "checked": "2026-08-21", "note": "Covers the finite core only. The sphere embedding and the algorithm's Property 1 are external inputs, which the record states." } ], "sources": [ { "label": "Zenodo: A Counterexample to Prescribed-Cycle Recovery in Barnette Graphs", "url": "https://zenodo.org/records/21890733", "kind": "research" }, { "label": "Lean development: lennrt/palomar-formalizations", "url": "https://github.com/lennrt/palomar-formalizations", "kind": "research" } ], "tags": [ "lean", "formalization", "graph-theory", "book-embedding", "algorithms", "construction" ], "added": "2026-08-25" }, { "id": "2026-08-26-nevanlinna-half-plane", "title": "Counterexample to Nevanlinna's half-plane omitted-values question", "claim": "An explicit meromorphic function omits three values in a half-plane without being of bounded type there, answering in the negative a question of Nevanlinna open for more than a century.", "field": "mathematics", "date": "2026-08-26", "lab": "Independent", "model": "GPT-5.6 Sol Ultra", "humans": [ "Quanyu Tang", "Bokai Cui", "Wei He", "Tao Hu", "Yanyang Li", "Ke Wang", "Zijun Yu" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "A meromorphic function on a domain is of bounded type if it is a quotient of two bounded holomorphic functions. Nevanlinna asked whether omitting three values on a half-plane forces bounded type there. The paper constructs a function omitting \\(0\\), \\(1\\) and \\(\\infty\\) whose \\(a\\)-point divisor fails the Blaschke condition in each half-plane, which is exactly the negation, and notes that postcomposition gives the same conclusion for any prescribed triple of distinct values on the Riemann sphere. The construction runs through a modular covering, a comparison domain and a Farey-edge boundary with Schwarz reflection.", "novelty_check": "The question is a named one in classical function theory, stated by Nevanlinna and described in the paper's own abstract as open for over a century, with the Blaschke condition and bounded type both standard notions carrying their own literature. The paper devotes a subsection to chronology and priority, which fixes what it believes was known when. No prior counterexample appears. The result is a new construction rather than a retrieval.", "caveats": "A preprint days old at entry, unrefereed and not formalized, and no independent check is recorded. Autonomy graded ai-led on an unusually specific disclosure, which the paper puts in a numbered subsection headed Declaration of AI usage rather than in a footnote: the core mathematical construction and proof were generated during an autonomous run of GPT-5.6 Sol on OpenAI's ultra setting, the run lasted 7 hours 14 minutes and 14 seconds, the candidate proof was then subjected to human checking, mathematical auditing and revision, and the initial draft of the manuscript was also machine-generated and lightly revised. Graded ai-led rather than autonomous because the problem was posed and the output audited by the seven authors; the run itself is described as autonomous, and that distinction is the whole of the difference.", "registrations": [ { "registry": "vibemathed", "id": "nevanlinna-s-half-plane-omitted-values-problem", "url": "https://vibemathed.com/problem/nevanlinna-s-half-plane-omitted-values-problem" }, { "registry": "mathdb", "id": "392094", "title": "Nevanlinna's half-plane problem", "url": "https://mathdb.com/p/392094/nevanlinna-s-half-plane-problem", "note": "Tracks the identical claimed counterexample to the same century-old problem." } ], "sources": [ { "label": "arXiv: Three omitted values and non-Blaschke point divisors in half-planes", "url": "https://arxiv.org/abs/2608.26062", "kind": "research" } ], "tags": [ "complex-analysis", "value-distribution", "counterexample", "construction" ], "added": "2026-08-31", "revisions": [ { "date": "2026-09-10", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration. No grade moved.", "url": "https://mathdb.com/p/392094/nevanlinna-s-half-plane-problem" } ] }, { "id": "2026-08-18-stein-riesz-weak-type", "title": "Dimension-free weak-type bound for the vector Riesz transform", "claim": "The best constant in the weak-type (1,1) bound for the vector Riesz transform on n-dimensional space is at most 2 whatever the dimension, answering a question Stein raised in 1986.", "field": "mathematics", "date": "2026-08-18", "lab": "Independent", "model": "GPT-5.6 Sol and Claude Opus 5.0, with Danus and Rethlas agents", "humans": [ "Yuyuan Ouyang", "Daniel Spector", "Cody B. Stockdale" ], "year_posed": 1986, "verification": "author-verified", "autonomy": "collaborative", "detail": "Dimension-free bounds for Riesz transforms are known in the strong-type range, and Stein asked at the 1986 ICM whether the weak-type (1,1) endpoint behaves the same way. The paper answers yes with an explicit constant: the best weak-type (1,1) constant for the vector Riesz transform is at most 2, independent of n. The proof goes through a new decomposition of the input data rather than through the torus-transference route the models first proposed, which the authors judged more complicated than necessary and replaced with a direct argument in Euclidean space.", "novelty_check": "Stein posed the question at the 1986 ICM and the paper cites it directly, along with the authors' own 2021 Communications in Contemporary Mathematics paper on the dimensional weak-type (1,1) bound, which fixes the prior state of the art as dimensional rather than dimension-free. Dimension-free strong-type bounds are long established and are not what is claimed here. No prior dimension-free weak-type constant appears in the literature. The result is new.", "caveats": "A preprint, unrefereed and not formalized, with no independent check recorded. Autonomy graded collaborative rather than ai-led, and the disclosure is detailed enough to justify the weaker reading: the statement opens by saying the proof strategy was developed by large language models in dialogue with the authors, but the same paragraph records that the first attempt failed and produced only partial results, that a second attempt yielded a solution the authors judged significantly more complicated than what was published, that the human authors then assessed feasibility and redirected the work into Euclidean space, and that they checked and rewrote the proofs where the presentation was unnatural, performed the literature review, and take full responsibility. Humans chose the route that was kept. The weaker defensible reading applies.", "registrations": [ { "registry": "vibemathed", "id": "stein-s-dimension-free-weak-1-1-riesz-transform-problem", "url": "https://vibemathed.com/problem/stein-s-dimension-free-weak-1-1-riesz-transform-problem" }, { "registry": "mathdb", "id": "383291", "title": "Stein's vector Riesz transform weak type problem", "url": "https://mathdb.com/p/383291/stein-s-vector-riesz-transform-weak-type-problem", "note": "Tracks the same problem this entry records a claimed resolution of. As of the September 2026 check the page still carries it as unverified, so it corroborates the problem's standing rather than the result." } ], "sources": [ { "label": "arXiv: A dimension-free weak-type (1,1) bound for the vector Riesz transform", "url": "https://arxiv.org/abs/2608.18068", "kind": "research" } ], "tags": [ "harmonic-analysis", "riesz-transforms", "argument" ], "added": "2026-08-31", "revisions": [ { "date": "2026-09-10", "kind": "correction", "note": "Recorded the MathDB record for this problem as a registration. The page still carries the problem as unverified, so no grade moved.", "url": "https://mathdb.com/p/383291/stein-s-vector-riesz-transform-weak-type-problem" } ] }, { "id": "2026-08-25-ramsey-algebraic-construction", "title": "Improved algebraic construction for off-diagonal Ramsey numbers", "claim": "A finite-geometry construction supplied by a language model, then reworked by the authors, improves the explicit lower bound for off-diagonal Ramsey numbers.", "field": "mathematics", "date": "2026-08-25", "lab": "Independent", "model": "ChatGPT 5.6", "humans": [ "Ferdinand Ihringer", "Sam Mattheus" ], "verification": "author-verified", "autonomy": "ai-assisted", "detail": "The authors set out to use \\(p\\)-rank arguments on constructions in finite geometry to push the explicit lower bound for off-diagonal Ramsey numbers. The first construction, a graph on the points of a projective space with adjacency given by a trace condition, came from the model together with a bound on its clique number resting on algebraic geometry. The authors then developed and refined it into the published construction, which gives a bound of the form \\(R(s,t) \\ge t^{(1-o(1))\\log s/\\log(\\log s + 1)}\\). The paper describes the underlying idea as a deterministic analogue of random sampling in finite-geometric settings.", "novelty_check": "Off-diagonal Ramsey lower bounds are a well-tracked ladder and the paper states the prior explicit bound it improves on, crediting Alon and Pudlak, and cites the recent off-diagonal literature it sits in. No registry carried this result at entry: it does not appear in vibemathed, MathDB, Palomar or ProofAtlas, and it reached this registry through a comment on the r/math weekly thread rather than through any catalogue. The construction is new; the p-rank framing it starts from is the authors' own and is credited as such.", "caveats": "A preprint, unrefereed and not formalized. Autonomy graded ai-assisted, which is what the disclosure supports and no more: the authors state that they had the idea to use p-rank arguments as the starting point, that the first construction and a bound on its clique number were provided by ChatGPT 5.6, and that they subsequently developed, refined and independently verified all arguments and take full responsibility. The model supplied one object inside a human-chosen frame, and the published construction is not the one it returned. Worth recording that the disclosure is a clearly headed AI Declaration at the end of the introduction rather than a remark buried in the body, which is better practice than several other entries here.", "sources": [ { "label": "arXiv: An improved algebraic construction for Ramsey numbers", "url": "https://arxiv.org/abs/2608.21769", "kind": "research" } ], "tags": [ "combinatorics", "ramsey-theory", "finite-geometry", "construction" ], "added": "2026-08-31" }, { "id": "2026-08-25-erdos-4-prime-gaps", "title": "Improved lower bound for large gaps between consecutive primes", "claim": "A new sieving construction raises the record lower bound on how large the gap between consecutive primes becomes infinitely often, the first advance on Erdős Problem #4 since 2018.", "field": "mathematics", "date": "2026-08-25", "lab": "Independent", "model": "GPT-5.6 Sol", "humans": [ "DottedCalculator", "Boris Alexeev" ], "year_posed": 1955, "verification": "independent", "autonomy": "ai-led", "detail": "Writing \\(G(T)\\) for the largest gap between consecutive primes below \\(T\\), the record since Ford, Green, Konyagin, Maynard and Tao (2018) has been \\(G(T) \\gg \\log T \\log_2 T \\log_4 T / \\log_3 T\\), where \\(\\log_k\\) is the \\(k\\)-fold iterated logarithm. The manuscript claims \\(G(T) \\gg \\log T \\log_2 T / \\log_4 T\\), a gain of \\(\\log_3 T / (\\log_4 T)^2\\), together with \\(Y(X) \\gg X \\log X / \\log_3 X\\) for the covering problem behind it. What is new is narrow and specific: the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt, while the hypergraph covering theorem and the Maynard weight are carried over unchanged from the 2018 paper. Ben Green, a co-author of the bound being beaten, describes the sieving procedure as different from the Erdős-Rankin one that underpinned every bound on the problem since 1938.", "novelty_check": "Erdős Problem #4 is tracked at erdosproblems.com/4 and carries a $10,000 prize; the prize question itself was settled in 2016 and the record bound since is Ford, Green, Konyagin, Maynard and Tao (2018). Green, who holds that record, read the manuscript and compared the new sieve against Erdős-Rankin (1938) directly, which fixes both what was known and what is added. No intervening record appears in the erdosproblems.com record for the problem or in the vibemathed entry. The result is a new construction, not a retrieval.", "caveats": "The manuscript has no human author: its title block and PDF metadata both name GPT 5.6 Sol, there is no acknowledgements section, and no human is credited anywhere in it. That is the argument for grading autonomy as autonomous; ai-led is taken instead because a human posed the problem, iterated on the output and asked for the argument to be made self-contained, and the weaker grade wins when the reading is arguable. The submitter, who is pseudonymous and wishes to stay so, states plainly that the mathematics was not theirs: 'I am not familiar with sieve theory.' Verification rests on Green becoming, in his own words, more or less convinced the argument is correct after consulting Tao and Maynard; that is expert reading, not refereeing. One attribution discrepancy is recorded rather than smoothed over: the manuscript says GPT 5.6 Sol, while the erdosproblems.com claim was filed as GPT 5.6 Pro. Boris Alexeev's Lean development transcribes the manuscript downstream rather than producing it, and this registry has not confirmed that it builds, so the grade stays at independent rather than formal. Readers on the r/math thread note that several later sections reproduce the 2018 paper with no new content, and Green calls the exposition horrific.", "independent_checks": [ { "who": "Ben Green, as quoted by vibemathed", "url": "https://vibemathed.com/problem/tilted-residue-class-construction-for-long-prime-free-intervals", "outcome": "A co-author of the 2018 record this beats, he called it an important claim that survives an initial plausibility check, then after conversations with Tao and Maynard said he became more or less convinced it is correct, while calling the exposition truly horrible." }, { "who": "Thomas Bloom (erdosproblems.com exposition)", "url": "https://www.erdosproblems.com/4", "outcome": "Wrote a heuristic sketch of the new sieving ideas, omitting routine calculations and concentration estimates he describes as substantial; erdosproblems.com lists the improvement." } ], "registrations": [ { "registry": "vibemathed", "id": "tilted-residue-class-construction-for-long-prime-free-intervals", "url": "https://vibemathed.com/problem/tilted-residue-class-construction-for-long-prime-free-intervals" } ], "sources": [ { "label": "Manuscript: Erdős_4_GPT_5.6_Sol.pdf", "url": "https://github.com/DottedCalculator/ai-math/blob/main/Erdos_4_GPT_5.6_Sol.pdf", "kind": "research" }, { "label": "Erdős Problems: problem #4", "url": "https://www.erdosproblems.com/4", "kind": "commentary" }, { "label": "vibemathed: a tilted residue-class construction for long prime-free intervals", "url": "https://vibemathed.com/problem/tilted-residue-class-construction-for-long-prime-free-intervals", "kind": "commentary" }, { "label": "Jared Duker Lichtman on the result (X)", "url": "https://x.com/jdlichtman/status/2094040463443673227", "kind": "announcement" } ], "tags": [ "number-theory", "erdos", "prime-gaps", "sieve-methods", "construction" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded Ben Green's assessment and Thomas Bloom's heuristic exposition, which together are the evidence the independently checked grade rests on. No grade moved.", "url": "https://vibemathed.com/problem/tilted-residue-class-construction-for-long-prime-free-intervals" } ] }, { "id": "2026-08-19-caratheodory-umbilic", "title": "Counterexample to the smooth Carathéodory conjecture on umbilic points", "claim": "An explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point, so the \\(C^\\infty\\) form of Carathéodory's 1922 conjecture is false.", "field": "mathematics", "date": "2026-08-19", "lab": "Independent", "model": "Claude; Codex", "humans": [ "Levent Alpöge", "John-Paul Smith" ], "year_posed": 1922, "verification": "claimed", "autonomy": "ai-assisted", "detail": "Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in \\(\\mathbb{R}^3\\) carries at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and that theorem is untouched. The counterexample is explicitly a \\(C^\\infty\\) object: an explicit support function produces a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point. The same family disproves the smooth Loewner conjecture, whose member at \\(k = 1\\) has an isolated trace-free Hessian zero of the wrong index. The gap between the smooth and the real-analytic case is the whole content of the result.", "novelty_check": "Carathéodory's conjecture is a named problem with a long literature and a standing partial theorem, Hamburger's real-analytic case from 1940-41, and it appears as Problem 8.1 in Ghomi's published problem list. The claim is a counterexample in the smooth category only, which is consistent with rather than contradicting Hamburger. No prior smooth counterexample appears in the problem list or in the vibemathed record. The construction is new work, not a retrieval of something already in the literature.", "caveats": "Announced in an X post rather than a preprint, and unrefereed. The Lean formalization is sorry-free at a commit but not merged, and vibemathed records its statement as read rather than compiled and audited, so nothing here is machine-checked end to end for this registry's purposes; the grade stays at claimed. Autonomy is graded ai-assisted rather than higher on a deliberately strict reading of two separate model roles, neither of which is the discovery: Alpöge's announcement credits John-Paul Smith and Claude with checking the construction, which makes the model a verifier of a human construction, and the Lean development was, in its author's words, developed with Codex and parallel proof-review agents, which is formalization of a human result. Only the smooth case falls; the classical real-analytic form of the conjecture remains true.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture", "outcome": "At the pinned formal-conjectures commit 7aa855b, the formalized hypothesis is the classical smooth statement rather than a weakened one, and the 8,013-line proof tree carries no sorry, no declared axioms and no native_decide. Read, not compiled; the commit is unmerged and both upstream pull requests are drafts." } ], "registrations": [ { "registry": "vibemathed", "id": "mathbb-c-infty-caratheodory-conjecture", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture" } ], "sources": [ { "label": "Levent Alpöge, announcement (X)", "url": "https://x.com/__alpoge__/status/2089971359921156203", "kind": "announcement" }, { "label": "vibemathed: the C-infinity Carathéodory conjecture on umbilic points", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture", "kind": "commentary" } ], "tags": [ "geometry", "differential-geometry", "counterexample", "lean", "construction" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's audit finding the formal statement faithful to the smooth conjecture. The proof was read rather than compiled and remains unmerged upstream, so no grade moved.", "url": "https://vibemathed.com/problem/mathbb-c-infty-caratheodory-conjecture" } ] }, { "id": "2026-08-23-elliptic-rank-31", "title": "Elliptic curves over the rationals of rank at least 30 and at least 31", "claim": "Two explicit elliptic curves over the rationals carry 30 and 31 independent rational points, raising the Mordell-Weil rank record twice in four days.", "field": "mathematics", "date": "2026-08-23", "lab": "Independent", "model": "Claude", "humans": [ "Levent Alpöge", "Ava Howell" ], "verification": "claimed", "autonomy": "ai-assisted", "detail": "Whether the Mordell-Weil rank of an elliptic curve over the rationals is bounded is open, and progress is measured by explicit records: rank at least 28 from 2006, raised to at least 29 by Elkies and Klagsbrun in 2024. Curve 273 on the elliptic-rank leaderboard reaches rank at least 30 on 2026-08-20 and curve 302 reaches at least 31 on 2026-08-23, each witnessed by explicit independent points. The method is the human Elkies-Klagsbrun elliptic-K3 family search, with the model assisting inside it. Both entries are recorded as two tiers: the lower bound is what the explicit points establish, while exact rank is claimed only conditionally on the Birch-Swinnerton-Dyer conjecture and the generalized Riemann hypothesis.", "novelty_check": "The rank record ladder is well documented and tabulated by Dujella, which makes the prior state of the art unambiguous: Elkies rank at least 28 in 2006 and Elkies and Klagsbrun rank at least 29 in 2024. Both new curves are given explicitly with their a-invariants, conductor, discriminant and regulator, so the claim is checkable against that table rather than resting on an assertion. No intervening record appears. These are new curves, not a rediscovery.", "caveats": "Recorded only as leaderboard database entries: there is no preprint, no transcript and no refereeing, and nothing is formalized. The two records are kept in one entry because they are the same method and the same people four days apart. Independence of the points is asserted through the leaderboard's stated general practice of exact 2-descent rather than reproduced on the page, and the exact-rank claim is conditional on BSD and GRH with no numeric derivation published for either curve. What the model contributed is documented only by the leaderboard's own commentary field, which reads in full: 'BSD + GRH certified to rank 31, found by Claude, Levent Alpöge, and Ava Howell.' Autonomy is graded ai-assisted on that thin evidence, because the search framework is the humans' and no stronger reading is defensible. Neither record answers the open question of whether ranks are unbounded.", "registrations": [ { "registry": "vibemathed", "id": "elliptic-curve-rank-record-thirty-one", "url": "https://vibemathed.com/problem/elliptic-curve-rank-record-thirty-one" }, { "registry": "vibemathed", "id": "elliptic-curve-rank-record-thirty", "url": "https://vibemathed.com/problem/elliptic-curve-rank-record-thirty" } ], "sources": [ { "label": "Elliptic curve rank leaderboard: curve 302, rank at least 31", "url": "https://elliptic-rank.icarm.cloud/curve/302", "kind": "research" }, { "label": "Elliptic curve rank leaderboard: curve 273, rank at least 30", "url": "https://elliptic-rank.icarm.cloud/curve/273", "kind": "research" }, { "label": "vibemathed: a rank-31 record for an elliptic curve over Q", "url": "https://vibemathed.com/problem/elliptic-curve-rank-record-thirty-one", "kind": "commentary" }, { "label": "vibemathed: record rank for an elliptic curve over Q", "url": "https://vibemathed.com/problem/elliptic-curve-rank-record-thirty", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: An elliptic curve of rank at least 30", "url": "https://news.ycombinator.com/item?id=49374873" } ], "tags": [ "number-theory", "elliptic-curves", "record", "search" ], "added": "2026-09-03" }, { "id": "2026-08-29-dean-conjecture-k5", "title": "Dean's conjecture for k = 5, cycles of length divisible by five", "claim": "A preprint claims every finite simple graph of minimum degree at least five contains a cycle whose length is divisible by five, the last open case of Dean's 1988 conjecture.", "field": "mathematics", "date": "2026-08-29", "lab": "Independent", "model": "GPT-5.6 Sol; Claude Opus 5; GLM 5.3 Flash", "humans": [ "Elias Botsford" ], "year_posed": 1988, "verification": "claimed", "autonomy": "ai-led", "detail": "Dean's conjecture asserts that every finite simple graph of minimum degree at least \\(k\\) contains a cycle of length divisible by \\(k\\). The cases \\(k = 3\\) and \\(k = 4\\) were known, and Luo, Ma and Zhao proved every \\(k\\) at least 6, which left \\(k = 5\\) as the single open case; if this proof stands, the conjecture holds for all \\(k\\) at least 3. Nine finite configuration propositions in the bipartite and triangle-free branches are computer-assisted, with verifier sources and certificate data archived in a separate computational supplement.", "novelty_check": "The landscape is fixed by an independent recent paper rather than by the claimant: Luo, Ma and Zhao (arXiv:2601.13552) state in their own abstract that the conjecture was known for \\(k \\in \\{3, 4\\}\\) and prove it for every \\(k\\) at least 6, which identifies \\(k = 5\\) as the remaining case at the time of writing. No competing resolution of \\(k = 5\\) appears in that literature or in the vibemathed record. The claim is new work on an identified gap.", "caveats": "A Zenodo preprint days old at entry, unrefereed, with no independent check on record. The computer-assisted part is the narrow part: nine finite configuration families have verifiers and certificates, and certificate runs are reported as passing, but the reductions from arbitrary graphs to those finite state spaces are ordinary prose arguments that are not machine-checked and that only the submitter has audited. That is why the grade is claimed despite the artifact. Autonomy is graded ai-led rather than autonomous because a human submitted, framed and archived the work; the Zenodo record names Elias Botsford as the sole creator, a detail the registry rows carrying this result omit.", "independent_checks": [ { "who": "vibemathed.com certificate replay", "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5", "outcome": "Downloaded the Zenodo computational supplement, matched all 86 files against its SHA-256 manifest and ran the driver end to end: all 47 certificate runs passed. This covers the nine finite proposition families only, not the prose reductions from arbitrary graphs to them." } ], "registrations": [ { "registry": "vibemathed", "id": "dean-s-conjecture-for-k-5", "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5" }, { "registry": "mathdb", "id": "374541", "title": "Dean's conjecture on cycles divisible by the minimum degree", "url": "https://mathdb.com/p/374541/dean-s-conjecture-on-cycles-divisible-by-the-minimum-degree", "note": "Names the same three models this entry credits and records k = 5 as the last open case, which is the framing the entry rests on." } ], "sources": [ { "label": "Zenodo: cycles of length divisible by five in graphs of minimum degree five", "url": "https://zenodo.org/records/22182448", "kind": "research" }, { "label": "Zenodo: computational supplement, version 1.0.1", "url": "https://doi.org/10.5281/zenodo.22167084", "kind": "research" }, { "label": "vibemathed: Dean's conjecture for k = 5", "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5", "kind": "commentary" } ], "tags": [ "graph-theory", "combinatorics", "computer-assisted", "argument" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-10", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration. It names the same three models and confirms k = 5 as the last open case. No grade moved.", "url": "https://mathdb.com/p/374541/dean-s-conjecture-on-cycles-divisible-by-the-minimum-degree" }, { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's full replay of the computational supplement, which passes. The graph reductions that carry the result beyond the finite cases remain unread by anyone independent, so no grade moved.", "url": "https://vibemathed.com/problem/dean-s-conjecture-for-k-5" } ] }, { "id": "2026-08-31-stable-forking", "title": "Counterexample to the stable forking conjecture", "claim": "A simple first-order theory is constructed in which forking cannot always be witnessed by a stable formula, answering a question of Hart, Kim and Pillay open since 1996.", "field": "mathematics", "date": "2026-08-31", "lab": "Independent", "model": "GPT-5.6 Sol", "humans": [ "James Freitag", "Scott Mutchnik" ], "year_posed": 1996, "verification": "author-verified", "autonomy": "collaborative", "detail": "The stable forking conjecture asserts that in a simple theory, whenever a does not fork with b over C, there is a formula in the type of a over Cb that forks over C and whose parameter-free form is stable. The counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised through an abstract independence relation using the Kim-Pillay criteria, and the random graph is encoded into that relation so that it has the order property, which is what breaks stability.", "novelty_check": "The conjecture is a named open question in classification theory, attributed in the paper to Hart, Kim and Pillay (1996) and described there as long-standing. The construction runs through the standard route for such results, proving simplicity and then characterising forking independence via the Kim-Pillay criteria, so its relation to the existing literature is explicit rather than assumed. Model-theoretic study of noncommutative rings has precedent, which the authors cite; no prior counterexample appears.", "caveats": "A preprint days old at entry, unrefereed, not formalized, and with no independent check on record. Autonomy is graded collaborative rather than ai-led on the authors' own account of the division of labour: they prompted the model with a detailed series of prompts that encoded the known restrictions such a counterexample must satisfy, and specifically directed it to the standard strategy of proving simplicity and applying the Kim-Pillay criteria, so the framing was substantially theirs. They state that the proofs of the relevant facts and aspects of the setup were written entirely by the authors and that no AI tool was used in writing the manuscript, with AI used for proofreading under manual evaluation. They also record their view that the counterexample would have been unlikely to be found in the near term without generative AI.", "registrations": [ { "registry": "vibemathed", "id": "a-counterexample-to-the-stable-forking-conjecture", "url": "https://vibemathed.com/problem/a-counterexample-to-the-stable-forking-conjecture" }, { "registry": "mathdb", "id": "375925", "title": "The stable forking conjecture", "url": "https://mathdb.com/p/375925/the-stable-forking-conjecture" } ], "sources": [ { "label": "arXiv: a counterexample to the stable forking conjecture", "url": "https://arxiv.org/abs/2609.00436", "kind": "research" }, { "label": "vibemathed: a counterexample to the stable forking conjecture", "url": "https://vibemathed.com/problem/a-counterexample-to-the-stable-forking-conjecture", "kind": "commentary" } ], "tags": [ "model-theory", "logic", "counterexample", "construction" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-15", "kind": "correction", "note": "Recorded the MathDB record for this conjecture as a registration. No grade moved.", "url": "https://mathdb.com/p/375925/the-stable-forking-conjecture" } ] }, { "id": "2026-08-26-spin-systems-girth-five", "title": "Rapid mixing for spin systems on graphs of girth at least five", "claim": "Glauber dynamics for proper \\(q\\)-colourings mixes rapidly once \\(q\\) exceeds the maximum degree by any fixed factor, provided the graph has girth at least five.", "field": "computer-science", "date": "2026-08-26", "lab": "Independent", "model": "GPT-5.6 Sol Ultra", "humans": [ "Xiaoyu Chen", "Kuikui Liu" ], "year_posed": 1995, "verification": "author-verified", "autonomy": "collaborative", "detail": "For every fixed positive \\(\\delta\\), the Glauber dynamics on proper \\(q\\)-colourings is shown to mix rapidly when \\(q\\) is at least \\((1 + \\delta)\\) times the maximum degree, the graph has girth at least five, and the maximum degree is large in terms of \\(\\delta\\). The result extends to general multi-spin systems satisfying a local spectral contraction condition, including the anti-ferromagnetic Potts model. The engine is a new spectral local-to-global principle for multi-spin systems on graphs of girth at least five, together with a Fourier analysis for Glauber dynamics on a star.", "novelty_check": "The target is Jerrum's long-standing question about the colouring threshold, and the paper situates itself explicitly against the recent chain it builds on: the Bochner-Bakry-Emery proof of the spectral-gap criterion of Chen, Chen, Chen, Yin and Zhang, and the trickle-down and spectral independence literature it names. The girth-five restriction is stated as a hypothesis rather than removed, so the claim is a partial advance on a known frontier and is positioned as such. No prior result at this threshold under these hypotheses appears.", "caveats": "A preprint days old at entry, unrefereed, not formalized, and with no independent check on record. The result is partial: girth at least five and a large maximum degree are genuine hypotheses, not technicalities. Autonomy is graded collaborative on the authors' own narrative, which describes several rounds of interaction rather than a single handoff: they first asked the model to redo known proofs by the Bochner identity, then brought it the open question, at which point it proposed an affirmative proof strategy; the human authors then verified the argument, generalized the proof with further assistance, and streamlined the paper.", "registrations": [ { "registry": "vibemathed", "id": "rapid-mixing-for-spin-systems-on-graphs-of-girth-at-least-five", "url": "https://vibemathed.com/problem/rapid-mixing-for-spin-systems-on-graphs-of-girth-at-least-five" } ], "sources": [ { "label": "arXiv: a spectral local-to-global principle for spin systems on graphs with girth at least five", "url": "https://arxiv.org/abs/2608.25491", "kind": "research" }, { "label": "vibemathed: rapid mixing for spin systems on graphs of girth at least five", "url": "https://vibemathed.com/problem/rapid-mixing-for-spin-systems-on-graphs-of-girth-at-least-five", "kind": "commentary" } ], "tags": [ "algorithms", "sampling", "markov-chains", "combinatorics", "argument" ], "added": "2026-09-03" }, { "id": "2026-08-20-fast-dynamo-three-torus", "title": "A smooth random fast dynamo on the three-torus", "claim": "A random smooth divergence-free velocity field on the three-torus is constructed whose magnetic field grows exponentially at a rate bounded below independently of the resistivity.", "field": "mathematics", "date": "2026-08-20", "lab": "Independent", "model": "ChatGPT 5.6 Sol Ultra", "humans": [ "Keefer Rowan" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "The construction is a random, time-dependent, divergence-free velocity field on the three-torus, refreshing independently on finite time blocks and obeying deterministic smooth bounds in space and time, which exhibits fast dynamo behaviour: for every sufficiently small fixed resistivity the almost sure exponential growth rate of the magnetic field solving the linear resistive induction equation is at least one half. The argument turns on an algebraic structure in Fourier space of the induction equation solution operator that lets the expected growth of the logarithmic size of three chosen Fourier modes be propagated, reducing the problem to a simple recursion and avoiding the infinite-dimensional dynamics usually involved.", "novelty_check": "The paper positions itself against Arnold's fast-dynamo problem and the dynamo literature it cites, and the arXiv comment records the provenance directly as AI discovered, human written. vibemathed records the resolution as a variant rather than the original problem, which is the honest reading: what is built is a smooth random field with the growth property, not a resolution of the fast-dynamo problem as classically posed. No prior construction of this kind appears in the cited literature.", "caveats": "A preprint, unrefereed, not formalized, with no independent check on record, and a variant of the classical problem rather than the problem itself. Autonomy is graded ai-led on an explicit disclosure: the author writes that the central proof idea was generated essentially autonomously by ChatGPT 5.6 Sol Ultra from a prompt adapted from an earlier preprint, and that the manuscript was written and verified by the author. Some simplifications over the machine-generated manuscript are the author's, including a simpler set of velocity fields and a more direct iteration argument, and the model was also used to explain sections of the original argument back to the author and to copy-edit.", "registrations": [ { "registry": "vibemathed", "id": "smooth-random-fast-dynamo-on-the-three-torus", "url": "https://vibemathed.com/problem/smooth-random-fast-dynamo-on-the-three-torus" } ], "sources": [ { "label": "arXiv: an AI-discovered smooth random fast dynamo on T³", "url": "https://arxiv.org/abs/2608.20105", "kind": "research" }, { "label": "vibemathed: smooth random fast dynamo on the three-torus", "url": "https://vibemathed.com/problem/smooth-random-fast-dynamo-on-the-three-torus", "kind": "commentary" } ], "tags": [ "analysis", "pde", "probability", "construction" ], "added": "2026-09-03" }, { "id": "2026-08-19-big-line-big-clique", "title": "First open case of the big-line-big-clique conjecture", "claim": "Every finite point set of size at least \\(10^{11055931}\\) contains four collinear points or six points that pairwise see each other, settling the first open case of the Kára-Pór-Wood conjecture.", "field": "mathematics", "date": "2026-08-19", "lab": "Independent", "model": "GPT-5.6 Sol Pro", "humans": [ "Édouard Bonnet" ], "year_posed": 2005, "verification": "author-verified", "autonomy": "ai-led", "detail": "The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all \\(k\\) and \\(l\\) there is an \\(n\\) such that every finite point set of at least \\(n\\) points contains \\(l\\) collinear points or \\(k\\) points that pairwise see each other, where two points see each other if no other point of the set lies between them. The paper proves the case \\(l = 4\\), \\(k = 6\\), the first case left open, with the explicit and very large threshold \\(10^{11055931}\\). The proof combines a lower bound on distance to 5-colorability, a Turán-type stability and edge-density argument, and matching harmonic upper and lower bounds.", "novelty_check": "The conjecture is named and dated to Kára, Pór and Wood in 2005, and the paper identifies \\(l = 4\\), \\(k = 6\\) as the first case left open by the existing partial results, which fixes the prior state of the art. The threshold is explicit, so the statement is falsifiable as written rather than asymptotic hand-waving. No prior proof of this case appears in the discrete geometry literature or in the vibemathed record.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record, and vibemathed lists it as a candidate pending review. The threshold is astronomically large and the result settles one case rather than the conjecture. Stijn Cambie has since brought the threshold for the same case down to 880 points in a three-page note, itself AI-assisted and marked not for publication. Autonomy is graded ai-led on the paper's own AI disclosure, which is unusually specific about the sequence: after one fruitless attempt and what the author calls the now customary generic encouragement, a relatively detailed proof of Theorem 1 was provided by GPT-5.6 Sol Pro after pondering for 222 minutes.", "registrations": [ { "registry": "vibemathed", "id": "big-line-big-clique-four-collinear-or-six-clique", "url": "https://vibemathed.com/problem/big-line-big-clique-four-collinear-or-six-clique" }, { "registry": "mathdb", "id": "315965", "title": "Big line big clique conjecture", "url": "https://mathdb.com/p/315965/big-line-big-clique-conjecture", "note": "Tracks the same Bonnet preprint resolving the (6, 4) case that this entry records." } ], "sources": [ { "label": "arXiv: large finite point sets have 4 collinear points or a 6-clique", "url": "https://arxiv.org/abs/2608.19468", "kind": "research" }, { "label": "vibemathed: the Kára-Pór-Wood big-line-big-clique conjecture", "url": "https://vibemathed.com/problem/big-line-big-clique-four-collinear-or-six-clique", "kind": "commentary" }, { "label": "arXiv: Cambie, Four collinear points or six visible points", "url": "https://arxiv.org/abs/2609.21035", "kind": "commentary" } ], "tags": [ "combinatorics", "discrete-geometry", "argument" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-10", "kind": "correction", "note": "Recorded the MathDB record for this result as a registration. No grade moved.", "url": "https://mathdb.com/p/315965/big-line-big-clique-conjecture" }, { "date": "2026-09-24", "kind": "correction", "note": "Recorded Stijn Cambie's improvement of the threshold for the same case to 880 points. No grade moved.", "url": "https://arxiv.org/abs/2609.21035" } ] }, { "id": "2026-08-19-linear-extensions-below-2n", "title": "Counting linear extensions below the 2ⁿ barrier", "claim": "A deterministic algorithm counts the linear extensions of an arbitrary \\(n\\)-element poset in time \\(O^*(1.89^n)\\), breaking the \\(2^n\\) barrier for the general problem.", "field": "computer-science", "date": "2026-08-19", "lab": "Independent", "model": "Claude Opus 5; ChatGPT 5.6 Sol", "humans": [ "Keigo Oka" ], "year_posed": 2013, "verification": "author-verified", "autonomy": "ai-led", "detail": "A linear extension of a finite poset is a total order respecting it. The algorithm counts them exactly for an arbitrary \\(n\\)-element poset in \\(O^*(1.89^n)\\) time, resolving a question posed by Koivisto at Dagstuhl in 2013. It refines an argument of Kozma for two-dimensional posets: a chain partition handles posets far from an antichain, and otherwise a maximum antichain is fixed and only the first comparable element above each of its elements is recorded, with a decoding lemma enumerating the resulting patterns from their multiplicities. Once a pattern is fixed each antichain element has a release condition and at most one deadline, so the dynamic program need only store how many elements are released in each deadline class.", "novelty_check": "The question is attributed to a specific venue and date, Koivisto at Dagstuhl 2013, and the \\(2^n\\) barrier for the general problem is the standing baseline the paper names. The refinement is stated against a specific prior argument, Kozma's for two-dimensional posets, which makes the delta explicit. A verification script ships as an ancillary file with the preprint. No prior sub-\\(2^n\\) algorithm for the general problem appears.", "caveats": "A preprint, unrefereed, with no independent check on record, and vibemathed lists it as a candidate pending review; the ancillary verification script is not a proof artifact and nothing is formalized. Autonomy is graded ai-led on an unusually layered disclosure: the core mathematical ideas behind the new part of the algorithm and proof are stated to have been discovered by Claude Opus 5, while the research prompt supplied to it was itself generated by ChatGPT 5.6 Sol, which the author instructed to model the prompt on OpenAI's publicly released one for the cycle double cover work, so the prompt-generation stage was AI-assisted as well. Generative tools were also used for literature search, drafting and verification assistance.", "registrations": [ { "registry": "vibemathed", "id": "counting-linear-extensions-below-two-to-the-n", "url": "https://vibemathed.com/problem/counting-linear-extensions-below-two-to-the-n" } ], "sources": [ { "label": "arXiv: breaking the 2ⁿ barrier for counting linear extensions with a short elementary algorithm", "url": "https://arxiv.org/abs/2608.19505", "kind": "research" }, { "label": "vibemathed: counting linear extensions below the 2ⁿ barrier", "url": "https://vibemathed.com/problem/counting-linear-extensions-below-two-to-the-n", "kind": "commentary" } ], "tags": [ "algorithms", "combinatorics", "exact-algorithms" ], "added": "2026-09-03" }, { "id": "2026-08-25-sparse-convex-body-domination", "title": "Sparse domination implies convex body domination", "claim": "Any bilinear form with a sparse bound has a convex body sparse bound for its vector-valued extension, answering a question about the relation between the two notions.", "field": "mathematics", "date": "2026-08-25", "lab": "Independent", "model": "GPT-5.6 Sol Pro", "humans": [ "Aapo Laukkarinen", "Emiel Lorist" ], "year_posed": 2017, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "If a bilinear form admits an (r,s)-sparse bound, then its coordinate-wise extension to functions valued in complex n-space admits an (r,s)-convex body sparse bound. The proof rests on a randomization argument, and the result is established both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application the authors deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.", "novelty_check": "Convex body domination was introduced by Nazarov, Petermichl, Treil and Volberg, whom vibemathed records as posing the question in 2017, and sparse domination is a standard tool with its own literature, cited in the paper alongside the Bellman-function and sparse-revolution surveys. The implication between the two notions is what was open; the paper proves it in both the fixed-lattice and arbitrary-cube settings rather than one. No prior proof of the implication appears.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-assisted, the weakest reading consistent with the disclosure, because the paper's AI disclosure statement claims only prototyping and not discovery: GPT-5.6 Sol Pro, accessed through ChatGPT, was used to prototype proof strategies for this note. Nothing in the disclosure attributes the final argument to the model.", "registrations": [ { "registry": "vibemathed", "id": "sparse-domination-implies-convex-body-domination", "url": "https://vibemathed.com/problem/sparse-domination-implies-convex-body-domination" } ], "sources": [ { "label": "arXiv: sparse domination implies convex body domination", "url": "https://arxiv.org/abs/2608.24802", "kind": "research" }, { "label": "vibemathed: sparse domination implies convex body domination", "url": "https://vibemathed.com/problem/sparse-domination-implies-convex-body-domination", "kind": "commentary" } ], "tags": [ "analysis", "harmonic-analysis", "argument" ], "added": "2026-09-03" }, { "id": "2026-08-27-large-systoles", "title": "Hyperbolic surfaces with large systoles in every large genus", "claim": "For every sufficiently large genus there is a closed hyperbolic surface whose systole is at least \\(\\log g - 12\\log\\log g\\), raising the known asymptotic constant from 2/9 to 1.", "field": "mathematics", "date": "2026-08-27", "lab": "Independent", "model": "GPT-5.6 Sol", "humans": [ "Yifei Cai" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "For every sufficiently large genus \\(g\\) the note constructs a closed hyperbolic surface with systole at least \\(\\log g - 12\\log\\log g\\), which gives a lower bound of 1 for the limit inferior of the maximum systole over genus-\\(g\\) surfaces divided by \\(\\log g\\), improving the previously known bound of 2/9. It continues the author's earlier work on the diameter of finite covers, using the same constant-twist pants decomposition framework applied to systoles instead.", "novelty_check": "The improvement is stated against a specific numerical baseline, the previously known bound of 2/9 for the same limit inferior, so the delta is explicit. The note names its own predecessor, arXiv:2608.12887, and reuses that framework, which makes clear what is new here as against what was already built. No stronger bound appears in the cited literature or in the vibemathed record.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record, and vibemathed records the resolution as partial. Autonomy is graded ai-led on the paper's declaration on the use of AI, which credits the model with the proof but the framework to the author: starting from the constant-twist pants decomposition approach described in the note, GPT-5.6 Sol developed the first complete proof of the main theorem through an extended discussion with the author.", "registrations": [ { "registry": "vibemathed", "id": "large-systoles-in-every-sufficiently-large-genus", "url": "https://vibemathed.com/problem/large-systoles-in-every-sufficiently-large-genus" } ], "sources": [ { "label": "arXiv: a note on surfaces with large systoles", "url": "https://arxiv.org/abs/2608.26660", "kind": "research" }, { "label": "vibemathed: large systoles in every sufficiently large genus", "url": "https://vibemathed.com/problem/large-systoles-in-every-sufficiently-large-genus", "kind": "commentary" } ], "tags": [ "geometry", "hyperbolic-geometry", "construction" ], "added": "2026-09-03" }, { "id": "2026-09-01-mckean-entropy-production", "title": "Entropy production of the Boltzmann equation is not always monotone", "claim": "Explicit radially symmetric mixtures of Maxwellians show the entropy production of the Boltzmann equation need not decrease monotonically, answering a question McKean asked in 1966.", "field": "mathematics", "date": "2026-09-01", "lab": "Independent", "model": "GPT-5.6 Sol; Claude", "humans": [ "Luis Silvestre" ], "year_posed": 1966, "verification": "author-verified", "autonomy": "ai-led", "detail": "In 1966 McKean asked whether the entropy production of the Boltzmann equation must be monotone decreasing in time. The paper shows it need not be, even in the space-homogeneous setting, for the Boltzmann collision operator with a constant angular cross section and kinetic parameter \\(\\gamma\\) in the interval from 0 to 1. That range recovers the classical case of hard spheres and the simplest case of Maxwell molecules, so the counterexample reaches the two physically standard models rather than an exotic corner. The examples are radially symmetric mixtures of Maxwellians.", "novelty_check": "The question is named, dated and attributed to McKean in 1966 in the paper's own opening, and the parameter range is chosen precisely so that the two classical models, hard spheres and Maxwell molecules, are covered. The paper separately flags the corresponding question for the Landau equation as still open, which shows the boundary of the claim is being drawn deliberately. No prior counterexample appears.", "caveats": "A preprint days old at entry, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-led on the paper's disclaimer on the use of AI tools, which records a chain of tools and a substantial human rewrite: a first version of the proof was obtained by ChatGPT 5.6 Sol, run in Codex in Ultra mode with access to the source of one of the author's own earlier papers and other notes, and it was then rewritten first using Claude Code and afterwards reinterpreted and restructured by the author.", "registrations": [ { "registry": "vibemathed", "id": "mckean-entropy-production-conjecture", "url": "https://vibemathed.com/problem/mckean-entropy-production-conjecture" } ], "sources": [ { "label": "arXiv: the entropy production is not always monotone for hard spheres or for Maxwell molecules", "url": "https://arxiv.org/abs/2609.01753", "kind": "research" }, { "label": "vibemathed: McKean entropy-production conjecture", "url": "https://vibemathed.com/problem/mckean-entropy-production-conjecture", "kind": "commentary" } ], "tags": [ "analysis", "pde", "mathematical-physics", "counterexample" ], "added": "2026-09-03" }, { "id": "2026-08-25-generically-stable-keisler", "title": "Equivalence of generic stability notions for Keisler measures", "claim": "Three candidate definitions of generic stability for a Borel-definable global Keisler measure are shown to be equivalent, closing a research objective left open by earlier work.", "field": "mathematics", "date": "2026-08-25", "lab": "Independent", "model": "ChatGPT 5.5; ChatGPT 5.6 Sol; Kimi K3; Claude Fable 5", "humans": [ "Gabriel Conant", "Kyle Gannon", "James E. Hanson" ], "year_posed": 2020, "verification": "author-verified", "autonomy": "ai-led", "detail": "For a first-order theory in discrete or continuous logic and a Borel-definable global Keisler measure, the paper shows three conditions equivalent: that the measure is a frequency interpretation measure, that it is definable and its canonical random extension is generically stable in the randomization theory, and that it is self-averaging. The forward implications were already established by the same authors for discrete theories; the paper's contribution is the two reverse implications, which is what makes the notion of generic stability robust and settles a long-term objective from that earlier work.", "novelty_check": "The prior state is unusually clean here because it is the authors' own: the implications from frequency interpretation to generic stability of the random extension to self-averaging were established in their previous work for discrete theories, and the reverse implications are what remained. The paper states that division explicitly, so what is new is not in doubt. No competing proof of the reverse implications appears.", "caveats": "A preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-led on the paper's AI acknowledgment, which reports both discovery and cross-checking by several models followed by substantial human rewriting: a proof of the main theorem was initially obtained from a ChatGPT 5.5 query focused on the discrete case, the authors were also able to find proofs independently using Kimi K3 and Claude Fable 5, and the arguments were then heavily reorganized and rewritten by the authors with further assistance from ChatGPT 5.6 Sol.", "registrations": [ { "registry": "vibemathed", "id": "equivalence-of-generic-stability-notions-for-keisler-measures", "url": "https://vibemathed.com/problem/equivalence-of-generic-stability-notions-for-keisler-measures" } ], "sources": [ { "label": "arXiv: generically stable Keisler measures", "url": "https://arxiv.org/abs/2608.24605", "kind": "research" }, { "label": "vibemathed: equivalence of generic stability notions for Keisler measures", "url": "https://vibemathed.com/problem/equivalence-of-generic-stability-notions-for-keisler-measures", "kind": "commentary" } ], "tags": [ "model-theory", "logic", "argument" ], "added": "2026-09-03" }, { "id": "2026-09-01-saxl-common-neighbour", "title": "Common neighbour conjectures for Saxl graphs fail at every base size", "claim": "Primitive permutation groups are constructed whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour, disproving the Burness-Giudici conjecture and its extension at every base size.", "field": "mathematics", "date": "2026-09-01", "lab": "Independent", "model": "Codex; ChatGPT Pro; Claude", "humans": [ "Aluna Rizzoli", "Adam R. Thomas" ], "year_posed": 2020, "verification": "formal", "autonomy": "collaborative", "detail": "For a finite permutation group a base is a set of points with trivial pointwise stabiliser, and the generalised Saxl graph records which pairs of points lie together in a base of minimum size. Burness and Giudici conjectured that any two vertices of the Saxl graph of a primitive group of base size two have a common neighbour, and Freedman, Huang, Lee and Rekvényi extended this to arbitrary base size. Both are disproved: for each integer B at least 2 the paper constructs infinitely many primitive groups of base size B whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size two there are three further infinite families, one each of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan-Scott types, and in the affine and product families the Saxl graphs have diameter exactly three. The result also answers Problem 21.29 of the Kourovka Notebook.", "novelty_check": "Both conjectures are named, attributed and dated in the paper: Burness and Giudici for base size two in 2020, and Freedman, Huang, Lee and Rekvényi for the extension to arbitrary base size. The counterexamples are exhibited as explicit infinite families rather than existence claims, and their spread across three of the five O'Nan-Scott types is stated precisely. The Kourovka Notebook problem answered here is 21.29, which is not among the eight problems covered by this registry's existing Kourovka entry. No prior counterexample appears.", "caveats": "A preprint days old at entry, unrefereed, with no independent human check on record. The formal grade rests on Theorem 1.2, the statement that counterexamples exist at every base size, which is verified in Lean 4 with no sorry placeholders and no project-specific axioms, with the source in the paper's repository; the three additional base-size-two families and the diameter computations are not covered by that formalization, and the Lean statements have not been audited by a third party against the paper. Autonomy is graded collaborative on a detailed disclosure describing back-and-forth rather than handoff: the project began with Codex support aimed at proving the conjecture for soluble affine groups, the authors then worked with Codex, ChatGPT Pro and Claude to find further examples and constructions, search the literature and draft the manuscript, Codex implemented much of the Magma, GAP, Python and C++ code, and the Lean formalisation was produced primarily by Codex with the authors reviewing its theorem statements and verifying the completed formalisation.", "registrations": [ { "registry": "vibemathed", "id": "common-neighbour-conjectures-for-saxl-graphs", "url": "https://vibemathed.com/problem/common-neighbour-conjectures-for-saxl-graphs" } ], "sources": [ { "label": "arXiv: common neighbour conjectures for Saxl graphs fail at every base size", "url": "https://arxiv.org/abs/2609.01367", "kind": "research" }, { "label": "vibemathed: common neighbour conjectures for Saxl graphs", "url": "https://vibemathed.com/problem/common-neighbour-conjectures-for-saxl-graphs", "kind": "commentary" } ], "tags": [ "group-theory", "combinatorics", "counterexample", "lean", "formalization" ], "added": "2026-09-03" }, { "id": "2026-08-29-metric-distortion-23282", "title": "Randomized metric distortion improved to 2.3282", "claim": "A randomized voting rule built on a new random-size stable lottery achieves metric distortion 2.3282, past a 2.5 barrier that existing arguments could not cross.", "field": "computer-science", "date": "2026-08-29", "lab": "Independent", "model": "GPT-5.6 Sol; Claude Opus 5.0", "humans": [ "Nisarg Shah" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "In metric social choice each voter ranks candidates by distance in an unknown metric space, a candidate's cost is its average distance to voters, and a randomized rule's distortion is the worst-case ratio between the expected cost of its lottery and that of the best candidate. Charikar, Ramakrishnan, Wang and Wu proved an upper bound of 2.753, and Frank and Ye independently improved it to 2.5 using an equal mixture of maximal lottery and Integrated Veto; existing arguments yield nothing better from any mixture of those two rules. The paper breaks that barrier with a new ingredient, a random-size stable lottery, reaching 2.3282.", "novelty_check": "The baseline chain is named and dated in the paper: 2.753 from Charikar, Ramakrishnan, Wang and Wu in JACM 2024, then 2.5 independently by Frank (arXiv:2608.17863) and Ye (arXiv:2608.21202) weeks before this work. The paper also states why 2.5 was a barrier rather than merely the current best, namely that no mixture of the two rules involved yields a better bound, which is what the new lottery is introduced to get past. No stronger bound appears.", "caveats": "A preprint days old at entry, unrefereed, not formalized, with no independent check on record, and vibemathed records it as a partial result since the optimal distortion remains unknown. Autonomy is graded ai-led on a disclosure that describes the model producing the key ideas under human guidance: all proofs in the document were obtained using GPT-5.6 Sol with guidance from the author, who verified them for correctness and expanded and simplified the exposition with the aid of GPT-5.6 Sol and Claude Opus 5. The author records two specific steps where the model autonomously identified the ingredient to add, first a stable lottery and then a lottery whose existence it proved by a minimax argument.", "registrations": [ { "registry": "vibemathed", "id": "improving-randomized-metric-distortion-to-2-3282", "url": "https://vibemathed.com/problem/improving-randomized-metric-distortion-to-2-3282" } ], "sources": [ { "label": "arXiv: improving randomized metric distortion to 2.3282", "url": "https://arxiv.org/abs/2608.29308", "kind": "research" }, { "label": "vibemathed: improving randomized metric distortion to 2.3282", "url": "https://vibemathed.com/problem/improving-randomized-metric-distortion-to-2-3282", "kind": "commentary" } ], "tags": [ "algorithms", "social-choice", "game-theory", "argument" ], "added": "2026-09-03" }, { "id": "2026-08-27-intuitionistic-higher-order-truth", "title": "Not every Heyting algebra is the subterminal lattice of a topos", "claim": "The free Heyting algebra on two generators cannot be the lattice of subterminal objects of an elementary topos, answering the question in the negative.", "field": "mathematics", "date": "2026-08-27", "lab": "Independent", "model": "ChatGPT 5.6 Sol", "humans": [ "Lingyuan Ye", "Yiqi Xu" ], "verification": "author-verified", "autonomy": "ai-assisted", "detail": "The question is whether every Heyting algebra arises as the lattice of subterminal objects of an elementary topos, which would say that intuitionistic propositional logic is fully realised by higher-order truth. The answer is no, and the witness is concrete: the free Heyting algebra on two generators cannot be such a lattice.", "novelty_check": "The question is a clean yes-or-no about a standard construction relating Heyting algebras to elementary toposes, and the paper answers it with a specific named algebra rather than an abstract obstruction, which makes the counterexample checkable. vibemathed records no attributed poser or year, so the entry carries no year_posed rather than guessing one. No prior negative answer appears in the categorical logic literature cited.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-assisted, the weakest reading the disclosure supports, because the paper claims help rather than authorship of the ideas: the mathematical results in the document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by the authors, who take full responsibility for its contents. The disclosure does not separate which results came from the model.", "registrations": [ { "registry": "vibemathed", "id": "failure-of-higher-order-truth-within-intuitionistic-propositional-logic", "url": "https://vibemathed.com/problem/failure-of-higher-order-truth-within-intuitionistic-propositional-logic" } ], "sources": [ { "label": "arXiv: failure of higher-order truth within intuitionistic propositional logic", "url": "https://arxiv.org/abs/2608.26874", "kind": "research" }, { "label": "vibemathed: failure of higher-order truth within intuitionistic propositional logic", "url": "https://vibemathed.com/problem/failure-of-higher-order-truth-within-intuitionistic-propositional-logic", "kind": "commentary" } ], "tags": [ "logic", "category-theory", "counterexample" ], "added": "2026-09-03" }, { "id": "2026-08-21-scl-relator-invariance", "title": "Stable commutator length of a relator is not a one-relator group invariant", "claim": "Two explicit words give isomorphic one-relator groups whose relators have different stable commutator lengths, answering a question of Heuer and Loh in the negative.", "field": "mathematics", "date": "2026-08-21", "lab": "Independent", "model": "Claude Opus 5; Harmonic Aristotle", "humans": [ "Artem Semidetnov" ], "year_posed": 2019, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "For the two explicit words \\(r = aabABabABBAbaabABBAb\\) and \\(r' = aabABabABabABBAbaBAb\\), both lying in the commutator subgroup of the free group on \\(a\\) and \\(b\\), the one-relator groups presented by \\(r = 1\\) and by \\(r' = 1\\) are isomorphic, while \\(r\\) and \\(r'\\) have different stable commutator lengths. That combination is exactly the negative answer to the question of Heuer and Loh, since it shows the stable commutator length of the relator is not determined by the group it presents.", "novelty_check": "The question is attributed to Heuer and Loh and dated 2019 in the vibemathed record, and the counterexample is a pair of concrete words whose properties can be checked directly rather than an existence argument. The repository accompanying the paper reproduces the computation, so both halves of the claim, the isomorphism and the difference in stable commutator length, are independently re-runnable in principle. No prior counterexample appears.", "caveats": "A short preprint, unrefereed, with no independent check on record. The formalization is partial and does not cover the result: the repository contains a Lean formalisation of the six identities used in the argument, produced with the assistance of Harmonic's Aristotle, not of the full claim, so the grade stays at author-verified rather than formal. Autonomy is graded ai-assisted, the weaker of the two readings the paper's own wording admits, since the body says Claude Desktop with Opus 5 designed and orchestrated an exhaustive search through orbits of words of bounded length while the acknowledgement says it assisted in designing and orchestrating that search.", "registrations": [ { "registry": "vibemathed", "id": "the-stable-commutator-length-of-a-relator-is-not-a-one-relator-group-invariant", "url": "https://vibemathed.com/problem/the-stable-commutator-length-of-a-relator-is-not-a-one-relator-group-invariant" } ], "sources": [ { "label": "arXiv: the stable commutator length of a relator is not a one-relator group invariant", "url": "https://arxiv.org/abs/2608.21465", "kind": "research" }, { "label": "vibemathed: the stable commutator length of a relator is not a one-relator group invariant", "url": "https://vibemathed.com/problem/the-stable-commutator-length-of-a-relator-is-not-a-one-relator-group-invariant", "kind": "commentary" } ], "tags": [ "group-theory", "geometric-group-theory", "counterexample", "computer-assisted" ], "added": "2026-09-03" }, { "id": "2026-08-21-bounded-mass-property", "title": "Counterexample to the bounded mass property on the Hopf threefold", "claim": "The bounded mass property fails on the Hopf threefold, answering a question of Boucksom, Guedj and Lu.", "field": "mathematics", "date": "2026-08-21", "lab": "Independent", "model": "Rethlas agent (GPT-5.6 Sol)", "humans": [ "Mingchen Xia", "Kewei Zhang" ], "year_posed": 2025, "verification": "author-verified", "autonomy": "ai-led", "detail": "A compact complex manifold has the bounded mass property if, for one and equivalently every Hermitian form, the masses of the \\(n\\)-th power of that form plus a \\(dd^c\\)-exact perturbation stay uniformly bounded over all smooth potentials keeping the form positive. The paper shows this fails on the Hopf threefold, the quotient of punctured complex three-space by the map sending \\(z\\) to \\(e^{-1}z\\), which answers a question of Boucksom, Guedj and Lu in the negative.", "novelty_check": "The question is recent and attributed: Boucksom, Guedj and Lu posed it in 2025, so the window in which a prior answer could exist is short and the paper names the source directly. The counterexample is a single named manifold, the Hopf threefold, rather than an abstract existence claim, which makes it checkable against the definition. No prior counterexample or positive resolution appears.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-led on a one-sentence disclosure that credits the construction itself to an agent rather than to assistance: the initial counterexample was constructed with the Rethlas agent, improved by Felix Ye, using the gpt-5.6-sol model. The disclosure covers the initial counterexample only and does not say how much of the final write-up is the authors' own.", "registrations": [ { "registry": "vibemathed", "id": "bounded-mass-property-for-compact-complex-manifolds", "url": "https://vibemathed.com/problem/bounded-mass-property-for-compact-complex-manifolds" } ], "sources": [ { "label": "arXiv: a counterexample to the bounded mass property", "url": "https://arxiv.org/abs/2608.21053", "kind": "research" }, { "label": "vibemathed: bounded mass property for compact complex manifolds", "url": "https://vibemathed.com/problem/bounded-mass-property-for-compact-complex-manifolds", "kind": "commentary" } ], "tags": [ "geometry", "complex-geometry", "counterexample" ], "added": "2026-09-03" }, { "id": "2026-08-20-pauli-fractional-colouring", "title": "Counterexamples to the fractional colouring conjecture for Pauli shadow tomography", "claim": "The fractional chromatic number of the anticommutation graph of large-expectation Pauli observables is not bounded by a constant over \\(\\varepsilon^2\\), removing a proposed route to triply efficient shadow tomography.", "field": "physics", "date": "2026-08-20", "lab": "Independent", "model": "GPT-5.6 Sol", "humans": [ "Jędrzej Stempin", "Santiago Llorens", "Felix Huber" ], "year_posed": 2025, "verification": "author-verified", "autonomy": "collaborative", "detail": "Fractional graph colourings are used in shadow tomography of Pauli observables, and it is desirable that any experimentally interesting set of Pauli operators have a small fractional chromatic number for its anticommutation graph. Conjecture 13 of King, Gosset, Kothari and Babbush states that for the set of Pauli observables whose expectation value in a given state has magnitude at least \\(\\varepsilon\\), the fractional chromatic number of the induced anticommutation graph is \\(O(\\varepsilon^{-2})\\). The paper gives counterexamples, so no such constant exists; had the conjecture held it would have implied a triply efficient Pauli shadow tomography algorithm for any subset of Pauli observables given an efficient fractional colouring algorithm.", "novelty_check": "The conjecture is pinned to a specific numbered statement in a specific published paper, Conjecture 13 of King, Gosset, Kothari and Babbush in PRX Quantum 6, 010336 (2025), which leaves no ambiguity about what is being refuted or when it was posed. The consequence that fails with it, a triply efficient algorithm for arbitrary Pauli subsets, is stated in the same source. No prior counterexample appears.", "caveats": "A preprint, unrefereed, not formalized, with no independent check on record, and vibemathed lists it as a candidate pending review. Autonomy is graded collaborative rather than ai-led because the disclosure is a single sentence with no account of the division of labour between the model and the three authors, and the weaker grade wins when the reading is arguable. The sentence, in full, is: GPT Sol 5.6 was used to derive the main results (Theorem A and Theorem B) of this paper.", "registrations": [ { "registry": "vibemathed", "id": "fractional-colouring-pauli-shadow-tomography", "url": "https://vibemathed.com/problem/fractional-colouring-pauli-shadow-tomography" } ], "sources": [ { "label": "arXiv: counterexamples to the fractional coloring conjecture for triply efficient shadow tomography", "url": "https://arxiv.org/abs/2608.20113", "kind": "research" }, { "label": "vibemathed: the fractional colouring conjecture for triply efficient Pauli shadow tomography", "url": "https://vibemathed.com/problem/fractional-colouring-pauli-shadow-tomography", "kind": "commentary" } ], "tags": [ "quantum-information", "combinatorics", "counterexample" ], "added": "2026-09-03" }, { "id": "2026-08-24-ancheta-massey-linear-coding", "title": "Optimal linear encoding rate for lossy compression of Bernoulli sources", "claim": "Ancheta's answer to Massey's question about linear encoders for lossy compression is extended from the fair-coin case to every source bias below one half.", "field": "computer-science", "date": "2026-08-24", "lab": "Independent", "model": "GPT-5.6 Sol", "humans": [ "Yihong Wu" ], "year_posed": 1978, "verification": "author-verified", "autonomy": "ai-led", "detail": "Linear encoders achieve the entropy for lossless compression of a Bernoulli source, but for lossy compression linearity is known to be strictly suboptimal against the rate-distortion function. Massey asked in 1978 whether the optimal rate for linear encoding is achieved simply by compressing a fraction of the bits linearly and losslessly and estimating the rest by zero. Ancheta answered yes for bias one half in the same year, and this note extends that to every bias below one half. The key step bounds the entropy of the posterior distribution conditioned on an affine subspace in terms of its marginals.", "novelty_check": "The question and its partial answer are both named, dated and cited: Massey posed it in 1978 and Ancheta settled the \\(p = 1/2\\) case that same year, so what remained was every other bias. The note also records, as a correction to its own novelty, that a literature search aided by Codex found several proof ingredients already present in or deducible from prior work, and devotes a section to that discussion. That is a partial-novelty flag surfaced by the author rather than by a later reader.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record. The novelty is qualified by the author's own hindsight statement that several ingredients of the proof have appeared in, or can be deduced from, the prior literature, discussed in the note's Section 4.1; what is claimed as new is the extension itself and a simplified self-contained presentation. Autonomy is graded ai-led on the declaration of AI use, which credits discovery to the model and simplification to the author: the first version of the proof was discovered by GPT-5.6 Sol in an interactive process guided by the author, who subsequently simplified the program and developed the self-contained proof presented in the note.", "registrations": [ { "registry": "vibemathed", "id": "entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a", "url": "https://vibemathed.com/problem/entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a" } ], "sources": [ { "label": "arXiv: entropy of Bernoulli measures conditioned on affine subspaces and a problem of Ancheta-Massey", "url": "https://arxiv.org/abs/2608.22837", "kind": "research" }, { "label": "vibemathed: entropy of Bernoulli measures conditioned on affine subspaces", "url": "https://vibemathed.com/problem/entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a", "kind": "commentary" } ], "tags": [ "information-theory", "coding-theory", "argument" ], "added": "2026-09-03" }, { "id": "2026-08-25-froberg-quintics-septics", "title": "Fröberg's conjecture for quintics and septics in four variables", "claim": "Fröberg's predicted Hilbert series is proved for ideals generated by any number of general forms of degree five or seven in four variables.", "field": "mathematics", "date": "2026-08-25", "lab": "Independent", "model": "GPT-5.6 Sol; Claude Fable 5; Grok 4.6", "humans": [ "Qihang Wang", "Dongming Zhang" ], "year_posed": 1985, "verification": "author-verified", "autonomy": "collaborative", "detail": "Over a field of characteristic zero in four variables, the paper proves Fröberg's predicted Hilbert series for ideals generated by \\(r\\) general forms of equal degree \\(d\\), for every \\(r\\) at least 1, in the two cases \\(d = 5\\) and \\(d = 7\\). Against the classical cases \\(r\\) at most 5 and the equal-degree theorem of Boij, Dannetun and Lundqvist through degree \\(d + 2\\), the ranges needing new input are \\(r\\) between 6 and 11 for quintics and 6 and 21 for septics. Each slice reduces to finitely many endpoint ranks of Macaulay multiplication matrices: ten exact endpoint computations from twenty-one sparse forms for quintics, and fifteen from a nested family of 120 integral forms for septics, with every endpoint certificate recording an explicit maximal minor that is nonzero modulo 2 and hence a nonzero integer.", "novelty_check": "The paper states exactly which generator-count ranges were already covered and by what, the classical cases r at most 5 and the Boij-Dannetun-Lundqvist equal-degree theorem through degree d + 2, and therefore which ranges required new input. That makes the contribution a bounded and stated delta rather than a broad claim on Fröberg's conjecture, which remains open in general. No prior treatment of these ranges appears.", "caveats": "A preprint, unrefereed, with no independent check on record, and vibemathed records the resolution as partial: this is two degrees in four variables, not Fröberg's conjecture. The certificates are explicit and the nonzero-modulo-2 minors make the endpoint computations checkable, but nothing is machine-checked, so the grade stays at author-verified. Autonomy is graded collaborative on a broad disclosure of automated assistance that spans the whole pipeline without separating human from model contribution: the workflow used GPT-5.6 Sol, Claude Fable 5 and Grok 4.6 for the formulation of mathematical ideas, generation of conjectures and proof strategies, derivation and checking of intermediate steps, construction of examples and exact certificates, comparison of literature and candidate proof approaches, and organization of arguments.", "independent_checks": [ { "who": "vibemathed.com recomputation", "url": "https://vibemathed.com/problem/froberg-s-conjecture-for-quintics-and-septics-in-four-variables", "outcome": "Rebuilt the quintic Macaulay matrices from the forms printed in the paper and reproduced all ten rows of Table 2 over three fields with its own elimination, confirmed each rank gives Fröberg's predicted dimension, and ran the authors' verifiers, which fail when an input is altered. The reduction of each range to its endpoints was not checked, and the septic forms were checked only through the authors' verifier." } ], "registrations": [ { "registry": "vibemathed", "id": "froberg-s-conjecture-for-quintics-and-septics-in-four-variables", "url": "https://vibemathed.com/problem/froberg-s-conjecture-for-quintics-and-septics-in-four-variables" } ], "sources": [ { "label": "arXiv: Fröberg's conjecture for quintics and septics in four variables", "url": "https://arxiv.org/abs/2608.24797", "kind": "research" }, { "label": "vibemathed: Fröberg's conjecture for quintics and septics in four variables", "url": "https://vibemathed.com/problem/froberg-s-conjecture-for-quintics-and-septics-in-four-variables", "kind": "commentary" } ], "tags": [ "algebra", "commutative-algebra", "computer-assisted" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's independent recomputation of the quintic certificates and its run of the septic verifier. The endpoint reduction is unchecked, so no grade moved.", "url": "https://vibemathed.com/problem/froberg-s-conjecture-for-quintics-and-septics-in-four-variables" } ] }, { "id": "2026-08-19-erdos-501-independence", "title": "Erdős Problem #501 shown independent of ZFC, with both directions in Lean", "claim": "The first question of Erdős Problem #501 is proved independent of ZFC, and both directions are formalized in Lean 4 as statements about the theory ZFC itself.", "field": "mathematics", "date": "2026-08-19", "lab": "Independent", "model": "Sol; Claude", "humans": [ "Elliot Glazer" ], "year_posed": 1961, "verification": "formal", "autonomy": "collaborative", "detail": "Erdős Problem #501 asks, for a family of bounded sets of outer measure less than 1 indexed by the reals, whether there must be an infinite independent set, meaning an infinite set of reals none of which lies in the set indexed by another. The answer is independent of ZFC: the continuum hypothesis gives a counterexample by Hechler's 1972 construction, while adding more than continuum-many random reals gives a positive answer. Both directions are formalized, and the independence is stated inside Mathlib's own first-order logic, with the theory ZFC and a sentence Erdős501 defined in a file importing Mathlib only, so that non-provability and non-refutability are themselves proved from the standard axioms. A further target certifies the rendering is faithful by showing the sentence is equivalent in Mathlib's ZFSet to the formalized statement of the question.", "novelty_check": "The problem is tracked at erdosproblems.com/501 and traced in the repository to Erdős 1961, Problem II.9 and Erdős-Hajnal 1971. The repository checks its own novelty claim against the database directly: in a snapshot of teorth/erdosproblems dated 2026-08-17 covering 1217 problems, the problems already marked independent, not provable or not disprovable all carry formal_status unformalized, while #501 is listed as open. That is the basis for the repository's claim that this is the first Erdős problem whose resolution is a formally verified independence result. The second question was settled by Newelski, Pawlikowski and Seredyński in 1987 and is formalized here without the boundedness hypothesis.", "caveats": "Announced through a repository rather than a paper, and unrefereed. The credit is shared and mostly human, which is why autonomy is graded collaborative rather than higher: Hechler supplied one direction in 1972, Newelski, Pawlikowski and Seredyński settled the second question in 1987, Sungchul Lee derived a positive answer from a real-valued measurable cardinal with assistance from GPT-5.5 Pro, and Nat Sothanaphan observed that the two halves together give independence. What Glazer and Sol added is the removal of the large cardinal assumption. The formal grade covers the Lean development, which builds under CI and proves its targets from the standard axioms; the Lean statements were audited against the informal problem by vibemathed, which checked the faithfulness target against Formal Conjectures' statement of the problem and found the same predicates in the same shape.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/erdos-501-infinite-independent-sets", "outcome": "Checked the repository's faithfulness theorem against FormalConjectures/ErdosProblems/501.lean and found the same three predicates in the same shape, anchoring the statement in a community-reviewed repository; confirmed no sorry outside the challenge files and only the standard axioms in the shipped audit." }, { "who": "erdosproblems.com", "url": "https://www.erdosproblems.com/501", "outcome": "The canonical tracker records that Glazer, using GPT 5.6 Sol, Fable 5 and Opus 4.8, proved the first question independent of ZFC." } ], "registrations": [ { "registry": "vibemathed", "id": "erdos-501-infinite-independent-sets", "url": "https://vibemathed.com/problem/erdos-501-infinite-independent-sets" } ], "sources": [ { "label": "GitHub: Erdős Problem #501 in Lean 4", "url": "https://github.com/ElliotGlazer/erdos501", "kind": "research" }, { "label": "Erdős Problems: problem #501", "url": "https://www.erdosproblems.com/501", "kind": "commentary" }, { "label": "vibemathed: Erdős Problem #501, infinite independent sets", "url": "https://vibemathed.com/problem/erdos-501-infinite-independent-sets", "kind": "commentary" } ], "tags": [ "set-theory", "erdos", "lean", "formalization", "forcing" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's audit anchoring the Lean statement to Formal Conjectures, and erdosproblems.com's record of the independence result. No grade moved.", "url": "https://vibemathed.com/problem/erdos-501-infinite-independent-sets" } ] }, { "id": "2026-08-22-erdos-270-affine-transcendence", "title": "Transcendence in the affine case of Erdős Problem 270", "claim": "The series in Erdős Problem 270 is claimed transcendental for every positive integer-valued affine choice of the defining function.", "field": "mathematics", "date": "2026-08-22", "lab": "Independent", "model": "GPT-5.6 Sol (Codex)", "year_posed": 1980, "verification": "claimed", "autonomy": "ai-led", "detail": "Erdős Problem 270 concerns the series whose \\(n\\)-th term is 1 over the product of the \\(f(n)\\) consecutive integers starting at \\(n+1\\). For the affine choice \\(f(n) = an + b\\) this is the sum of \\(n!/((a+1)n + b)!\\), and the manuscript claims that constant is transcendental for every \\(a\\) at least 1 and \\(b\\) at least \\(1 - a\\), settling the positive integer-valued affine subclass. It also determines the algebraic structure of the family at each fixed slope, and ships a Lean development covering part of the argument.", "novelty_check": "Two pieces of prior work bound what is left to prove, and both are recorded in the repository rather than found later. Problem 270 as Erdős and Graham posed it, for every f tending to infinity, was already answered in the negative by Crmarić and Kovac in 2025, who showed some such f makes the series sum to any prescribed value; what survives is the non-decreasing case, inside which the affine family sits. Separately, the short irrationality proof for the base constant is Crmarić and Kovac's, posted by Kovac on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows from known results. So the checkable parts were already known and the new content is the affine extension.", "caveats": "Announced through a repository rather than a paper, unrefereed, with no independent check on record, and vibemathed lists it as a candidate pending review. The Lean development is explicitly partial: the repository separates what is fully verified from what is not yet verified, so the transcendence claim as a whole is not machine-checked and the grade stays at claimed. The novelty is narrow for the reason given above, since the pieces a reader can most easily check are the ones already in the literature. Autonomy is graded ai-led on the repository's own AI disclosure, which is unusually sweeping: generative AI performed nearly all of the mathematical work, independently rediscovering the elementary denominator argument, developing the affine extensions and the transcendence-theoretic arguments, conducting the literature search, writing the manuscript and producing the Lean formalization, with the repository owner directing and supervising. No human author is named on the work.", "registrations": [ { "registry": "vibemathed", "id": "transcendence-in-the-affine-case-of-erdos-problem-270", "url": "https://vibemathed.com/problem/transcendence-in-the-affine-case-of-erdos-problem-270" } ], "sources": [ { "label": "GitHub: algebraic independence in the affine case of Erdős Problem 270", "url": "https://github.com/clambro/erdos-270-transcendence", "kind": "research" }, { "label": "Erdős Problems: problem #270", "url": "https://www.erdosproblems.com/270", "kind": "commentary" }, { "label": "vibemathed: transcendence in the affine case of Erdős Problem 270", "url": "https://vibemathed.com/problem/transcendence-in-the-affine-case-of-erdos-problem-270", "kind": "commentary" } ], "tags": [ "number-theory", "erdos", "transcendence", "lean" ], "added": "2026-09-03" }, { "id": "2026-08-06-kreiss-constant-separation", "title": "Separation between the ordinary and strong Kreiss constants", "claim": "A negative solution to the inverse generator problem on Hilbert spaces yields an operator whose Cayley transforms satisfy the ordinary Kreiss condition but are neither strongly Kreiss bounded nor power bounded.", "field": "mathematics", "date": "2026-08-06", "lab": "Independent", "model": "ChatGPT 5.6 Pro; Claude Fable", "humans": [ "Emiel Lorist", "Martin Meyries", "Mark Veraar" ], "year_posed": 2025, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "The paper constructs a bounded operator with dense range on a Hilbert space that generates a bounded, strongly stable C0-semigroup while its inverse generates no C0-semigroup, which answers the inverse generator problem in the negative. It also builds an exponentially stable generator with 0 in the resolvent set whose inverse semigroup is unbounded and grows at least double logarithmically; for that generator every Cayley transform satisfies the ordinary Kreiss resolvent condition but is neither strongly Kreiss bounded nor power bounded, which separates the two notions. A numerical consequence follows: the Crank-Nicolson scheme is unstable in operator norm both for every fixed step size over long times and under mesh refinement at any fixed final time. All counterexamples are deduced from a common finite-dimensional construction.", "novelty_check": "The separation question between ordinary and strong Kreiss constants is attributed by vibemathed to Chalmoukis and Tsikalas and dated 2025, so the window for a prior answer is short. The inverse generator problem is the older and better-known target, and the paper answers it in the negative on Hilbert spaces, the setting where it was sharpest. Both claims descend from one explicit finite-dimensional construction whose entries are given by explicit formulas, so the construction can be inspected directly. No prior counterexample appears.", "caveats": "A preprint, unrefereed, with no independent check on record; it has been revised, with v1 posted 2026-08-06 and v3 on 2026-08-26, and the date here is the first public posting. The formalization is partial and covers the finite-dimensional core rather than the claims of the entry: the proof of Theorem 1.1 has been formalized in Lean 4 in a public repository, while the semigroup and Kreiss deductions drawn from it are not machine-checked, so the grade stays at author-verified rather than formal. Autonomy is graded ai-assisted on a disclosure that keeps the model in a search and formalization role: ChatGPT 5.6 Pro was used to explore various Schauder basis counterexamples to the inverse generator problem, and the Lean 4 formalization was done using Claude Fable.", "registrations": [ { "registry": "vibemathed", "id": "separation-of-ordinary-and-strong-kreiss-constants", "url": "https://vibemathed.com/problem/separation-of-ordinary-and-strong-kreiss-constants" } ], "sources": [ { "label": "arXiv: a solution to the inverse generator problem and related questions", "url": "https://arxiv.org/abs/2608.06272", "kind": "research" }, { "label": "GitHub: Lean 4 formalization of Theorem 1.1", "url": "https://github.com/zoowirt/inverse-generator-lean", "kind": "research" }, { "label": "vibemathed: separation between the ordinary and strong Kreiss constants", "url": "https://vibemathed.com/problem/separation-of-ordinary-and-strong-kreiss-constants", "kind": "commentary" } ], "tags": [ "analysis", "operator-theory", "counterexample", "lean" ], "added": "2026-09-03" }, { "id": "2026-08-31-boolean-multiplicative-complexity-mul4", "title": "Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication", "claim": "Multiplying two four-term binary polynomials needs exactly nine AND gates even when nonlinear intermediate wires may be reused, with the whole result formalized in Lean 4.", "field": "computer-science", "date": "2026-08-31", "lab": "Independent", "model": "GPT-5.6 Sol; Claude Opus 5", "humans": [ "Gregory Morse" ], "year_posed": 2014, "verification": "formal", "autonomy": "ai-assisted", "detail": "Classical lower bounds show that multiplying two degree-three polynomials over the field with two elements needs nine scalar products in bilinear or quadratic models, but they leave open the unrestricted Boolean case, where an XOR-AND circuit may reuse nonlinear intermediate wires and equality is taken modulo the idempotence relations, so a multiplication can lower algebraic degree. The paper proves that the map producing the seven coefficients of the product of two four-term binary polynomials has unrestricted XOR-AND multiplicative complexity exactly nine. This resolves, for a natural vector-valued quadratic function, the Boyar-Find question of whether a quadratic-circuit lower bound survives unrestricted nonlinear reuse. The proof is structural rather than exhaustive, and the same argument gives multiplicative complexity six for three-term multiplication.", "novelty_check": "The open question is attributed to Boyar and Find and dated 2014, and the paper is explicit about what the classical bounds do and do not settle: they cover bilinear and quadratic models, not unrestricted Boolean circuits with nonlinear reuse. That distinction is the entire content of the contribution and is stated before the result. No prior unrestricted lower bound for this function appears.", "caveats": "A preprint days old at entry, unrefereed, with no independent human check on record, and vibemathed records the resolution as partial since this settles one function rather than the general question. The formal grade is well supported for a self-published result: the Lean 4 formalization covers the Boolean-ANF semantics, the unrestricted circuit model, the seven- and eight-gate exclusions and the explicit nine-gate upper bound, uses no project-specific axiom and no native decision procedure, pins Lean v4.32.1 and a locked mathlib revision, and its continuous integration performs the build, an axiom audit and a declaration replay with a weekly fresh source replay. What is missing is third-party auditing that the formalized statements say what the paper says. Autonomy is graded ai-assisted, matching the disclosed roles, with GPT-5.6 Sol at an extra-high setting and Claude Opus 5 acting as a referee.", "independent_checks": [ { "who": "vibemathed.com rebuild", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4", "outcome": "Rebuilt release n4-arxiv-v2 from source on the pinned toolchain, ran the axiom audit and replayed the whole environment with leanchecker: all six headline theorems use only the three standard axioms, with no sorry, project axiom, native_decide or bv_decide. The statements were not audited against the Boyar-Find question." } ], "registrations": [ { "registry": "vibemathed", "id": "unrestricted-multiplicative-complexity-mul4", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4" }, { "registry": "palomar", "id": "PALOMAR-2026-09-21-000005", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-21-000005&version=1", "repository": "GregoryMorse/unrestricted-boolean-mul", "commit": "56837b8ed4ffe6390a7a84d9a77f1489bfd9d872", "theorems": [ "UnrestrictedBooleanMul.mc_mul_zero", "UnrestrictedBooleanMul.mc_mul_one", "UnrestrictedBooleanMul.mc_mul_two", "UnrestrictedBooleanMul.mc_mul_three", "UnrestrictedBooleanMul.N4.no_eight_gate_circuit", "UnrestrictedBooleanMul.N4.mc_mul_four" ], "checked": "2026-09-21", "note": "Registered by the author. Palomar's record states that the open unrestricted and bilinear lower bounds for five and six terms are not claims of this submission." } ], "sources": [ { "label": "arXiv: unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication", "url": "https://arxiv.org/abs/2608.30238", "kind": "research" }, { "label": "vibemathed: unrestricted Boolean multiplicative complexity of Mul4", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4", "kind": "commentary" } ], "tags": [ "circuit-complexity", "algorithms", "lean", "formalization" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's rebuild of the Lean development from source, which confirms the axiom audit. It checks the build, not whether the statements match the paper, so no grade moved.", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4" }, { "date": "2026-09-24", "kind": "correction", "note": "Recorded the author's Palomar registration, PALOMAR-2026-09-21-000005, alongside the vibemathed record.", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-21-000005&version=1" } ] }, { "id": "2026-08-27-entanglement-supporting-functional", "title": "Supporting affine functionals for entanglement of formation need not exist", "claim": "An explicit two-qubit state has no global supporting affine functional for the entanglement of formation, contradicting an assumption used in several papers.", "field": "physics", "date": "2026-08-27", "lab": "Independent", "model": "Claude Fable 5", "humans": [ "A. S. Holevo", "M. E. Shirokov" ], "verification": "author-verified", "autonomy": "ai-assisted", "detail": "Several articles assume that the convex roof structure of the entanglement of formation together with finite-dimensional subsystems guarantees a global supporting affine functional at every state, meaning a Hermitian operator whose trace against the state gives the entanglement of formation and which lower-bounds it everywhere else. The paper gives an explicit counterexample at a degenerate state, in the simplest case where both subsystems are qubits. The construction rests on the equivalence between existence of such a supporting functional at a state and Lipschitz lower semicontinuity of the entanglement of formation there, and uses Wootters' closed formula to locate the state.", "novelty_check": "The target is not a named conjecture but an assumption in circulation, which the paper identifies as appearing in several articles by the authors themselves among others, so the thing being corrected is stated before it is refuted. The counterexample is at the smallest possible size, two qubits, where Wootters' formula gives the entanglement of formation in closed form and the claim can therefore be checked directly rather than argued asymptotically. No prior counterexample appears.", "caveats": "A short preprint, unrefereed, not formalized, with no independent check on record. Autonomy is graded ai-assisted, the weakest reading the disclosures support, because every mention of the model in the paper is of assistance in a search step inside a human-built argument: the authors use Wootters' formula and the help of Claude Fable 5 to find a state where the property fails, note that this is done very quickly with the model, and attribute one crucial corollary to work obtained with its help. The equivalence with Lipschitz lower semicontinuity, which is what makes the search possible, is the authors' own.", "independent_checks": [ { "who": "vibemathed.com re-derivation", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation", "outcome": "Implemented Wootters' concurrence from scratch and re-derived the counterexample in exact arithmetic from the statement rather than the paper's method: the entanglement of formation fails to be Lipschitz lower semicontinuous at the stated state, the failure the paper claims. The paper's equivalence between that property and the existence of a supporting affine functional was not checked." } ], "registrations": [ { "registry": "vibemathed", "id": "supporting-affine-functionals-for-entanglement-of-formation", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation" } ], "sources": [ { "label": "arXiv: on supporting affine functionals for entanglement of formation", "url": "https://arxiv.org/abs/2608.27363", "kind": "research" }, { "label": "vibemathed: supporting affine functionals for entanglement of formation", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation", "kind": "commentary" } ], "tags": [ "quantum-information", "entanglement", "counterexample" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's exact-arithmetic re-derivation of the counterexample state. The equivalence it relies on is the paper's own and unchecked, so no grade moved.", "url": "https://vibemathed.com/problem/supporting-affine-functionals-for-entanglement-of-formation" } ] }, { "id": "2026-08-19-delavina-waller-wiener", "title": "The DeLaViña-Waller conjecture on the Wiener index", "claim": "Every connected graph on \\(2d+1\\) vertices of diameter \\(d\\) at least 3 is claimed to have Wiener index at most that of the cycle on the same number of vertices, with the equality cases determined.", "field": "mathematics", "date": "2026-08-19", "lab": "Independent", "model": "GPT-5.6 Sol; Claude Fable 5", "humans": [ "Mingchang Liu" ], "year_posed": 2008, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "The conjecture of DeLaViña and Waller states that a finite simple connected graph on \\(2d+1\\) vertices with diameter exactly \\(d\\) at least 3 has Wiener index, the sum of all pairwise distances, at most that of the odd cycle on \\(2d+1\\) vertices. The preprint claims a proof and pins down the equality cases: the odd cycle for every \\(d\\) at least 3, plus the double star \\(D(2, 3)\\) when \\(d = 3\\) and the nine-vertex tree \\(S(2, 3, 3)\\) when \\(d = 4\\).", "novelty_check": "The conjecture is named, attributed to DeLaViña and Waller and dated 2008 in the vibemathed record, which is where this candidate is carried. The claim is sharpened rather than merely asserted, since the equality cases are enumerated and include two sporadic small-diameter graphs alongside the cycle, and those exceptions are the sort of detail a re-derivation can check. No prior proof appears in that record.", "caveats": "A Zenodo preprint, unrefereed, not formalized, with no independent check on record, and vibemathed lists it as a candidate pending review with a low significance rating. The Zenodo metadata carries no abstract, so the statement here is reconstructed from the registry record rather than from the deposit's own description, and a reader should open the deposit before relying on it. Autonomy is graded ai-assisted on the only account of the model's role available, which is the registry's one-line summary that the models assisted in developing proof strategies and checking computations; no AI disclosure written by the author has been located.", "registrations": [ { "registry": "vibemathed", "id": "the-delavina-waller-conjecture-on-the-wiener-index", "url": "https://vibemathed.com/problem/the-delavina-waller-conjecture-on-the-wiener-index" }, { "registry": "mathdb", "id": "374919", "title": "DeLaViña-Waller conjecture on the Wiener index", "url": "https://mathdb.com/p/374919/delavina-waller-conjecture-on-the-wiener-index-of-graphs-of", "note": "A background page for the conjecture itself. It does not mention the preprint recorded here and makes no reference to AI involvement, so it establishes the problem's standing in the literature and nothing about this result." } ], "sources": [ { "label": "Zenodo: the DeLaViña-Waller conjecture on the Wiener index", "url": "https://doi.org/10.5281/zenodo.22015517", "kind": "research" }, { "label": "vibemathed: the DeLaViña-Waller conjecture on the Wiener index", "url": "https://vibemathed.com/problem/the-delavina-waller-conjecture-on-the-wiener-index", "kind": "commentary" } ], "tags": [ "graph-theory", "combinatorics", "argument" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-10", "kind": "correction", "note": "Recorded the MathDB record for this conjecture as a registration. It is the background page for the problem and does not mention this preprint, so no grade moved.", "url": "https://mathdb.com/p/374919/delavina-waller-conjecture-on-the-wiener-index-of-graphs-of" } ] }, { "id": "2026-08-19-kasami-apn-triple-counts", "title": "Partial proof of the Kasami APN triple-count conjecture, verified in Lean", "claim": "A conjecture from a 2019 cryptography olympiad on triple counts for the Kasami almost perfect nonlinear function is proved in four residue cases and verified exhaustively for small field degrees, with every proof machine-checked.", "field": "mathematics", "date": "2026-08-19", "lab": "Independent", "model": "Claude Fable 5; Harmonic Aristotle", "humans": [ "Gábor P. Nagy", "Attila Vajda" ], "year_posed": 2019, "verification": "formal", "autonomy": "ai-led", "detail": "For the Kasami almost perfect nonlinear function on a binary field, the conjecture concerns a set built from the function's second difference and asserts that, for any two distinct nonzero elements, the number of triples in that set satisfying a fixed linear relation is exactly \\(2^{2n-3}\\). The paper proves this when \\(k\\) modulo \\(n\\) lies in the set \\(\\{1, 2, n-2, n-1\\}\\), including a complete proof for \\(k = 2\\) via quadratic-form theory and an exact root-count reduction, and verifies the conjecture exhaustively by computer for every admissible pair with \\(n\\) at most 13. It also corrects one hypothesis in the supporting facts supplied with the original problem, concerning the Müller-Cohen-Matthews permutation.", "novelty_check": "The conjecture was proposed at the NSUCRYPTO 2019 cryptographic olympiad, with the proposer not publicly disclosed, and the companion repository supplied with the problem records which supporting facts were already formally verified, so the starting point is unusually well defined. Working from that inventory the authors identify one hypothesis in it as essential and correct it, which is itself evidence the prior material was checked rather than assumed. The result is a partial proof of an open conjecture, not a retrieval.", "caveats": "A preprint, unrefereed, with no independent human check on record, and the result is explicitly partial: four residue classes of k modulo n plus exhaustive verification for n at most 13, not the conjecture. The formal grade rests on the authors' statement that all new results and proofs have been formally verified in Lean 4 by Aristotle from Harmonic, with provenance details in the acknowledgements; vibemathed records the Lean development as checked with its statement unaudited, so nobody outside has confirmed the formalized statements match the paper. Autonomy is graded ai-led on a strong and specific disclosure: every proof in the paper was obtained by prompting Claude Fable 5 with the conjecture statement, background hints and a pointer to the companion repository, and the human authors supplied the framing and the subsequent verification.", "registrations": [ { "registry": "vibemathed", "id": "kasami-apn-function-triple-count-conjecture", "url": "https://vibemathed.com/problem/kasami-apn-function-triple-count-conjecture" }, { "registry": "mathdb", "id": "383508", "title": "Kasami APN function conjecture", "url": "https://mathdb.com/p/383508/kasami-apn-function-conjecture" } ], "sources": [ { "label": "arXiv: on a conjecture on the Kasami APN function", "url": "https://arxiv.org/abs/2608.18584", "kind": "research" }, { "label": "vibemathed: a conjecture on triple counts for the Kasami APN function", "url": "https://vibemathed.com/problem/kasami-apn-function-triple-count-conjecture", "kind": "commentary" } ], "tags": [ "number-theory", "finite-fields", "cryptography", "lean", "formalization" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-15", "kind": "correction", "note": "Recorded the MathDB record for this conjecture as a registration. No grade moved.", "url": "https://mathdb.com/p/383508/kasami-apn-function-conjecture" } ] }, { "id": "2026-08-31-pisot-cantor-equidistribution", "title": "Criteria and two quadratic instances for Bugeaud's Problem 10.61", "claim": "Covering criteria for a 1967 equidistribution problem of Mendès France are proved and applied to two quadratic Pisot cases, with Lean checking; the general problem stays open.", "field": "mathematics", "date": "2026-08-31", "lab": "Independent", "model": "Claude Fable 5; Claude Opus 5", "humans": [ "Ralf Stephan" ], "year_posed": 1967, "verification": "claimed", "autonomy": "ai-led", "detail": "Bugeaud's Problem 10.61, due to Mendès France in 1967, asks whether for a Pisot number \\(\\alpha\\) and the associated Cantor set of base-\\(\\alpha\\) expansions with digits 0 and 1, no point of that Cantor set makes the sequence of its multiples by powers of \\(\\alpha\\) uniformly distributed modulo one. What the repository establishes is narrower than the problem: covering criteria for it, proved only for quadratic setups where \\(\\alpha\\) is a real root of a quadratic whose conjugate has modulus below one, together with two instances of that kind. Material for arbitrary degree is present but conditional, showing for one family that the large real root is Pisot with a conjugate-modulus bound.", "novelty_check": "The repository is explicit that the problem remains open and that what is proved are criteria and instances rather than the general statement, so the novelty question is about the criteria rather than the headline. Every compared statement that concludes Problem 10.61 does so in a quadratic setup, and both worked instances are quadratic, because the covering criterion is proved only there; the arbitrary-degree material does not carry any compared statement above degree two. That boundary is recorded in the registry entry as well as in the repository.", "caveats": "Announced through a repository rather than a paper, unrefereed, with no independent check on record. The grade is claimed rather than formal despite a Lean development, because what is machine-checked covers the quadratic criteria and instances rather than the problem, vibemathed records the Lean work as checked with its statement unaudited, and the covering criterion additionally leaves cases open even within the quadratic range. Anyone citing this should treat Problem 10.61 as open. Autonomy is graded ai-led following the registry's record of the models as the source of the results, with the repository owner supplying framing and publication; no author-written AI disclosure has been located.", "registrations": [ { "registry": "vibemathed", "id": "bugeaud-problem-10-61", "url": "https://vibemathed.com/problem/bugeaud-problem-10-61" }, { "registry": "palomar", "id": "PALOMAR-2026-08-31-000013", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-08-31-000013&version=2", "repository": "rwst/Pisot-Cantor-61", "commit": "d61132ffcdb748b641f9de59b7acfffc860796b2", "theorems": [ "BB61.QuadSetup.equidistributed_iff_exists_invariant_measure", "BB61.QuadSetup.forall_not_equidistributed_iff_exists_trigCertificate", "MeasureTheory.le_integral_of_partitionPressure_le", "BB61.forall_not_equidistributed_of_partitionPressure_lt", "BB61.forall_not_equidistributed_of_transferBound", "BB61.QuadSetup.routeAExponent_mul_hMin", "BB61.slope_neg_iff", "BB61.logb_coverTotal_balancedDepth_le", "BB61.QuadSetup.exists_avoided_interval", "BB61.QuadSetup.not_equidistributed_of_routeAExponent_lt_one", "BB61.QuadSetup.routeAExponent_lt_one_iff_quadratic", "BB61.isPisot_family", "BB61.norm_le_familyConjBound", "BB61.routeA_family_lt_one", "BB61.routeA_two_add_sqrt5", "BB61.two_add_sqrt5_not_equidistributed", "BB61.GapThree.problem_10_61_two_add_sqrt3_axiom_free" ], "checked": "2026-09-06", "note": "What typechecks is the criteria and the two quadratic instances, not Problem 10.61. BB61.QuadSetup.routeAExponent_lt_one_iff_quadratic is where that boundary is visible: the covering criterion is characterized only for quadratic setups, and the arbitrary-degree material carries no compared statement to the problem. Every compared statement depends on propext, Classical.choice and Quot.sound and nothing else. Palomar also carries the submitter's own disclosure that he has not verified informal-to-formal fidelity, and that the challenge file governs wherever the prose disagrees with it." } ], "sources": [ { "label": "GitHub: Pisot-Cantor-61, paper and Lean development", "url": "https://github.com/rwst/Pisot-Cantor-61", "kind": "research" }, { "label": "vibemathed: Bugeaud Problem 10.61", "url": "https://vibemathed.com/problem/bugeaud-problem-10-61", "kind": "commentary" } ], "tags": [ "number-theory", "equidistribution", "lean", "partial-result" ], "added": "2026-09-03", "revisions": [ { "date": "2026-09-07", "kind": "check", "note": "Palomar registered the Lean development as PALOMAR-2026-08-31-000013, mechanically confirming that it typechecks against seventeen stated theorems at a pinned commit. The grade stays at claimed: what is checked is the quadratic criteria and instances, and Problem 10.61 remains open." } ] }, { "id": "2026-09-04-fermat-last-theorem-lean", "title": "Fermat's Last Theorem formalized end to end in Lean 4", "claim": "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.", "field": "mathematics", "date": "2026-09-04", "lab": "Anthropic", "model": "Internal Anthropic research model, described in the announcement as roughly comparable to Claude Fable 5.1", "humans": [ "Tianyi Peng" ], "year_posed": 1637, "verification": "formal", "autonomy": "ai-led", "detail": "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.", "novelty_check": "This is a formalization, not a new theorem, and the entry is graded as such. Wiles and Taylor-Wiles proved Fermat's Last Theorem in 1995; nothing mathematically new is claimed here, and Kevin Buzzard states directly that 'mathematically this work of anthropic tells us essentially nothing' and that 'the formalization just faithfully follows the early literature on the proof and adds nothing'. What is new is the scale of autoformalization: prior partial formalizations existed, notably the ongoing Imperial College FLT project this work builds on, and no complete machine-checked proof of the theorem existed before. It completes the last open item of Wiedijk's list of 100 theorems.", "caveats": "The result is a formalization of an existing proof and should not be read as new mathematics. Verification is graded formal on the machine checks rather than on the announcement: the comparator check ties the proved statement to Mathlib's own, and a second independent kernel replayed it. What no tool establishes, and the repository says so itself, is that each intermediate theorem means what its name suggests, since names are machine-generated and 'where a name and a statement disagree the statement is what was proved'. Buzzard reports the codebase is over 13.4 million lines and that the proof as formalized only works for exponents p at least 17, with smaller cases covered by earlier work. Reproducing the checks is expensive: the build takes hours at 96 jobs with a peak of 153 GB of memory, and the comparator run about 15 hours. Autonomy is graded ai-led rather than autonomous: the announcement describes Claude working 'largely autonomously' but also records high-level human steering from Tianyi Peng, and the strictest defensible reading applies.", "independent_checks": [ { "who": "Kevin Buzzard, Imperial College London, who leads the FLT formalization project this work builds on", "url": "https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-has-beaten-me-to-it/", "outcome": "Ran the comparator and reports the proof checks out. States he 'manually inspected every line of the code base which (according to Claude) was not a mathematical definition or proof of a theorem', looking for soundness exploits, and that random reads of the mathematics were clearly developing FLT the classical way. Judges the mathematical contribution to be nil." } ], "registrations": [ { "registry": "mathdb", "id": "316214", "title": "Fermat's Last Theorem", "url": "https://mathdb.com/p/316214/fermat-s-last-theorem", "note": "Tracks the theorem itself rather than this formalization of it." } ], "sources": [ { "label": "GitHub: anthropics/fermats-last-theorem", "url": "https://github.com/anthropics/fermats-last-theorem", "kind": "research" }, { "label": "Anthropic: Formalizing Fermat's Last Theorem in Lean (PDF)", "url": "https://www-cdn.anthropic.com/9e431dff043da6538d99d6c2d231b670aa3da263.pdf", "kind": "research" }, { "label": "Anthropic: Formalizing Fermat's Last Theorem", "url": "https://www.anthropic.com/research/formalizing-fermats-last-theorem", "kind": "announcement" }, { "label": "Kevin Buzzard: FLT, Anthropic has beaten me to it", "url": "https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-has-beaten-me-to-it/", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: Formalizing Fermat's Last Theorem", "url": "https://news.ycombinator.com/item?id=49568506" }, { "label": "Hacker News: Fermat's Last Theorem in Lean 4", "url": "https://news.ycombinator.com/item?id=49568697" } ], "tags": [ "number-theory", "lean", "formalization", "fermat" ], "added": "2026-09-07", "revisions": [ { "date": "2026-09-15", "kind": "correction", "note": "Recorded the MathDB record for the underlying theorem as a registration. No grade moved.", "url": "https://mathdb.com/p/316214/fermat-s-last-theorem" } ] }, { "id": "2026-08-28-percolation-critical-point", "title": "Lean proof that the percolation probability vanishes at the critical point in every dimension", "claim": "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.", "field": "mathematics", "date": "2026-08-28", "lab": "Anthropic", "model": "Anthropic Claude models, versions not stated", "humans": [ "Justin Leder" ], "verification": "claimed", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The result was known for dimension two by Harris and Kesten and for dimension at least eleven by the lace expansion of Hara and Slade and of Fitzner and van der Hofstad; the repository cites Grimmett's Percolation and Duminil-Copin's 2018 ICM conjecture for the standing of the open cases 3 to 10. The proved statement is Kozma and Nitzan's Conjecture 3, posed in 2024 and open at the time of writing. The repository is explicit about what is not claimed: their Conjectures 1, 2 and 4, anything about slabs, site percolation or other lattices, and continuity of the percolation probability as a formal statement. No competing claim on Conjecture 3 was found in the four registries or on arXiv.", "caveats": "The artifact is not published where it says it is, and that is the first thing to check before citing it. The percolation directory does not exist on the main branch of anthropics/formal-math, whose project table lists only the zeta23 formalization; no branch carries it and the repository's own commit history for that path is empty. It survives only at the pinned commit recorded in the sources here, and the README's own repository link and AUDIT.md both point at a main-branch path that now returns 404. Verification is graded claimed rather than formal despite a passing comparator run, for two reasons: no human has read it, and the statement fidelity question is load-bearing here in a way it is not for a formalization of a known theorem. The README says so itself, that the work 'has not yet been refereed by human mathematicians or by anyone independent of the author; the only review so far was carried out by AI systems', and that readers should check that Challenge.lean states the intended theorem. There is no registration at Palomar. Autonomy is ai-led on the repository's own wording, that the Lean was written by an AI system 'working autonomously under the direction of Justin Leder; no human wrote or edited the Lean code': the direction is human, so the strictest defensible reading is not autonomous.", "independent_checks": [ { "who": "vibemathed.com statement and structure audit", "url": "https://vibemathed.com/problem/absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2", "outcome": "Read Challenge.lean against Grimmett's definitions and found the lattice, the Bernoulli measure, \\(\\theta\\) and \\(p_c\\) formalized as intended; counted no sorry outside the two placeholders, no axiom declarations and no native_decide across 251 files; confirmed the comparator compares exactly the two theorems with the standard axioms. Not rebuilt, and no mathematician has read the argument." } ], "registrations": [ { "registry": "vibemathed", "id": "absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2", "url": "https://vibemathed.com/problem/absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2", "note": "Registered on 2026-09-03 under a different title, against the same pinned commit 795efb86 this entry cites. Rated lean-verified but review pending." } ], "sources": [ { "label": "GitHub: anthropics/formal-math, percolation project at the pinned commit 795efb86", "url": "https://github.com/anthropics/formal-math/tree/795efb86f191735c5481675763537cfb4ff37e55/percolation", "kind": "research" }, { "label": "Audit record for the percolation project", "url": "https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/AUDIT.md", "kind": "research" }, { "label": "arXiv: Kozma and Nitzan, gluing inequalities for percolation", "url": "https://arxiv.org/abs/2401.12397", "kind": "research" } ], "tags": [ "probability", "percolation", "lean", "formalization", "argument" ], "added": "2026-09-07", "revisions": [ { "date": "2026-09-10", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. It was registered under a different title against the same pinned commit already cited here. No grade moved.", "url": "https://vibemathed.com/problem/absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2" }, { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's audit of the formal statement against Grimmett's definitions, which finds it faithful. No human has read the argument and the artifact is still absent from the main branch, so no grade moved.", "url": "https://vibemathed.com/problem/absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2" } ] }, { "id": "2026-09-03-koethe-conjecture-matrix-form", "title": "Lean disproof of Krempa's matrix form of the Koethe conjecture", "claim": "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.", "field": "mathematics", "date": "2026-09-03", "lab": "OpenAI / Epoch AI", "model": "GPT-6 Astra (pre-release)", "humans": [ "Tom Adamczewski" ], "year_posed": 1930, "verification": "formal", "autonomy": "autonomous", "detail": "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.", "novelty_check": "The Koethe conjecture has stood since 1930 and is stated as open in Google DeepMind's Formal Conjectures at the commit the harness pinned, which is what the model was given. Krempa's 1972 equivalences are the standard literature on the problem and are cited in the repository. No prior disproof of the matrix form, and no prior resolution of the conjecture in any of its equivalent forms, was found in the four registries or the surrounding literature. What is new is the counterexample and its Lean proof, not the equivalence, which is classical.", "caveats": "The machine-checked statement is Krempa's matrix form, not Koethe's original question, and the step from one to the other is a 1972 equivalence in the literature rather than part of what was checked. Anyone citing this as a disproof of the Koethe conjecture is relying on that equivalence, and an algebraist should confirm it holds in the form used. The repository is prepared for submission to Palomar but no registration had landed at entry, and no expert in ring theory is on record as having read the construction. The verification grade rests on the mechanical check of the matrix-form statement, which vibemathed independently cloned and confirmed at 3,331 lines with no sorry outside the statement stub and no axiom declarations. Autonomy is graded autonomous on the repository's explicit record that the run was one attempt per statement with 'no human saw or steered the proof search', the only human input being the pre-existing benchmark statement; the repository also discloses that the whole repository was machine-written by AI assistants at Adamczewski's direction.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/kothe-conjecture", "outcome": "From a clone at b052755: the compared statement is byte-identical to Formal Conjectures' general matrix variant at 9cbe1d3c, and Mathlib's TwoSidedIdeal.matrix makes it Krempa's matrix form as intended; no sorry outside the stub, no axiom declarations, no native_decide. The equivalence with Koethe's original statement is not formalized." } ], "registrations": [ { "registry": "vibemathed", "id": "kothe-conjecture", "url": "https://vibemathed.com/problem/kothe-conjecture" } ], "sources": [ { "label": "GitHub: tadamcz/koethe, Lean disproof and Palomar submission repository", "url": "https://github.com/tadamcz/koethe", "kind": "research" }, { "label": "vibemathed: Koethe Conjecture", "url": "https://vibemathed.com/problem/kothe-conjecture", "kind": "commentary" } ], "tags": [ "algebra", "ring-theory", "lean", "formalization", "counterexample", "construction" ], "added": "2026-09-07", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's audit finding the compared statement to be Formal Conjectures' matrix form verbatim. No ring theorist has read it yet, and no grade moved.", "url": "https://vibemathed.com/problem/kothe-conjecture" } ] }, { "id": "2026-09-03-erdos-sos-conjecture", "title": "Erdos-Sos conjecture proved in Lean, in a form marginally weaker than the classical statement", "claim": "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.", "field": "mathematics", "date": "2026-09-03", "lab": "OpenAI / Epoch AI", "model": "GPT-6 Astra (pre-release)", "humans": [ "Tom Adamczewski" ], "year_posed": 1962, "verification": "formal", "autonomy": "autonomous", "detail": "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.", "novelty_check": "The conjecture dates to 1962 and is Erdos Problem 548. The weaker statement with (k-2)n edges is an easy induction, and the repository records that the conjecture was previously proved for large k by Ajtai, Komlos, Simonovits and Szemeredi, work that has not appeared in full. The formalized statement is the benchmark's, stated as open in Formal Conjectures at the pinned commit. No prior full formal proof was found in the four registries. The novelty question here is less about priority than about scope, and the scope limit is recorded in the caveats rather than left implicit.", "caveats": "The formalized statement is very slightly weaker than the classical conjecture, and the repository documents the gap precisely rather than glossing it: when (k-1)n is odd, more than (k-1)n/2 edges is already satisfied with one edge fewer than at least (k-1)n/2 + 1 requires, so in that parity case the compared theorem assumes half an edge more. The classical statement implies the formalized one and not conversely. The repository notes that the proof's internal counting lemma derives the sharp classical bound, but only the weaker advertised form is what the comparator compares, so the machine-checked claim is the weaker one and this entry is worded to it. Thomas Bloom, who maintains erdosproblems.com, has since written a proof exposition that he says should contain enough detail to convince anyone following along, and the site now marks the problem proved; the Palomar submission had not landed at entry. Autonomy is autonomous on the same disclosure as the Koethe entry: one attempt, no human seeing or steering the proof search.", "independent_checks": [ { "who": "Thomas Bloom (erdosproblems.com exposition)", "url": "https://www.erdosproblems.com/548", "outcome": "Wrote a proof exposition of the model's argument, describing it as surprisingly simple and elementary and the sketch as detailed enough to convince a reader following along; erdosproblems.com marks the problem proved (Lean). vibemathed separately records that Bloom reviewed the benchmark statement the Lean proof is compared against." } ], "registrations": [ { "registry": "vibemathed", "id": "erdos-problem-548-erdos-sos-conjecture", "url": "https://vibemathed.com/problem/erdos-problem-548-erdos-sos-conjecture" }, { "registry": "mathdb", "id": "392272", "title": "Erdos-Sos conjecture", "url": "https://mathdb.com/p/392272/erdos-sos-conjecture" } ], "sources": [ { "label": "GitHub: tadamcz/erdos548, Lean proof and Palomar submission repository", "url": "https://github.com/tadamcz/erdos548", "kind": "research" }, { "label": "vibemathed: Erdos Problem 548, the Erdos-Sos conjecture", "url": "https://vibemathed.com/problem/erdos-problem-548-erdos-sos-conjecture", "kind": "commentary" } ], "tags": [ "graph-theory", "combinatorics", "erdos", "lean", "formalization", "argument" ], "added": "2026-09-07", "revisions": [ { "date": "2026-09-15", "kind": "correction", "note": "Recorded the MathDB record for this conjecture as a registration. No grade moved.", "url": "https://mathdb.com/p/392272/erdos-sos-conjecture" }, { "date": "2026-09-24", "kind": "check", "note": "Recorded Thomas Bloom's proof exposition on erdosproblems.com, which now marks the problem proved. No grade moved.", "url": "https://www.erdosproblems.com/548" } ] }, { "id": "2026-09-02-qma2-oracle-separation", "title": "A quantum oracle separating QMA(2) from QMA", "claim": "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.", "field": "computer-science", "date": "2026-09-02", "lab": "MIT / Columbia / University of Washington", "model": "ChatGPT 5.6 Sol", "humans": [ "John Bostanci", "Sabee Grewal", "Jonas Haferkamp", "Andrew Huang", "Yeongwoo Hwang", "Anand Natarajan", "Chinmay Nirkhe" ], "year_posed": 2009, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "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.", "novelty_check": "Whether QMA(2) collapses to QMA has been tracked since Watrous posed the no-disentanglers conjecture, reported by Aaronson, Beigi, Drucker, Fefferman and Shor in 2009. The paper cites the prior partial results it improves on: Harrow, Natarajan and Wu obtained a quasipolynomial lower bound when the error is exponentially small, and Akibue, Kato and Tani proved an exponential lower bound for the more restrictive strong disentanglers. No prior oracle separation was found. The result is new, and is an oracle separation rather than an unrelativized one.", "caveats": "An unrefereed preprint days old at entry, with no formalization and no independent check. The separation is relative to a quantum oracle and does not settle whether QMA(2) equals QMA in the unrelativized world. Autonomy is graded ai-assisted rather than ai-led on the strictest defensible reading of the paper's own account: the authors state that 'the proof idea underlying the main theorem was generated using ChatGPT 5.6 Sol', but also that their initial prompts directed the model to the work of She and Yuen and that they 'subsequently verified, simplified, and developed the argument presented here, and take full responsibility for its correctness'. Humans chose the problem, supplied the key reference and did the development. No transcript has been released, so the phrase 'minimal additional guidance' cannot be checked.", "registrations": [ { "registry": "vibemathed", "id": "a-quantum-oracle-separation-between-mathsf-qma-2-and-mathsf-qma", "url": "https://vibemathed.com/problem/a-quantum-oracle-separation-between-mathsf-qma-2-and-mathsf-qma" } ], "sources": [ { "label": "arXiv: A quantum oracle separation between QMA(2) and QMA", "url": "https://arxiv.org/abs/2609.02865", "kind": "research" }, { "label": "vibemathed: A quantum oracle separation between QMA(2) and QMA", "url": "https://vibemathed.com/problem/a-quantum-oracle-separation-between-mathsf-qma-2-and-mathsf-qma", "kind": "commentary" } ], "tags": [ "quantum-computing", "complexity-theory", "oracle-separation", "argument" ], "added": "2026-09-07" }, { "id": "2026-09-02-daykin-frankl-conjecture", "title": "Confirmation of the Daykin-Frankl conjecture from a language-model proof", "claim": "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.", "field": "mathematics", "date": "2026-09-02", "lab": "Independent", "model": "GPT-5.6 Sol Pro", "humans": [ "Kada Williams" ], "year_posed": 1983, "verification": "author-verified", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The conjecture is named and dated to 1983, due to David E. Daykin and Peter Frankl, and no prior proof was found in the four registries or the surrounding literature on the Boolean lattice. The note presents itself as a first proof rather than a rediscovery, and the strengthened product inequality it proves does not appear in the prior literature located. The result is new.", "caveats": "A four-page unrefereed preprint, checked by its human author and by nobody independent, with no formalization. The human role as stated is verification and communication rather than discovery, which is why autonomy is graded ai-led: the model produced the proof content and the author checked and wrote it up. The model is named in the note's body rather than the abstract. What 'verify' amounts to here is not spelled out, and for a short combinatorial argument the distinction between a careful check and a read matters; treat the proof as needing an independent reading.", "registrations": [ { "registry": "vibemathed", "id": "daykin-frankl-conjecture", "url": "https://vibemathed.com/problem/daykin-frankl-conjecture" } ], "sources": [ { "label": "arXiv: Confirmation of the Daykin-Frankl conjecture", "url": "https://arxiv.org/abs/2609.03087", "kind": "research" }, { "label": "vibemathed: The Daykin-Frankl conjecture on convex subsets of the Boolean lattice", "url": "https://vibemathed.com/problem/daykin-frankl-conjecture", "kind": "commentary" } ], "tags": [ "combinatorics", "boolean-lattice", "order-theory", "argument" ], "added": "2026-09-07" }, { "id": "2026-09-02-trautman-conjecture", "title": "A smooth counterexample to the Trautman conjecture", "claim": "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.", "field": "mathematics", "date": "2026-09-02", "lab": "Oklahoma State University", "model": "ChatGPT Plus, version not named", "humans": [ "Sean N. Curry" ], "year_posed": 1998, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "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.", "novelty_check": "The conjecture is attributed to Andrzej Trautman and dated 1998, and the paper's introduction sets out its origins and the standard nonembeddability constructions it modifies. No prior counterexample was found in the four registries or in the CR geometry literature located; the point of the paper is that the existing nonembeddable examples do not preserve the closed canonical section, which is exactly what the modification supplies. The result is new.", "caveats": "A single-author unrefereed preprint days old at entry, labelled by its author a preliminary version, with no independent check and no formalization. The AI disclosure sits in the acknowledgements rather than the abstract, which is the pattern worth noting: a reader of the abstract alone would not know a model was involved. Autonomy is graded ai-assisted on the strictest defensible reading. The acknowledgements say 'The counterexample construction presented below was discovered through experimentation using ChatGPT Plus on August 18, 2026' and that ChatGPT also drafted the manuscript and helped proofread, but the framing is the author's experimentation, the problem was posed to him by Pawel Nurowski and Rod Gover, and he states he 'independently checked all calculations and arguments presented here and takes full responsibility'. A case for ai-led could be made from the word discovered; the weaker grade wins.", "registrations": [ { "registry": "vibemathed", "id": "a-smooth-counterexample-to-the-trautman-conjecture", "url": "https://vibemathed.com/problem/a-smooth-counterexample-to-the-trautman-conjecture" } ], "sources": [ { "label": "arXiv: A Smooth Counterexample to the Trautman Conjecture", "url": "https://arxiv.org/abs/2609.03198", "kind": "research" }, { "label": "vibemathed: A smooth counterexample to the Trautman conjecture", "url": "https://vibemathed.com/problem/a-smooth-counterexample-to-the-trautman-conjecture", "kind": "commentary" } ], "tags": [ "differential-geometry", "cr-geometry", "counterexample", "construction" ], "added": "2026-09-07" }, { "id": "2026-09-03-catalan-constant-irrationality", "title": "Disputed claim that Catalan's constant is irrational", "claim": "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.", "field": "mathematics", "date": "2026-09-03", "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", "humans": [ "Zhi-Wei Sun" ], "verification": "disputed", "autonomy": "ai-assisted", "detail": "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.", "novelty_check": "The irrationality of Catalan's constant is a well-known open problem with no claimed resolution in the four registries before this preprint. The method is explicitly derivative of Calegari, Dimitrov and Tang's 2024 work, which the paper cites, and the claimed novelty is the weighting construction rather than the framework. No competing claim was found. The question of priority is not the issue here; the correctness of the argument is, and it is disputed.", "caveats": "Graded disputed on a specific, unresolved technical objection rather than on general scepticism. The objection, raised publicly and cited here, is that the tail recurrence of equation 1.4 is inconsistent with the factor used in equation 2.12, so the zero count the argument needs in order to force the degree bound fails. The same critique notes that the paper replaces the arithmetic content of the model it follows with a combinatorial minimization plus numerics that no human has checked, and points to internal inconsistencies in the write-up. The preprint remains at version one with no revision or withdrawal at entry. A second, independent critique by Dane Wachs, with published scripts, shows numerically that a quantity the main derivation needs to be negative of order \\(B^2\\) is positive of that order, so the proof of the main theorem is incomplete; it concludes nothing about whether the constant is irrational, which is why the grade is disputed rather than refuted. Autonomy is graded ai-assisted, and the paper's own account is the reason: the model 'made many attempts but we failed again and again' until 'the author realized that we should use suitable weights', so the decisive idea is the author's. The only stated verification of the whole proof is by a language model, which is not an independent check by any human.", "independent_checks": [ { "who": "Dane Wachs (critical note with computations)", "url": "https://arxiv.org/abs/2609.22339", "outcome": "Found a slip in the printed proof of the rank theorem that is repairable, and verified that statement exactly at 44 parameter pairs, but showed numerically that a quantity the main derivation needs to be at most \\(-0.00966B^2 + o(B^2)\\) is in fact between \\(+1.49B^2\\) and \\(+1.79B^2\\), concluding that the published derivation does not justify the estimate and the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of \\(G\\)." } ], "sources": [ { "label": "arXiv: Catalan's constant is irrational", "url": "https://arxiv.org/abs/2609.04176", "kind": "research" }, { "label": "Hacker News: technical objection to the argument, equation 1.4 against equation 2.12", "url": "https://news.ycombinator.com/item?id=49560333", "kind": "challenge" }, { "label": "arXiv: Wachs, A note on a recent claimed proof of the irrationality of Catalan's constant", "url": "https://arxiv.org/abs/2609.22339", "kind": "challenge" } ], "tags": [ "number-theory", "irrationality", "disputed", "argument" ], "added": "2026-09-07", "revisions": [ { "date": "2026-09-24", "kind": "challenge", "note": "Cited Dane Wachs's note showing numerically that the main estimate fails as derived, so the proof is incomplete. The grade stays disputed: the note shows a gap, not that the claim is false.", "url": "https://arxiv.org/abs/2609.22339" } ] }, { "id": "2026-09-08-navier-stokes-forced-blowup", "title": "Finite-time blowup for Navier-Stokes with smooth forcing, Clay alternatives C and D", "claim": "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.", "field": "mathematics", "date": "2026-09-08", "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", "year_posed": 2000, "verification": "claimed", "autonomy": "ai-led", "detail": "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 \\(\\mathbb{R}^3\\), and smooth periodic data and forcing admitting no global smooth solution on \\(\\mathbb{R}^3/\\mathbb{Z}^3\\). 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.", "novelty_check": "The alternatives settled here are Clay's own (C) and (D), stated in Fefferman's 2000 problem description, which the repository cites and links directly; the underlying regularity question goes back to Leray in 1934. The prior literature on forced or modified blowup is substantial but does not reach full three-dimensional Navier-Stokes with smooth forcing: blowup is known for the hypodissipative equations with a force (arXiv:2407.06776), for solutions with linear growth at infinity (arXiv:2103.12237), and for model equations (arXiv:1811.09394), and Tao's averaged Navier-Stokes construction is of the same character. No prior claim on (C) or (D) themselves was found in the four registries or in that search. The live novelty question is not priority over the classical literature but priority within the past year: OpenAI's own account credits the basic idea of the program to Diego Cordoba and Luis Martinez-Zoroa, and how much is owed to that program is publicly disputed, which the caveats record.", "caveats": "What this settles, and what it does not, needs stating carefully. Fefferman's problem description asks for a proof of any one of four statements, and the breakdown alternatives (C) and (D) are two of them, so on the problem as officially posed this is a candidate solution to the Millennium Prize Problem rather than something adjacent to it. The Clay Mathematics Institute said as much on 2026-09-11, that the problem has 'apparently been settled', while noting that its own process for evaluating the work and assigning credit is deliberately unhurried. What the result is not is a resolution of the unforced regularity question, alternatives (A) and (B), which is what the phrase 'the Navier-Stokes problem' is usually taken to name and what a reader of the press coverage would most likely assume was meant. The repository's README names the alternatives precisely and links Clay's own description, which is more care than most of that coverage took. Verification is graded claimed rather than formal despite four theorems carrying no sorry and exactly the three standard axioms, for the reason the repository itself supplies: its formalization.yaml records the review status as self-assessed, and no mathematician outside OpenAI is on record as having read the argument. Clay's own acceptance process additionally requires journal publication and two years of community acceptance. The statement-fidelity worry is smaller here than on a self-stated formalization, because the Comparator challenge statements were adapted from DeepMind's independent Formal Conjectures formalization rather than authored alongside the proof, and vibemathed has since diffed the adaptation: the two compared theorems and the conditions they depend on are byte-identical to DeepMind's file, the only difference being nine deleted lines carrying alternatives (A) and (B), which OpenAI does not claim. The model attribution is split, and both halves are recorded above because they may describe different steps: the announcement describes an unnamed next-generation model more capable than GPT-6 Astra, reported as roughly 10,000 coordinating agents over 88 hours, while the repository's own metadata credits GPT-6 Astra through Codex under an agent method, which most plausibly describes who wrote the Lean rather than who found the proof. Autonomy is ai-led rather than autonomous because the disclosure describes an agent framework without stating that the run was unsteered, so the strictest defensible reading is not autonomous. Priority is contested in public and unresolved at entry: the entangled Alpoge and Buckmaster results released the same week drew allegations, on r/math and in the comments of Terence Tao's blog, that the Cordoba and Martinez-Zoroa program was worked from for roughly a year before a denial that several commenters read as evasive, and Andreas Thom has publicly described asking OpenAI whether his own private conversations on an adjacent problem were drawn on and being told they were not, without further explanation. None of that bears on whether the Lean compiles.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/navier-stokes-millennium-prize-problem-finite-time-breakdown-with-smooth-forcing", "outcome": "Diffed ComparatorChallenges/NavierStokes.lean against google-deepmind/formal-conjectures at 8bf45ed: both compared theorems and their decay, periodicity and energy conditions are byte-identical, with alternatives (A) and (B) removed. Counted no sorry outside the placeholders and no axiom declarations. Not rebuilt; no mathematician has read the argument." } ], "registrations": [ { "registry": "vibemathed", "id": "navier-stokes-millennium-prize-problem-finite-time-breakdown-with-smooth-forcing", "url": "https://vibemathed.com/problem/navier-stokes-millennium-prize-problem-finite-time-breakdown-with-smooth-forcing", "note": "Recorded there as a candidate with review pending." } ], "sources": [ { "label": "GitHub: openai/NavierStokesAndEuler, the Lean 4 formalizations", "url": "https://github.com/openai/NavierStokesAndEuler", "kind": "research" }, { "label": "OpenAI: Finite time blowup for Navier-Stokes (PDF)", "url": "https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf", "kind": "research" }, { "label": "Clay Mathematics Institute: the official problem description, alternatives (C) and (D) on page 2", "url": "https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf", "kind": "research" }, { "label": "Formal Conjectures: the Lean statement of the Navier-Stokes problem the Comparator challenges adapt", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Millennium/NavierStokes.lean", "kind": "research" }, { "label": "OpenAI: On the Navier-Stokes Millennium Prize Problem", "url": "https://openai.com/index/navier-stokes-solution/", "kind": "announcement" }, { "label": "Clay Mathematics Institute: statement on the Navier-Stokes announcement", "url": "https://www.claymath.org/news/navier-stokes-announcement/", "kind": "commentary" }, { "label": "Terence Tao on the blowup results released the same week", "url": "https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/", "kind": "commentary" }, { "label": "vibemathed: finite-time breakdown with smooth forcing", "url": "https://vibemathed.com/problem/navier-stokes-millennium-prize-problem-finite-time-breakdown-with-smooth-forcing", "kind": "commentary" }, { "label": "Quanta: AI has solved one of math's $1 million Millennium Prize problems", "url": "https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/", "kind": "coverage" }, { "label": "Nature: OpenAI claims huge maths breakthrough on a famed Millennium Problem", "url": "https://www.nature.com/articles/d41586-026-02842-5", "kind": "coverage" }, { "label": "Science: How an AI math breakthrough ignited a controversy", "url": "https://www.science.org/content/article/how-ai-math-breakthrough-ignited-controversy", "kind": "coverage" }, { "label": "Buckmaster and Alpoge: public statement on the priority of the forced-blowup program (PDF)", "url": "https://cims.nyu.edu/~tristanb/statement.pdf", "kind": "challenge" }, { "label": "Constantin, Ignatova and Vicol: the forcing in any construction with these features can be neither locally vanishing nor real-analytic in space", "url": "https://arxiv.org/abs/2609.20803", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: On the Navier-Stokes Millennium Prize Problem", "url": "https://news.ycombinator.com/item?id=49613262" }, { "label": "Hacker News: How an AI math breakthrough ignited a controversy", "url": "https://news.ycombinator.com/item?id=49624163" } ], "tags": [ "pde", "fluid-dynamics", "navier-stokes", "singularity", "lean", "formalization", "argument" ], "added": "2026-09-10", "revisions": [ { "date": "2026-09-14", "kind": "correction", "note": "Corrected what the Clay prize covers. This entry originally said the Millennium Prize attaches only to alternatives (A) and (B) and that the result therefore was not the Millennium problem. Fefferman's problem description asks for a proof of any one of four statements, and the breakdown alternatives (C) and (D) are two of them, so the result is a candidate solution to the problem as posed. Added the Clay Mathematics Institute's statement of 2026-09-11. Neither grade moved.", "url": "https://www.claymath.org/news/navier-stokes-announcement/" }, { "date": "2026-09-22", "kind": "challenge", "note": "Cited the first independent technical engagement with the construction: Constantin, Ignatova and Vicol prove that a solution with the announced anisotropic Type II bounds and a collapsing axisymmetric core is regular at the putative singular point under a real-analytic force, so the force can neither vanish near that point nor be real-analytic in space. It constrains the construction rather than refuting it, and no grade moved.", "url": "https://arxiv.org/abs/2609.20803" }, { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's diff showing the compared statements are DeepMind's Formal Conjectures formalization verbatim. Nobody outside OpenAI has read the argument, so no grade moved.", "url": "https://vibemathed.com/problem/navier-stokes-millennium-prize-problem-finite-time-breakdown-with-smooth-forcing" } ] }, { "id": "2026-09-08-euler-unforced-blowup", "title": "Finite-time blowup for the unforced Euler equations from smooth compactly supported data", "claim": "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.", "field": "mathematics", "date": "2026-09-08", "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", "detail": "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 \\(\\mathbb{R}^3\\), 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 \\(C^1\\) 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.", "novelty_check": "The gap this closes is specific and the prior literature marks it clearly. Elgindi proved finite-time singularity for axisymmetric no-swirl Euler on \\(\\mathbb{R}^3\\) at \\(C^{1,\\alpha}\\) regularity, and Elgindi and Pasqualotto did the analogous thing for Boussinesq, in both cases below smooth. Chen and Hou proved blowup for smooth finite-energy data by computer-assisted analysis, but in an axially periodic cylinder with an impermeable wall, so with a boundary (PNAS, 2025). What remained open was exactly the case claimed here: smooth, compactly supported data on \\(\\mathbb{R}^3\\) with no boundary and no forcing. No prior claim on that case was found in the four registries or in the search behind this note. The companion result released the same week by Alpoge and Buckmaster is finite-time blowup for Euler with smooth forcing, which is a weaker and different statement, and the two should not be read as competing claims on the same theorem.", "caveats": "Verification is graded claimed rather than formal despite two theorems with no sorry and exactly the three standard axioms. The repository's own formalization.yaml records its review status as self-assessed, and no mathematician outside OpenAI is on record as having read the argument at entry. Statement fidelity carries more weight on this entry than on the companion Navier-Stokes one, though less than this entry first said. The Euler challenge module does record an outside source: its header, present since its first commit on 2026-09-08, says it is adapted from Formal Conjectures' Navier-Stokes statement specialized to zero viscosity and zero force, and vibemathed compared it line by line and found the retained conditions match. That anchors the first compared theorem, the breakdown alternative. The second, exists_compact_smooth_euler_singularity, is OpenAI's own richer form of the paper's Theorem 1.1 with no outside counterpart, and a reader checking this should start by reading that statement against the paper rather than by rerunning the build. The model attribution is split and both halves are recorded above: the announcement describes an unnamed next-generation model, while the repository metadata credits GPT-6 Astra through Codex under an agent method, which most plausibly describes who wrote the Lean rather than who found the proof. Autonomy is ai-led rather than autonomous because the disclosure describes an agent framework without stating the run was unsteered. The priority dispute around the same week's releases is recorded on the Navier-Stokes entry; how much of it reaches this unforced result is not clear from the public record, and the credit OpenAI gives to Diego Cordoba and Luis Martinez-Zoroa is stated for the forced program rather than for this one.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/finite-time-blowup-for-the-3d-incompressible-euler-equations-unforced", "outcome": "Compared ComparatorChallenges/Euler.lean line by line with Formal Conjectures' Navier-Stokes file at 8bf45ed with viscosity and force set to zero and found the retained conditions match for the first theorem; the second theorem is OpenAI's own. Counted no sorry and no axiom declarations under Euler/. Not rebuilt; no mathematician has read the argument." } ], "registrations": [ { "registry": "vibemathed", "id": "finite-time-blowup-for-the-3d-incompressible-euler-equations-unforced", "url": "https://vibemathed.com/problem/finite-time-blowup-for-the-3d-incompressible-euler-equations-unforced" } ], "sources": [ { "label": "GitHub: openai/NavierStokesAndEuler, the Lean 4 formalizations", "url": "https://github.com/openai/NavierStokesAndEuler", "kind": "research" }, { "label": "OpenAI: Finite time blowup for the Euler equation (PDF)", "url": "https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf", "kind": "research" }, { "label": "OpenAI: On the Navier-Stokes Millennium Prize Problem, the announcement covering both results", "url": "https://openai.com/index/navier-stokes-solution/", "kind": "announcement" }, { "label": "Chen and Hou: singularity formation in 3D Euler with smooth initial data and boundary, the closest prior result", "url": "https://www.pnas.org/doi/10.1073/pnas.2500940122", "kind": "commentary" }, { "label": "Terence Tao on the Alpoge and Buckmaster forced-blowup results of the same week, a different claim", "url": "https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/", "kind": "commentary" }, { "label": "Science: How an AI math breakthrough ignited a controversy", "url": "https://www.science.org/content/article/how-ai-math-breakthrough-ignited-controversy", "kind": "coverage" } ], "discussion": [ { "label": "Hacker News: On the Navier-Stokes Millennium Prize Problem", "url": "https://news.ycombinator.com/item?id=49613262" }, { "label": "Hacker News: How an AI math breakthrough ignited a controversy", "url": "https://news.ycombinator.com/item?id=49624163" } ], "tags": [ "pde", "fluid-dynamics", "euler-equations", "singularity", "lean", "formalization", "argument" ], "added": "2026-09-10", "revisions": [ { "date": "2026-09-22", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. It cites the same theorems and the same repository already sourced here. No grade moved.", "url": "https://vibemathed.com/problem/finite-time-blowup-for-the-3d-incompressible-euler-equations-unforced" }, { "date": "2026-09-24", "kind": "correction", "note": "Corrected the caveat, which said the repository records no outside source for the Euler challenge statement. Its header has named Formal Conjectures' Navier-Stokes file as the source since the first commit; only the second compared theorem is OpenAI's own.", "url": "https://github.com/openai/NavierStokesAndEuler/blob/main/ComparatorChallenges/Euler.lean" }, { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's line-by-line comparison of the first compared theorem with Formal Conjectures. No grade moved.", "url": "https://vibemathed.com/problem/finite-time-blowup-for-the-3d-incompressible-euler-equations-unforced" } ] }, { "id": "2026-09-14-nivat-conjecture", "title": "Nivat's conjecture claimed in its sharp form, with the submitter stating he cannot check it", "claim": "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.", "field": "mathematics", "date": "2026-09-14", "lab": "Independent", "model": "GPT-6 Pro (proof); GPT-6 Astra Ultra through Codex (Lean formalization)", "humans": [ "Boon Suan Ho" ], "year_posed": 1997, "verification": "claimed", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The conjecture dates to 1997 and is a named open problem in symbolic dynamics, still described as open in the literature the repository cites. No competing resolution of the sharp form appears in the four registries the watch sweep diffs. The repository itself notes that Bryna Kra published a guest post on Terence Tao's blog discussing Nivat's conjecture and AI-generated mathematics roughly three hours before its first commit, and states the proof work was complete before the author saw that post; the two are independent, and neither is evidence about the other.", "caveats": "Nobody has checked this, and the submitter says so first. His disclaimer reads: the proof was found entirely by GPT-6 Pro in response to his prompts, the paper was written by AI and has not been mathematically digested by humans, and he does not regard himself as qualified to digest the argument or give it the exposition it deserves. His stated role was prompting and arranging formal verification. That is a clearer disclosure than most entries here carry, and it is also the reason the grade is claimed rather than formal: a Palomar registration establishes that the Lean typechecks against the recorded statement, not that the statement is Nivat's conjecture. Autonomy is ai-led rather than autonomous because a human chose the problem and drove the prompting.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-14-000003", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-14-000003&version=1", "repository": "boonsuan/nivat" } ], "sources": [ { "label": "GitHub: boonsuan/nivat, the Lean 4 and mathlib formalization", "url": "https://github.com/boonsuan/nivat", "kind": "research" }, { "label": "Palomar registration for the Nivat formalization", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-14-000003&version=1", "kind": "research" }, { "label": "Bryna Kra on AI and deep theorems, guest post on Terence Tao's blog", "url": "https://terrytao.wordpress.com/2026/09/13/deep-theorems-were-scarce-and-difficult-and-so-became-an-effective-mechanism-to-identify-deep-thought-ai-has-broken-this-system/", "kind": "commentary" } ], "tags": [ "symbolic-dynamics", "combinatorics", "lean", "formalization", "argument" ], "added": "2026-09-15" }, { "id": "2026-09-03-smale-mean-value-k1", "title": "Counterexample to Smale's mean value conjecture at K = 1", "claim": "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.", "field": "mathematics", "date": "2026-09-03", "lab": "OpenAI / Epoch AI", "model": "GPT-6 Astra (pre-release)", "humans": [ "Tom Adamczewski" ], "year_posed": 1981, "verification": "formal", "autonomy": "autonomous", "detail": "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| \\le 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.", "novelty_check": "The conjecture dates to 1981 and is one of Smale's problems for the twenty-first century, described as open in the sources the repository cites. No competing counterexample appears in the four registries. The repository's own fidelity note is the substantive part of this check: the compared theorem is the explicit negation of the Formal Conjectures statement, and it observes that Lean's convention that division by zero yields zero makes the formal conjecture weaker than the informal one, so the counterexample refutes the conjecture as stated in the literature rather than only its formalization.", "caveats": "Days old at entry, unrefereed, with no expert in complex analysis on record as having read it. The witness is the weak part and the repository says so: a polynomial of very large unspecified degree that violates the bound by a small margin. That is consistent with Smale's own \\(K = 4\\) theorem and with the known asymptotics of the best constant, so nothing about it is obviously wrong, but a nonconstructive witness of unspecified degree is hard for a reader to check by hand. The grade was claimed until 2026-09-24 because no third party had audited the statement against the literature. That audit now exists twice, by vibemathed and by this registry, and both find the compared statement to be the negation of Formal Conjectures' mean_value_problem verbatim, a statement written independently of the prover, so the grade is formal. The formal grade certifies the disproof, not the machine-generated proof account, which no mathematician outside the run has read. Autonomy is autonomous on the same disclosure the Koethe and Erdos-Sos entries rest on: one attempt inside an evaluation harness, with no human seeing or steering the proof search.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1", "outcome": "From a clone at f99ab98: the compared statement is byte-identical to Formal Conjectures' mean_value_problem at 9cbe1d3c, with no sorry outside the challenge stub, no axiom declarations and no native_decide, and CI runs Comparator permitting only the three standard axioms." }, { "who": "whataifound.org (statement comparison)", "url": "https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/MeanValueProblem.lean", "outcome": "Compared Challenge.lean at commit 15e07155 with the Formal Conjectures file at 9cbe1d3c: the compared theorem is the negation of mean_value_problem with identical hypotheses and inequality. At that commit the repository's Lean build and statement-against-solution CI jobs pass; the one failing job checks a README link." } ], "registrations": [ { "registry": "vibemathed", "id": "smale-s-mean-value-conjecture-k-1", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1" } ], "sources": [ { "label": "GitHub: tadamcz/mean-value-problem, the Lean 4 disproof", "url": "https://github.com/tadamcz/mean-value-problem", "kind": "research" }, { "label": "vibemathed: Smale's mean value conjecture (K = 1)", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1", "kind": "commentary" } ], "tags": [ "complex-analysis", "polynomials", "lean", "formalization", "construction" ], "added": "2026-09-15", "revisions": [ { "date": "2026-09-24", "kind": "regraded", "note": "Upgraded from claimed to formally verified: the compared statement was audited against the literature, by vibemathed and here, and is Formal Conjectures' formalization verbatim, which clears the only reason the entry gave for withholding the formal grade.", "url": "https://vibemathed.com/problem/smale-s-mean-value-conjecture-k-1" } ] }, { "id": "2026-09-05-ibragimov-iosifescu", "title": "Disproof of the Ibragimov-Iosifescu conjecture for phi-mixing sequences", "claim": "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.", "field": "mathematics", "date": "2026-09-05", "lab": "OpenAI / Epoch AI", "model": "GPT-6 Astra (pre-release)", "humans": [ "Tom Adamczewski" ], "year_posed": 1971, "verification": "claimed", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The repository carries its own literature review documenting a search through 2026-09-05 for a prior resolution. It reports finding no earlier counterexample satisfying all the conjecture's hypotheses, and identifies a 2023 scholarly source still describing the conjecture as open. That review was not independently repeated for this entry. No competing claim appears in the four registries.", "caveats": "Unrefereed and days old at entry, with no probabilist on record as having read it. The grade is claimed rather than formal for the usual reason on this pipeline: the axiom check establishes that the Lean proves the recorded statement, not that the recorded statement is the conjecture as the literature states it, and no third party has audited that step. Autonomy is ai-led rather than autonomous, weaker than the sibling Smale entry: the disclosure says the model constructed the counterexample and wrote the proof over roughly twelve hours and 1054 turns, which describes an agentic run rather than the unsteered single attempt the benchmark entries rest on.", "registrations": [ { "registry": "vibemathed", "id": "ibragimov-iosifescu-varphi-mixing-clt-conjecture", "url": "https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture" } ], "sources": [ { "label": "GitHub: tadamcz/phi-mixing-clt, the Lean counterexample and literature review", "url": "https://github.com/tadamcz/phi-mixing-clt", "kind": "research" }, { "label": "vibemathed: Ibragimov-Iosifescu phi-mixing CLT conjecture", "url": "https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture", "kind": "commentary" } ], "tags": [ "probability", "central-limit-theorem", "lean", "formalization", "construction" ], "added": "2026-09-15" }, { "id": "2026-09-06-bollobas-nikiforov", "title": "Bollobas-Nikiforov conjecture claimed in full, in a weighted form", "claim": "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.", "field": "mathematics", "date": "2026-09-06", "lab": "Independent", "model": "GPT-6 Astra (ideation and manuscript); Grok 4.6 (Lean formalization); Claude Fable 5.1 (submission packaging)", "humans": [ "Gabriel Coutinho", "Yinchen Liu", "Thomas Jung Spier", "Quanyu Tang", "Shengtong Zhang" ], "year_posed": 2007, "verification": "claimed", "autonomy": "collaborative", "detail": "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.", "novelty_check": "The conjecture dates to 2007. The strongest previously published result found is Giacomelli (arXiv:2603.26379), a single-author paper with no AI disclosure, which proves the conjecture only for complete multipartite graphs and for dense K4-free graphs. This claim is materially larger: the general conjecture for every noncomplete graph, in a stronger weighted form, submitted some months later. The gap between the two is the reason to record this early and to state plainly that it has not been checked.", "caveats": "The jump from the published partial results to a claimed full resolution is the thing to verify first, and nobody outside the five authors has done it. The grade is claimed rather than formal because the Palomar record establishes that the Lean typechecks against the recorded statement, not that the recorded statement is the conjecture in its general form. Autonomy is collaborative rather than ai-led: the disclosure describes ideation as human and model back and forth among five named mathematicians rather than a model producing the argument, and the formalization step was a separate model working from an already written proof.", "independent_checks": [ { "who": "vibemathed.com statement audit", "url": "https://vibemathed.com/problem/bollobas-nikiforov-conjecture", "outcome": "At commit edb5259, read Challenge.lean and found the headline theorem stated entirely in Mathlib vocabulary (clique number, edge count, the two largest adjacency eigenvalues) with no hypothesis beyond non-completeness and at least two vertices, and the comparator run recorded under two kernels. Not rebuilt." } ], "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-07-000002", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-07-000002&version=1", "repository": "ShengtongZhang-alt/BN" }, { "registry": "vibemathed", "id": "bollobas-nikiforov-conjecture", "url": "https://vibemathed.com/problem/bollobas-nikiforov-conjecture" } ], "sources": [ { "label": "GitHub: ShengtongZhang-alt/BN, the Lean 4 and Mathlib formalization", "url": "https://github.com/ShengtongZhang-alt/BN", "kind": "research" }, { "label": "Giacomelli: the prior partial result, complete multipartite and dense K4-free graphs", "url": "https://arxiv.org/abs/2603.26379", "kind": "research" }, { "label": "vibemathed: Bollobas-Nikiforov conjecture", "url": "https://vibemathed.com/problem/bollobas-nikiforov-conjecture", "kind": "commentary" } ], "tags": [ "graph-theory", "spectral-graph-theory", "combinatorics", "lean", "formalization", "argument" ], "added": "2026-09-15", "revisions": [ { "date": "2026-09-24", "kind": "check", "note": "Recorded vibemathed's audit of the formal statement, which finds it expresses the conjecture in plain Mathlib terms. The audit is one curator's reading rather than a named expert's, so no grade moved.", "url": "https://vibemathed.com/problem/bollobas-nikiforov-conjecture" } ] }, { "id": "2026-09-03-prime-gaps-212", "title": "Bounded prime gaps improved to 212, with a Lean certificate of the deduction", "claim": "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.", "field": "mathematics", "date": "2026-09-03", "lab": "Axiom Math", "model": "AxiomProver", "humans": [ "Francois Charton", "Letong Hong", "Kenny Lau", "Ken Ono", "Guillaume Remy", "Ho Chung Siu", "Ashvin A. Swaminathan", "Jesse Thorner", "Yunzhou Xie" ], "year_posed": 1849, "verification": "claimed", "autonomy": "collaborative", "detail": "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.", "novelty_check": "The bound improves on published work the paper names in its own abstract: 246 from Polymath8b and 240 from Stadlmann. No earlier claim below 240 was found in the four registries. Two competing prime-gap claims appeared in the same window, an unreviewed 236 with no named model and no preprint, and an OpenAI 186 that is recorded separately here and is conditional on three unproved axioms; neither displaces this one on evidence.", "caveats": "Nothing outside the author group has checked this. The certificate covers the deduction from stated assumptions rather than the analytic inputs themselves, the same shape as the 246 entry already in the registry, so it is not a machine-checked proof of the bound from first principles. The Lean build was not independently reproduced for this entry. Autonomy is collaborative on the authors' own framing, that the mathematics is theirs and the AI contribution is the formal certificate. Whether this should supersede or extend 2026-08-18-prime-gaps-246 rather than sit beside it is an open editorial question: same lab, same tooling, a strictly better bound.", "registrations": [ { "registry": "vibemathed", "id": "bounded-prime-gaps-at-most-212", "url": "https://vibemathed.com/problem/bounded-prime-gaps-at-most-212" } ], "sources": [ { "label": "Axiom Math: a new bound for small gaps between primes (PDF)", "url": "https://primegaps.axiommath.ai/bgp212.pdf", "kind": "research" }, { "label": "vibemathed: bounded prime gaps at most 212", "url": "https://vibemathed.com/problem/bounded-prime-gaps-at-most-212", "kind": "commentary" } ], "tags": [ "number-theory", "prime-gaps", "sieve-methods", "lean", "formalization", "argument" ], "added": "2026-09-15" }, { "id": "2026-09-02-prime-gaps-186", "title": "Prime gaps at most 186, conditional on three unproved Lean axioms", "claim": "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.", "field": "mathematics", "date": "2026-09-02", "lab": "OpenAI", "model": "GPT-6 Astra", "year_posed": 1849, "verification": "claimed", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The claimed bound is below every published figure in the bounded-gaps line, which runs through Polymath8b's 246 and Stadlmann's 240, and below the Axiom Math 212 from the same window. No earlier claim at 186 or lower was found in the four registries. Priority is not really the open question here; whether the claim is established at all is, and the caveats address that.", "caveats": "The grade is capped by the artifact's own structure rather than by caution. Three load-bearing statements are axioms rather than theorems, so what the kernel establishes is an implication: granted those three, the bound follows. Until they are proved in Lean or cited to the literature, the development certifies a conditional statement, and the numerical work does not discharge the corresponding axiom. No human author is named on the repository, so nobody is on record as having reviewed it. Autonomy is ai-led rather than autonomous: the disclosure describes the model working to specific formalization instructions, including which inputs to retain as axioms, which is steering even without a named human.", "registrations": [ { "registry": "vibemathed", "id": "prime-gaps-at-most-186", "url": "https://vibemathed.com/problem/prime-gaps-at-most-186" } ], "sources": [ { "label": "GitHub: openai/PrimeGaps186, the conditional Lean formalization and numerical certificate", "url": "https://github.com/openai/PrimeGaps186", "kind": "research" }, { "label": "vibemathed: prime gaps at most 186", "url": "https://vibemathed.com/problem/prime-gaps-at-most-186", "kind": "commentary" } ], "discussion": [ { "label": "Hacker News: Prime Gaps at Most 186", "url": "https://news.ycombinator.com/item?id=49555257" } ], "tags": [ "number-theory", "prime-gaps", "sieve-methods", "lean", "formalization", "argument" ], "added": "2026-09-15", "revisions": [ { "date": "2026-09-22", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. Its own label reads Lean-checked, statement unaudited, which matches the conditional grade already here. No grade moved.", "url": "https://vibemathed.com/problem/prime-gaps-at-most-186" } ] }, { "id": "2026-09-17-talagrand-operator-cotype", "title": "Counterexample to Talagrand's operator cotype problem", "claim": "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.", "field": "mathematics", "date": "2026-09-17", "lab": "Independent", "model": "ChatGPT (GPT-5.6)", "humans": [ "Xinglong Wu" ], "verification": "claimed", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The disclosure and the attribution were read from the arXiv abstract directly rather than from an aggregator label, which matters this week: the 2026-09-23 watch run spot-checked 11 registry rows tagged as AI-discovered and found 9 of them carried no AI disclosure anywhere in the actual paper. This one does, verbatim. No prior counterexample to the operator cotype question was found in the four registries the watch reads. This is a distinct Talagrand problem from the convolution conjecture already in this registry at 2026-08-16-talagrand-convolution, which is a different question by different authors using a different system; the two should not be read as related results. The year Talagrand posed the operator cotype question is not stated in the abstract and is deliberately left unrecorded here rather than guessed.", "caveats": "A preprint days old at entry, unrefereed, with no formalization and no independent check on record, so the grade is claimed on the registry's default for an announced but unverified result. Autonomy is ai-led on the author's own one-sentence attribution taken at face value; as with any such statement there is no transcript or artifact that would let a reader separate discovery from the author's own framing of it, and the model field should be read as the author's attribution rather than as a verified fact. The sole source is the preprint itself.", "registrations": [ { "registry": "vibemathed", "id": "talagrand-operator-cotype-problem", "url": "https://vibemathed.com/problem/talagrand-operator-cotype-problem" } ], "sources": [ { "label": "arXiv: A Counterexample to Talagrand’s Operator Cotype Problem", "url": "https://arxiv.org/abs/2609.19731", "kind": "research" } ], "tags": [ "functional-analysis", "probability", "counterexample", "construction" ], "added": "2026-09-22", "revisions": [ { "date": "2026-09-24", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. It names the same model and collaborator as the preprint. No grade moved.", "url": "https://vibemathed.com/problem/talagrand-operator-cotype-problem" } ] }, { "id": "2026-09-07-complex-grothendieck-constant", "title": "Improved lower bound for the complex Grothendieck constant", "claim": "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.", "field": "mathematics", "date": "2026-09-07", "lab": "Independent", "model": "Odin Automatic AI Research Agent", "humans": [ "Shengtao Guo", "Ethan X. Fang", "Junwei Lu" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The abstract fixes the prior state of the art by name, Davie below and Haagerup above, and states the improvement as a fraction of that gap rather than as an unqualified advance, which is the form that makes the claim checkable. No competing improvement was found in the four registries the watch reads at entry. On 2026-09-21 Steven Heilman, Chris Jones and Giulio Malavolta posted a lower bound of 1.4, which supersedes this one; they describe this bound as an independent improvement by a different subtraction of Hermite projections. The AI attribution was read from the arXiv abstract directly rather than from an aggregator tag.", "caveats": "Odin is the same undocumented system credited on this registry's 2026-08-16-talagrand-convolution entry, by an overlapping author set, and the caveat recorded there applies here without change: Odin is not a publicly documented system, no vendor is named, no transcript or artifact is published, and there is no way to separate discovery from human reorganization. Read the model field as the authors' attribution rather than as a verified fact. Two further gaps are worth stating. The abstract links no code, but the first author's repository, found through the vibemathed record on 2026-09-24, publishes the Arb interval-arithmetic computations behind both the lower bound and the paper's restricted pointwise bound, with a single replay command; its README is explicit that the second is not an upper bound for the constant. With the artifact public, the editorial rule 4 cap no longer applies and the grade is author verified, matching the sibling Talagrand entry. It goes no further because nobody outside the author group is on record as having run the certificates.", "registrations": [ { "registry": "vibemathed", "id": "lower-bound-for-the-complex-grothendieck-constant", "url": "https://vibemathed.com/problem/lower-bound-for-the-complex-grothendieck-constant" } ], "sources": [ { "label": "arXiv: An Improved Lower Bound for the Complex Grothendieck Constant", "url": "https://arxiv.org/abs/2609.07000", "kind": "research" }, { "label": "GitHub: shengtaoguo/complex-grothendieck-certificates, the interval-arithmetic certificates", "url": "https://github.com/shengtaoguo/complex-grothendieck-certificates", "kind": "research" }, { "label": "arXiv: Heilman, Jones and Malavolta, Sharper bounds for the complex Grothendieck constant", "url": "https://arxiv.org/abs/2609.25462", "kind": "commentary" } ], "tags": [ "functional-analysis", "optimization", "interval-arithmetic", "construction" ], "added": "2026-09-22", "revisions": [ { "date": "2026-09-24", "kind": "correction", "note": "Recorded the vibemathed record as a registration and added the certificate repository it links as a research source.", "url": "https://github.com/shengtaoguo/complex-grothendieck-certificates" }, { "date": "2026-09-24", "kind": "regraded", "note": "Upgraded from claimed to author verified: the interval-arithmetic certificates the abstract asserted are now public, which lifts the missing-artifact cap and aligns this entry with the sibling Talagrand entry. Nobody outside the authors has run them yet.", "url": "https://github.com/shengtaoguo/complex-grothendieck-certificates" }, { "date": "2026-09-24", "kind": "correction", "note": "Recorded that the bound was superseded two weeks later by Heilman, Jones and Malavolta's lower bound of 1.4. The result stands as a step, and no grade moved.", "url": "https://arxiv.org/abs/2609.25462" } ] }, { "id": "2026-09-17-rogers-ramanujan-torus-knot", "title": "Huang-Jiang-Oblomkov conjecture proved for every torus-knot singularity, with a Lean formalization conditional on two literature inputs", "claim": "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.", "field": "mathematics", "date": "2026-09-17", "lab": "Axiom Math", "model": "AxiomProver", "humans": [ "Yifeng Huang", "Kenny Lau", "Ken Ono" ], "verification": "claimed", "autonomy": "ai-assisted", "detail": "The conjecture of Huang, Jiang and Oblomkov gives a geometric extension of the Rogers-Ramanujan and Andrews-Gordon identities for the torus-knot singularity \\(X^a = Y^b\\) with coprime exponents. The paper establishes a threefold equality between a normalized count of pairs of commuting nilpotent matrices over a finite field satisfying \\(A^a = B^b\\), 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.", "novelty_check": "The 2026-09-23 watch run searched arXiv and the web for the \\(a = 3\\) layer specifically and found two August 2026 papers by an overlapping author set, arXiv:2608.05480 and arXiv:2608.15219, which resolve individual exponent pairs rather than a full family; no prior resolution of the general coprime case was found. That search was not a systematic review of the Rogers-Ramanujan and torus-knot-singularity literature, and whether this paper generalizes rather than restates those two predecessors is the first thing a specialist should check. The year the HJO conjecture was posed is not stated in the abstract and is left unrecorded here rather than guessed.", "caveats": "Autonomy is ai-assisted rather than anything stronger because the abstract places AxiomProver on the formalization, not on the discovery: the proof strategy it names is a combination of existing human theorems. Verification is claimed rather than formal for the reason the abstract itself supplies, that the Lean development is conditional on two stated literature inputs, so what the kernel establishes is an implication rather than the theorem outright, the same structure as this registry's 2026-09-02-prime-gaps-186 entry. Whether those two inputs, the collinear commutation of slope operators and the compositional rational shuffle identity, are settled literature or are doing substantive unproved work is the question that decides how much the formalization is worth. A preprint days old at entry, unrefereed, with no mathematician review found as of 2026-09-22 and no third-party audit of the Lean statements against the paper.", "sources": [ { "label": "arXiv: Rogers-Ramanujan identities from the geometry of Xᵃ = Yᵇ", "url": "https://arxiv.org/abs/2609.20567", "kind": "research" }, { "label": "GitHub: AxiomMath/HJO, the Lean formalization", "url": "https://github.com/AxiomMath/HJO", "kind": "research" } ], "tags": [ "number-theory", "combinatorics", "q-series", "lean", "formalization", "argument" ], "added": "2026-09-22" }, { "id": "2026-09-16-browning-sawin-hypersurfaces", "title": "Browning-Sawin conjecture on random sign-coefficient hypersurfaces proved, with a Lean formalization assuming existing literature", "claim": "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.", "field": "mathematics", "date": "2026-09-16", "lab": "Axiom Math", "model": "AxiomProver", "humans": [ "Ken Ono", "Ashvin Swaminathan" ], "year_posed": 2025, "verification": "claimed", "autonomy": "ai-assisted", "detail": "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 \\(O_n(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.", "novelty_check": "The conjecture is attributed in the abstract to Browning and Sawin, whose 2025 preprint arXiv:2510.26191 is its source, and the 2026-09-23 watch run searched arXiv and the web for a prior proof and found none, making this the first claimed resolution. MathSciNet and Zentralblatt were not searched, and whether Browning or Sawin have themselves circulated a proof was not established.", "caveats": "Autonomy is ai-assisted rather than stronger because the only AI role the abstract states is the Lean formalization. A separate and fuller disclosure describing human-AI collaboration through literature search, numerical experimentation and the proposal and rejection of candidate strategies was reported by the 2026-09-21 watch run as coming from this paper, but it does not appear in the abstract that was read directly on 2026-09-22; it may sit in the body. Until someone locates it, the autonomy grade here rests only on the formalization sentence, and a maintainer who finds that fuller text may have grounds to revisit it. Verification is claimed rather than formal because the formalization is explicitly conditional on existing literature taken as input, so the kernel certifies an implication; whether the five classical results taken as Lean axioms are standard or are doing substantive proof work is unresolved. A preprint days old at entry, unrefereed, with no specialist review and no discussion found on any tracked venue as of 2026-09-22.", "sources": [ { "label": "arXiv: On a conjecture of Browning and Sawin on random hypersurfaces with sign coefficients", "url": "https://arxiv.org/abs/2609.18879", "kind": "research" }, { "label": "GitHub: AxiomMath/BrowningSawin, the Lean formalization", "url": "https://github.com/AxiomMath/BrowningSawin", "kind": "research" }, { "label": "arXiv: Browning and Sawin, Random Diophantine equations of large degree, the source of the conjecture", "url": "https://arxiv.org/abs/2510.26191", "kind": "research" } ], "tags": [ "algebraic-geometry", "probability", "arithmetic-statistics", "lean", "formalization", "argument" ], "added": "2026-09-22" }, { "id": "2026-09-15-ipm-forced-blowup", "title": "Finite-time blowup for the IPM equation with uniformly space-time smooth forcing", "claim": "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.", "field": "mathematics", "date": "2026-09-15", "lab": "Independent", "model": "Not named; the arXiv comment field states only that the proof is LLM-assisted", "humans": [ "Levent Alpöge", "Tristan Buckmaster", "Matei P. Coiculescu" ], "verification": "claimed", "autonomy": "ai-assisted", "detail": "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.", "novelty_check": "The paper frames itself in its own abstract as extending the Cordoba and Martinez-Zoroa IPM blowup from spatially smooth to uniformly space-time smooth forcing, so the prior state of the art is named rather than left implicit. Terence Tao's 2026-09-07 post describes the same group's work across IPM, Boussinesq and Euler and notes that the IPM case was already handled by Cordoba and Martinez-Zoroa, with the contribution here being a variant of their method that also extends to the other two equations. This is a different claim from the OpenAI results already in this registry at 2026-09-08-navier-stokes-forced-blowup and 2026-09-08-euler-unforced-blowup, which concern different equations, and the four should not be read as competing claims on one theorem.", "caveats": "The AI disclosure is thin and its location matters: it is not in the abstract but in the arXiv comment field, which reads in full, 57 pages, LLM-assisted and Lean formalized proof. No model or vendor is named anywhere, which is why the model field records the absence rather than a guess, and autonomy is graded ai-assisted as the weakest reading the phrase LLM-assisted will carry. The 2026-09-21 watch run recorded this result under the date 2026-09-08 and without the third author; both were corrected here against the arXiv record on 2026-09-22, which gives a submission date of 15 September 2026 and names Coiculescu alongside Alpoge and Buckmaster. The Lean formalization is asserted in the comment field but no repository is linked from the abstract, so verification is capped at claimed under editorial rule 4 and nothing here is machine-checked for this registry's purposes. The vibemathed record for this result, added as a registration on 2026-09-24, names models (Claude, and Codex with GPT-5.6 Sol) and links tristanbuckmaster/fluid_lean as the Lean proof, but that repository contains only the companion Boussinesq and Euler formalizations, and the paper itself still names no model; the model field keeps to the paper. Tao's post records that the authors released preliminary preprints before the work was polished, describing their own earlier write-up in severe terms, so a reader should expect the exposition to be in flux.", "registrations": [ { "registry": "vibemathed", "id": "ipm-blowup-spacetime-smooth-forcing", "url": "https://vibemathed.com/problem/ipm-blowup-spacetime-smooth-forcing", "note": "The record credits Claude and Codex with GPT-5.6 Sol and labels the proof Lean-verified, linking tristanbuckmaster/fluid_lean. At commit d0124689230b58b4f86e7b90ac59de06404b3b6b that repository holds the companion Boussinesq and Euler developments and no IPM project, so neither claim is corroborated by an artifact." } ], "sources": [ { "label": "arXiv: Extending the Cordoba-Martinez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing", "url": "https://arxiv.org/abs/2609.16470", "kind": "research" }, { "label": "arXiv: Cordoba and Martinez-Zoroa, the prior spatially-smooth-forcing result", "url": "https://arxiv.org/abs/2410.22920", "kind": "research" }, { "label": "Terence Tao on the forced-blowup results for IPM, Boussinesq and Euler", "url": "https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/", "kind": "commentary" } ], "tags": [ "pde", "fluid-dynamics", "singularity", "argument" ], "added": "2026-09-22", "revisions": [ { "date": "2026-09-24", "kind": "correction", "note": "Recorded the vibemathed record for this result as a registration. It names models and a Lean repository the paper does not, and the repository it links holds no IPM formalization, so the model field and the grade did not move.", "url": "https://vibemathed.com/problem/ipm-blowup-spacetime-smooth-forcing" } ] }, { "id": "2026-09-10-approval-committee-core", "title": "Existence of the core in approval-based committee elections", "claim": "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.", "field": "mathematics", "date": "2026-09-10", "lab": "Independent", "model": "GPT-6 Astra", "humans": [ "Patrick Becker", "Matthias Greger", "Dominik Peters" ], "verification": "claimed", "autonomy": "collaborative", "detail": "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.", "novelty_check": "The abstract states the question was the main open one in its area and that this settles it; the claim of prior status rests on that framing plus Epoch AI's listing of the problem among its FrontierMath open problems, rather than on a literature search run for this entry. Epoch AI has described the result as a Major Advance on that benchmark. Whether the authors are independent of Epoch AI, which would bear on any claim of independent verification, was not established.", "caveats": "Autonomy is graded collaborative rather than ai-led on the strictest defensible reading of the public account, which describes a lengthy interactive session rather than a clean split between an AI insight and a human check: Peters is reported as saying both that he doubts the team would have found the proof without Astra and that Astra does not appear able to solve the problem from a simple prompt. Each half of that statement pulls against a stronger grade in one direction, and collaborative is where they meet. The AI disclosure lives in the arXiv comment field rather than the abstract or a dedicated statement, so it is thin relative to what this registry prefers. Verification is claimed: a preprint at entry, unrefereed, with no formalization and no independent check on record. The 2026-09-21 and 2026-09-23 watch runs both recorded this result under the date 2026-09-19 and with surnames only; the date and the full author names were corrected here against the arXiv record on 2026-09-22, which gives a submission date of 10 September 2026. Whether Epoch AI publishes rerunnable verification for its open-problem write-ups, which would bear on the verification grade, is unresolved.", "sources": [ { "label": "arXiv: Existence of the Core in Approval-Based Committee Elections", "url": "https://arxiv.org/abs/2609.11912", "kind": "research" }, { "label": "Epoch AI: FrontierMath open problems, committee election", "url": "https://epoch.ai/frontiermath/open-problems/committee-election", "kind": "announcement" } ], "tags": [ "social-choice", "combinatorics", "algorithms", "argument" ], "added": "2026-09-22" }, { "id": "2026-09-09-small-undecidable-groups", "title": "A 3-generator 9-relator group with unsolvable word problem, and smaller Adian-Rabin families", "claim": "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.", "field": "mathematics", "date": "2026-09-09", "lab": "Independent", "model": "GPT-5.6 Sol Pro (construction); GPT-5.6 Sol in Codex (Lean formalization)", "humans": [ "Marc Kegel", "Shana Yunsheng Li", "Qiuyu Ren" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "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.", "novelty_check": "The paper fixes the prior state of the art by name: Borisov 1969 at 12 relators for the word problem, Tancer 2023 at deficiency 9 and Gordon 2022 at 13 relators for Adian-Rabin families, which gave topological unrecognizability of 9 copies and smooth unrecognizability of 12 copies of S2 x S2. The registry was searched on 2026-09-24 for Adian, Rabin, unrecognizable, word problem and 4-manifold with no hits, and a web search for recent fewest-relator results turned up only this paper and Tancer's 2023 preprint. Of the four registries the watch reads, only Palomar holds a record, and it is the authors' own formalization of this paper. The paper itself records a competing result in progress: Cameron Gordon has independently improved the smallest Adian-Rabin families along similar lines and obtained both smooth and topological unrecognizability of 8 copies, in a paper not yet out. That will leave the topological bound of 7 here as the stronger one, but will supersede the smooth bound of 9. The problem has no single origin year, so year_posed is left unrecorded.", "caveats": "Graded author verified rather than formal even though the three algebraic theorems carry a Lean proof registered on Palomar. The four-manifold corollary, which the paper's title leads with, is a conventional argument on top of the formalized algebra and is not machine-checked, as Palomar's own record says. The Lean statements were checked only by the authors: the Lean comparator accepts the proof and the axiom audit reports only the three standard principles, but no third party has audited the statements against the paper, and a Palomar registration establishes that the proof typechecks against the recorded statement, not that the statement is the paper's theorem. Autonomy is graded ai-led on an explicit disclosure in section 1.5: the main construction was essentially developed by GPT-5.6 Sol Pro, which proposed the two HNN extensions, the Tietze reductions and the two extra eliminations behind Theorem 1.3, while the authors independently checked and reorganized the arguments, verified the references and take responsibility for correctness. It is not autonomous because the authors chose the problem and directed the work. The model also flagged two corrections needed in Tancer's 2023 paper, which Tancer confirmed to the authors, and the reduction behind Theorem 1.3 uses the corrected form. An unrefereed preprint at entry.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-22-000002", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-22-000002&version=1", "repository": "32805433/Adian-Rabin", "commit": "fb4a401e8fbdba36e7b35c11fde2baa15592e185", "theorems": [ "Undecidability.exists_three_generator_nine_relator_group_with_unsolvable_word_problem", "Undecidability.exists_four_generator_eleven_relator_adian_rabin_family", "Undecidability.exists_two_generator_ten_relator_adian_rabin_family" ], "checked": "2026-09-22", "note": "Registered by the paper's own authors. These are Theorems 1.1, 1.3 and 1.4; the record states that the four-manifold consequences are outside the formalized scope." } ], "sources": [ { "label": "arXiv: Small undecidable groups and unrecognizable 4-manifolds", "url": "https://arxiv.org/abs/2609.10461", "kind": "research" }, { "label": "GitHub: 32805433/Adian-Rabin, the Lean 4 formalization of the algebraic results", "url": "https://github.com/32805433/Adian-Rabin", "kind": "research" }, { "label": "arXiv: Tancer, Simpler algorithmically unrecognizable 4-manifolds, the prior deficiency-9 bound", "url": "https://arxiv.org/abs/2310.07421", "kind": "research" } ], "tags": [ "group-theory", "logic", "construction", "lean", "formalization" ], "added": "2026-09-24" }, { "id": "2026-09-23-array-associated-reverse-transcriptases", "title": "Agent-run genome survey identifies array-associated reverse transcriptases in jumbo phages", "claim": "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.", "field": "biology", "date": "2026-09-23", "lab": "Anthropic", "model": "Claude Mythos 5", "humans": [ "Peter H. Yoon", "Januka S. Athukoralage", "Emmanuel Ameisen", "Eric Kauderer-Abrams", "Nicholas T. Perry", "Matthew G. Durrant" ], "verification": "claimed", "autonomy": "collaborative", "detail": "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.", "novelty_check": "The preprint states that the family belongs to no described reverse transcriptase class, and records the nearest prior work itself: the 2021 genome report for the jumbo phage MarsHill identified this reverse transcriptase and proposed an upstream non-coding RNA but described neither the repeats nor the partner gene, and arrays beside the unrelated UG27 reverse transcriptases were reported in a September 2026 bioRxiv preprint, which the authors read as the architecture arising twice. So the enzyme itself was not new; the claim is the array and partner that define the family. The registry was searched on 2026-09-24 for reverse transcriptase, retron, CRISPR and jumbo with no hits. A web search for array-associated reverse transcriptase returned only the announcement and its press coverage, and a search for jumbo-phage reverse transcriptases with tandem repeat arrays returned the DRT and retron literature, including the 2025 finding that phage infection triggers reverse transcription of a non-coding RNA into a repetitive gene. That mechanism builds repeats by reverse transcription rather than carrying them in the genome, which looks distinct, but a specialist should confirm it.", "caveats": "A lab announcement about the lab's own model, so it enters at claimed under editorial rule 1. The preprint is hosted on Anthropic's own site, not a preprint server, and is unreviewed. It carries no data or code availability statement, and the session transcripts it quotes are not published, so the account of which step the agents took rests on the lab's word. The authors state the biological limit plainly: they have not shown that the reverse transcriptase is active, that the unit RNAs are its substrates, or what the system does for the phage, so what is established is a sequence-level family with transcription data, not a characterized enzyme system. Autonomy is graded collaborative, the weaker reading under editorial rule 5. An agent made the observation that defines the family, reading DNA outside what the brief asked for, but humans wrote the brief, chose which of the 19 reports to pursue, defined the family in interactive sessions with Claude and ran the experiments, and the only record of the agents' part is transcripts the lab has not published. The announcement names no model; the model field comes from the preprint. Feng Zhang, quoted in the announcement, is thanked in the preprint for reviewing an early copy, so his remark is not independent commentary. A top Hacker News comment argues for a plainer framing, that Claude identified an undescribed genomic arrangement around a known retron-like reverse transcriptase, which matches the novelty check. The preprint also attributes the recognition to specific internal Mythos 5 signals that respond to repeated DNA; that is a claim about the model, not the biology, and is not graded here.", "sources": [ { "label": "Anthropic preprint: Autonomous AI agents discover reverse transcriptases with tandem repeat arrays", "url": "https://www-cdn.anthropic.com/22573675ada52a8ca8a97a1a4b4326b2f208a071.pdf", "kind": "research" }, { "label": "Korn et al. 2021, the MarsHill genome report that first identified the reverse transcriptase", "url": "https://doi.org/10.1128/jvi.02391-20", "kind": "research" }, { "label": "Anthropic: Claude discovers a novel enzyme system with CRISPR-like repeats", "url": "https://www.anthropic.com/news/claude-discovers-novel-enzyme-system", "kind": "announcement" } ], "discussion": [ { "label": "Hacker News: Claude discovers a novel enzyme system with CRISPR-like repeats", "url": "https://news.ycombinator.com/item?id=49820134" } ], "tags": [ "genomics", "bacteriophage", "multi-agent", "autonomous-discovery" ], "added": "2026-09-24" }, { "id": "2026-09-15-bit-php-resolution-over-parities", "title": "Exponential lower bound for the bit pigeonhole principle in unrestricted resolution over parities", "claim": "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.", "field": "computer-science", "date": "2026-09-15", "lab": "Independent", "model": "GPT-6 Astra (mathematics, version 1 formalization and drafting); Claude Fable 5.1 and Claude Opus 5 (revision 1 formalization)", "humans": [ "Kamil Braun" ], "verification": "claimed", "autonomy": "ai-led", "detail": "Resolution over parities, \\(\\mathrm{Res}(\\oplus)\\), extends resolution by letting clauses be disjunctions of affine equations over \\(\\mathbb{F}_2\\). 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 \\(\\mathrm{Res}(\\oplus)\\) refutation of the bit pigeonhole principle with \\(n+1\\) pigeons and \\(n = 2^l\\) holes has more than \\(\\exp(n/(32768\\,l^2))\\) 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 \\(\\mathrm{Res}(\\oplus)\\) size lower bounds, and their dependencies are formalized in Lean 4 in the author's repository.", "novelty_check": "The 2026-09-28 watch run surfaced this through vibemathed; this check read the paper's related-work section and searched arXiv and the web. The closest prior results found keep a restriction: Byramji and Impagliazzo (arXiv:2511.20023, November 2025) prove exponential bounds for bit pigeonhole principles only in bounded-depth \\(\\mathrm{Res}(\\oplus)\\), and the paper itself surveys regular and bounded-depth results by Efremenko, Garlík, Itsykson, Alekseev and others, each with a regularity or depth restriction. No earlier superpolynomial lower bound for unrestricted DAG-like \\(\\mathrm{Res}(\\oplus)\\) was found, which matches the abstract's framing. The year the question was first posed is not stated in the paper and is left unrecorded rather than guessed: Raz and Tzameret introduced the integer version in 2008 and Itsykson and Sokolov studied the \\(\\mathbb{F}_2\\) version from 2014. This was not a systematic review of the proof-complexity literature.", "caveats": "The author states that he does not have the expertise to check the mathematics and did not check it, and that his confidence rests on the Lean formalization. The paper also says what that formalization does not cover: the prose, the claims about the literature, and the fidelity of the formal definitions to the standard proof system. That last point decides what the Lean is worth, and no third party has audited the Lean definitions of \\(\\mathrm{Res}(\\oplus)\\) or of the bit pigeonhole principle; there is no Palomar registration. The review rounds recorded in the repository were conversations with Claude and ChatGPT, which the author labels as not independent. Autonomy is ai-led rather than autonomous on the strictest defensible reading: the author chose the problem and steered the research loop, while his disclosure attributes all mathematical derivations to the models, with GPT-6 Astra the main contributor to version 1. A self-published claim on a well-studied problem, unrefereed and under two weeks old at entry.", "registrations": [ { "registry": "vibemathed", "id": "superpolynomial-lower-bounds-for-bit-php-in-unrestricted-resolution-over-paritie", "url": "https://vibemathed.com/problem/superpolynomial-lower-bounds-for-bit-php-in-unrestricted-resolution-over-paritie" } ], "sources": [ { "label": "arXiv: An exponential lower bound for the bit pigeonhole principle in resolution over parities", "url": "https://arxiv.org/abs/2609.23015", "kind": "research" }, { "label": "GitHub: kbr-/math-research, the preprint sources and Lean formalization", "url": "https://github.com/kbr-/math-research", "kind": "research" }, { "label": "Byramji and Impagliazzo: the prior bounded-depth result", "url": "https://arxiv.org/abs/2511.20023", "kind": "research" } ], "tags": [ "proof-complexity", "pigeonhole-principle", "lean", "formalization", "argument" ], "added": "2026-09-28" }, { "id": "2026-09-16-four-color-theorem-lean", "title": "Four Color Theorem formalized in Lean 4 by AI agents, in two independent developments", "claim": "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.", "field": "mathematics", "date": "2026-09-16", "lab": "Independent", "model": "Claude Fable 5.1 and Claude Opus 5 (Emery); Claude Opus 5.5 (Barish)", "humans": [ "Chris Emery", "Robert D. Barish" ], "year_posed": 1852, "verification": "claimed", "autonomy": "ai-led", "detail": "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.", "novelty_check": "This is a formalization of a known theorem, not new mathematics, and is graded as such. Appel and Haken proved the theorem in 1976, Robertson, Sanders, Seymour and Thomas gave a simplified proof in 1997, and Gonthier and Werner machine-checked it in Coq in 2005; both Lean developments follow that Coq proof. A web search and the two READMEs found no earlier complete Lean 4 proof: Barish's README names Emery's as public before his own, and Emery's repository was created on 2026-09-16, ten days before Barish's. The 2026-09-28 watch run surfaced only Barish's port, through its Palomar registration.", "caveats": "No human has reviewed either development. Barish's README says the final comparison of the statements against Gonthier's was done by a separate Claude Opus 5.5 agent that had not written the proof, and Emery's asks readers not to take the theorem on his authority or the model's. The checks recorded are the authors' own runs; nothing here has been rebuilt. Graded claimed rather than formal, following this registry's practice for Palomar-registered proofs: a registration shows the proof typechecks against a recorded statement, not that the statement is the theorem, and no third party has audited either statement file. What would move the grade is small by design, since each statement is about 200 lines of elementary definitions. Autonomy is ai-led rather than autonomous because in both cases a human set the goal and constraints and ran the final builds, while the disclosures credit the models with essentially all of the code.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-27-000005", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-27-000005&version=1", "repository": "RBarish-UTokyo/FourColorTheorem-Lean4", "commit": "20fa34599f5189dd569d96dd22f306c94ab1c446", "theorems": [ "FourColor.RealPlane.four_color" ], "checked": "2026-09-27", "note": "Covers Barish's port only; Emery's development carries no registration." } ], "sources": [ { "label": "GitHub: corun1024/4ct, Chris Emery's Lean 4 proof", "url": "https://github.com/corun1024/4ct", "kind": "research" }, { "label": "GitHub: RBarish-UTokyo/FourColorTheorem-Lean4, Robert D. Barish's port", "url": "https://github.com/RBarish-UTokyo/FourColorTheorem-Lean4", "kind": "research" }, { "label": "GitHub: math-comp/fourcolor, Gonthier and Werner's Coq proof that both follow", "url": "https://github.com/math-comp/fourcolor", "kind": "research" } ], "tags": [ "graph-theory", "four-color-theorem", "lean", "formalization" ], "added": "2026-09-28" }, { "id": "2026-09-26-erdos-1220-not-provable", "title": "Erdős Problem #1220 formalized as not provable in ZFC, from a 1987 Shelah-Stanley consistency result", "claim": "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.", "field": "mathematics", "date": "2026-09-26", "lab": "Independent", "model": "Claude Opus 5.5 (Claude Code); Astra (OpenAI Codex)", "humans": [ "Ji Ho Bae" ], "year_posed": 1971, "verification": "author-verified", "autonomy": "ai-assisted", "detail": "Erdős Problem #1220, from Erdős and Hajnal in 1971, asks whether \\(\\lambda \\to (\\lambda, \\aleph_1)^2\\) holds for every singular cardinal \\(\\lambda\\) with \\(\\lambda\\) and \\(\\operatorname{cf}(\\lambda)\\) both \\(\\aleph_0\\)-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 \\(\\lambda = \\beth_{\\mathfrak{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.", "novelty_check": "The mathematics is not new: Shelah and Stanley established the consistency of a negative answer in 1987 (Ann. Pure Appl. Logic 36), and the repository credits them for it. erdosproblems.com/1220, read on 2026-09-28, still lists the problem as open, cites only a Shelah theorem on a special case, records no formalized statement and does not mention Shelah-Stanley, so the connection the repository draws between the 1987 result and the problem as tracked is not yet reflected there. No earlier Lean formalization was found. The closest precedent is this registry's 2026-08-19-erdos-501-independence, whose Flypitch port and ZFC encoding this work reuses.", "caveats": "Graded author verified rather than formal. The author reports comparator and nanoda accepting both targets under the three standard axioms, and a Palomar registration records the proof, but no third party has audited the Mathlib-level statement of the problem against erdosproblems.com's wording, and unlike the #501 entry there is no Formal Conjectures statement to anchor it to. The result is non-provability only, so the problem is settled in the sense the repository states, not by an outright answer. Autonomy is ai-assisted: the README credits the author with the problem, the literature connection, the mathematical direction and the review, and credits the models with writing Lean code under his direction, running builds and audits, and cross-reviewing. Unrefereed, and published as a repository rather than a paper.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-26-000001", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-26-000001&version=1", "repository": "jbaelaw/erdos1220-lean", "commit": "9cb81ffa48ffb766127c7e485ccef77bd4160868", "theorems": [ "erdos1220_not_provable", "erdos1220_sentence_faithful" ], "checked": "2026-09-26" } ], "sources": [ { "label": "GitHub: jbaelaw/erdos1220-lean", "url": "https://github.com/jbaelaw/erdos1220-lean", "kind": "research" }, { "label": "Erdős Problems: problem #1220", "url": "https://www.erdosproblems.com/1220", "kind": "commentary" } ], "tags": [ "set-theory", "erdos", "lean", "formalization", "forcing" ], "added": "2026-09-28" }, { "id": "2026-09-26-lovasz-cayley-polylog-degree", "title": "Hamilton cycles in connected Cayley graphs of polylogarithmic degree, toward the Lovász conjecture", "claim": "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.", "field": "mathematics", "date": "2026-09-26", "lab": "Independent", "model": "GPT 6 Pro (manuscript draft); Claude Opus 5.5 agents in Grok Build (Lean formalization)", "humans": [ "Domagoj Bradač", "Matija Bucić", "Micha Christoph", "Zach Hunter", "Oliver Janzer", "Alp Müyesser", "Shengtong Zhang" ], "year_posed": 1969, "verification": "claimed", "autonomy": "collaborative", "detail": "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.", "novelty_check": "The README's context section and a web search place the result in an active 2026 line of work: the \\(n^{1-c}\\) threshold of arXiv:2603.08675, Hamiltonicity of regular sublinear expanders (arXiv:2605.15043), a sublinear-expander approach to the Lovász conjecture (arXiv:2606.09742) and almost Hamilton cycles at polylogarithmic degree (arXiv:2609.30165). Each of those papers has at least one of the draft's named authors, so the draft reads as the continuation of that program rather than a result competing with it; the 2026-09-28 watch run's concern that it might be superseded by that literature confused the two. No earlier result giving full Hamiltonicity at polylogarithmic degree was found. The draft builds on three unpublished manuscripts listed in its bibliography, which are not public, so what is new in the draft relative to them cannot be checked.", "caveats": "The draft calls its own proof a consolidated proposed proof, not an independently verified theorem, and the PDF carries no author line: the six authors are named by the repository's maintainer, Shengtong Zhang, who directed the formalization, wrote no Lean and is not an author of the manuscript. No human has reviewed the Lean statement or proof; the statement was audited by the formalizing agent and by agent runs of Palomar's review policy. Graded claimed for that reason, since a Palomar registration shows the proof typechecks against the recorded statement, not that the statement is the theorem. Autonomy is collaborative on the disclosure's own wording, that GPT 6 Pro prepared the draft with significant input from the authors including unpublished human manuscripts; how much of the argument is the model's and how much the manuscripts' is not documented, and if it proves to be mostly the latter the grade should move down to ai-assisted. The formalization is entirely agent-written, by 38 parallel subagents under a coordinating agent. Unrefereed, and days old at entry.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-26-000003", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-26-000003&version=1", "repository": "ShengtongZhang-alt/lovasz-opus", "commit": "ee0085ddec79996279c7b52044659032e5cf8431", "theorems": [ "Lovasz.hamiltonian_of_polylog_degree" ], "checked": "2026-09-26" } ], "sources": [ { "label": "GitHub: ShengtongZhang-alt/lovasz-opus, the draft and its Lean formalization", "url": "https://github.com/ShengtongZhang-alt/lovasz-opus", "kind": "research" }, { "label": "Bedert, Draganić, Müyesser and Pavez-Signé: the prior n¹⁻ᶜ threshold", "url": "https://arxiv.org/abs/2603.08675", "kind": "research" } ], "tags": [ "graph-theory", "combinatorics", "lovasz-conjecture", "lean", "formalization", "argument" ], "added": "2026-09-28" }, { "id": "2026-08-27-norm-variation-ergodic-averages-lean", "title": "Norm-variation of multiple ergodic averages, including Tao's norm-convergence theorem, autoformalized in Lean in one week", "claim": "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.", "field": "mathematics", "date": "2026-08-27", "lab": "Independent", "model": "Coding agents, models not stated", "humans": [ "Floris van Doorn", "Polona Durcik", "Joris Roos", "Lenka Slavíková", "Christoph Thiele" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "The companion arXiv paper (2608.27321) is a blueprint for a forthcoming traditional paper by the same authors. It proves \\(r\\)-variation bounds in \\(L^2\\) 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.", "novelty_check": "The formalized mathematics is the human authors', not the models': the blueprint's abstract attributes the new estimate to the authors and says the formalization was completed largely automatically using frontier large language models. Only the blueprint's abstract, the repository and this registry were checked; no search for earlier formalizations of Tao's theorem was made, and that is the first gap to close. The 2026-09-28 watch run surfaced this through its Palomar registration.", "caveats": "Which models were used is not stated in the repository, the Palomar abstract or the arXiv abstract, which say only coding agents and frontier large language models, and the model field records exactly that. Graded author verified: the statements were formalized by hand by the authors and the proof carries a Palomar registration, but no third party has audited the statements or rebuilt the development. Autonomy is ai-led on the same reading as this registry's Fermat's Last Theorem entry, with the models writing essentially all of the proof code while the humans wrote the blueprint, formalized the statements and supervised; it is not autonomous because the mathematics being formalized is the authors'. The traditional paper the blueprint accompanies had not appeared at entry, and the new estimate is unrefereed.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-09-15-000008", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-15-000008&version=1", "repository": "roos-j/lean-nct", "commit": "3e09aa52c478ec69abb02eb250c5e5ed06653c3a", "theorems": [ "nCT.main_ergodic_theorem", "nCT.tao_norm_convergence", "nCT.main_twisted_theorem" ], "checked": "2026-09-15" } ], "sources": [ { "label": "arXiv: A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations", "url": "https://arxiv.org/abs/2608.27321", "kind": "research" }, { "label": "GitHub: roos-j/lean-nct", "url": "https://github.com/roos-j/lean-nct", "kind": "research" } ], "tags": [ "ergodic-theory", "harmonic-analysis", "lean", "formalization" ], "added": "2026-09-28" }, { "id": "2026-09-25-nine-loop-hexagon-amplitude", "title": "Six-particle MHV amplitude in planar N=4 super Yang-Mills computed at nine loops", "claim": "Claude computed the six-particle maximally-helicity-violating amplitude in planar \\(\\mathcal{N} = 4\\) super Yang-Mills theory at nine loops, one loop beyond the previous record, from a single prompt naming the problem.", "field": "physics", "date": "2026-09-25", "lab": "Anthropic", "model": "Claude Fable 5.1 (in Claude Science)", "humans": [ "Liam Fitzpatrick", "Siddharth Mishra-Sharma" ], "verification": "author-verified", "autonomy": "ai-led", "detail": "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.", "novelty_check": "Searched arXiv on 2026-10-01 (abstract query \"nine loops\" AND amplitude) and read the announcement and both data deposits. The previous record for this amplitude is eight loops, arXiv:2308.08199 (Dixon and Liu, 2023), built on the eight-loop form factor of arXiv:2204.11901. The only arXiv hit at nine or more loops, arXiv:1609.00007, determines loop integrands of four- and five-point quantities, not integrated six-point functions, and is not prior art. The result is not uniquely first: He, Jing and Li deposited the symbols of the six-gluon MHV amplitudes through nine loops on Zenodo on 2026-09-17 (doi:10.5281/zenodo.22800071), and Claude's own data page is dated 2026-09-16. Dixon writes that the He group used GPT-6-based assistance and computed the symbol, the part of the amplitude below the full function; their deposit itself carries no AI disclosure. Neither group has posted a paper, and no statement that the two symbols agree was found.", "caveats": "Verification is author-verified rather than independent. The one outside expert who checked the result, Lance Dixon, was asked to by Anthropic, received Claude usage credits, and writes in the announcement itself, while the independent grade asks for several unaffiliated people. The concurrent He-Jing-Li symbol is the obvious second check, but no comparison between the two has been published. Autonomy is ai-led rather than autonomous on the stricter reading. The post describes no human mathematical input beyond posing the problem, and Claude chose both methods, but both the bootstrap and the antipodal-duality route were developed by humans and used for the eight-loop result, so the contribution is carrying an established programme one order further rather than a new idea. The physicists also asked Claude which open problem it was most likely to manage before posing this one, so the choice of problem was partly the model's. Nothing has been refereed and there is no paper. The announcement is a guest post on the lab's blog by the science writer Matt von Hippel, whose earlier post had set the challenge. The Zenodo deposit is data with validation records, not a rerunnable pipeline, so reproducing the computation rather than checking its output would need independent code.", "independent_checks": [ { "who": "Lance Dixon (SLAC), at Anthropic's request and with Claude usage credits", "url": "https://www.anthropic.com/research/yes-claude-can-do-nine-loops", "outcome": "validated the nine-loop result, mostly by mapping the amplitude back to the three-point form factor" } ], "sources": [ { "label": "Anthropic: Yes, Claude can do nine loops", "url": "https://www.anthropic.com/research/yes-claude-can-do-nine-loops", "kind": "announcement" }, { "label": "Zenodo: nine-loop six-gluon MHV amplitude, large data files", "url": "https://zenodo.org/records/22949278", "kind": "research" }, { "label": "Cosmic9: symbol and function files with validation records", "url": "https://smsharma.io/cosmic-nine-loops/", "kind": "research" } ], "discussion": [ { "label": "Hacker News", "url": "https://news.ycombinator.com/item?id=49848033" } ], "tags": [ "scattering-amplitudes", "quantum-field-theory", "agents", "computation" ], "added": "2026-10-01" }, { "id": "2026-09-30-pinwheel-kernel-conjecture", "title": "Six-task counterexample to the Kernel Conjecture of pinwheel scheduling", "claim": "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 \\(2^k\\) Conjecture of Gąsieniec, Smith and Wild.", "field": "computer-science", "date": "2026-09-30", "lab": "Independent", "model": "GPT-6 Astra (in OpenAI Codex)", "year_posed": 2021, "verification": "formal", "autonomy": "ai-led", "detail": "In pinwheel scheduling one task is performed per day, and a task with period \\(a_i\\) must be performed at least once in every \\(a_i\\) 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 \\(2^k\\) 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.", "novelty_check": "Read Conjectures 2.2 and 2.3 and Proposition 2.4 of arXiv:2111.01784 (Gąsieniec, Smith and Wild, 2021; ALENEX 2022) on 2026-10-06 and confirmed that the instance contradicts them as stated, domination meaning componentwise at most. Web searches for the Kernel Conjecture and the \\(2^k\\) Conjecture, and an arXiv title scan of pinwheel scheduling papers from 2024 to 2026, found no earlier counterexample or resolution; the 2026 papers on pinwheel density thresholds address, by their titles, a different conjecture. That is not a systematic review of the scheduling literature.", "caveats": "No paper and no peer review: the evidence is a GitHub repository holding the certificates, a Python checker and a Lean development, all produced by the workflow that found the example. The human who directed it is not named. The repository credits the model with discovering the example, writing the searches, certificates and checker, and developing the formalization, and the human with setting the objective, supervising, and asking for the prior-work and formal checks, which is why autonomy is ai-led rather than autonomous. The Lean build was run by the authors on a cloud worker and replayed through the Nanoda kernel, also by the authors; nobody outside has compiled it or audited that the Lean statements match the conjecture. Graded formal because the finite claim is exactly rerunnable, and this registry reran it with separate code. The result bounds any universal six-task cap from below by 36; it does not show that 36 suffices.", "independent_checks": [ { "who": "whataifound.org (independent recomputation, Python)", "outcome": "Wrote a feasibility search sharing no code with the repository. From the all-zero age vector, \\((3, 4, 5, 20, 22, 32)\\) and \\((3, 4, 5, 20, 22, 35)\\) reach 50,881 and 56,284 states, the counts the two certificates report, and in both every walk dies out once dead ends are pruned; \\((3, 4, 5, 20, 22, 36)\\) keeps an infinite walk, and the repository's 36-day witness word meets every period. This checks the finite claim, not the Lean development." }, { "who": "Nanoda kernel replay (author-run, reports public)", "url": "https://github.com/MathIsEvenEasier/pinwheel-kernel-counterexample", "outcome": "Replayed the four endpoint theorems and their dependencies, 6,850 declarations, on 2026-10-01, admitting only propext and Quot.sound." } ], "registrations": [ { "registry": "vibemathed", "id": "the-pinwheel-kernel-conjecture-a-six-task-counterexample", "url": "https://vibemathed.com/problem/the-pinwheel-kernel-conjecture-a-six-task-counterexample" } ], "sources": [ { "label": "GitHub: MathIsEvenEasier/pinwheel-kernel-counterexample", "url": "https://github.com/MathIsEvenEasier/pinwheel-kernel-counterexample", "kind": "research" } ], "tags": [ "scheduling", "algorithms", "counterexample", "lean", "formalization", "construction" ], "added": "2026-10-06" }, { "id": "2026-10-02-squares-in-circle-packings", "title": "Optimal packings of one to seven unit squares in a circle, proved in Lean", "claim": "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.", "field": "mathematics", "date": "2026-10-02", "lab": "Independent", "model": "ChatGPT (GPT-6 Pro) and Claude Opus 5 and 5.5", "humans": [ "The-Anh Vu-Le" ], "verification": "formal", "autonomy": "ai-led", "detail": "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 \\(\\sqrt{2}/2\\), \\(\\sqrt{5}/2\\), \\(5\\sqrt{17}/16\\), \\(\\sqrt{2}\\), \\(\\sqrt{5/2}\\), \\(\\sqrt{q^*}\\) with \\(q^*\\) a quartic irrational near \\(2.851\\), and \\(\\sqrt{13}/2\\). The optimal packing is unique up to rotation and relabelling for \\(n \\le 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.", "novelty_check": "Read the repository's prior-work review, current to September 2026, and checked Erich Friedman's Squares in Circles page on 2026-10-06. One and two squares are folklore and four squares were Problem 6 of the 2026 International Mathematics Summer Camp. For three squares Montanher, Neumaier, Markót, Domes and Schichl (J. Global Optim., 2019) enclosed the radius by computer-assisted interval arithmetic, without an exact value or exact uniqueness. For five, six and seven squares the review finds no earlier proof: Friedman's page listed his 1997 packings only as the best known in its July 2026 snapshot, and now records three, five, six and seven squares as proved by The-Anh Vu-Le in September 2026. No earlier proof-assistant verification of any optimal packing of squares or circles in a circle was found. The year the question was first posed is left unrecorded rather than guessed.", "caveats": "Not peer-reviewed, and there is no paper; the proofs are written up as a textbook inside the repository. Graded formal on the Palomar registration, which checks the proofs against a separate statement file importing only Mathlib and replays them through three kernels. What that certifies is the statement in that file: the kernel checks the proofs, not that the definitions of square, disk and packing capture the intended problem, and no outside audit of those definitions was found. For seven squares, uniqueness holds only up to the heights of the three middle squares. The four-square uniqueness argument matches an unpublished July 2026 note by Wei Zhao that was also developed with Claude, and the repository itself says the two may not be independent. Autonomy is ai-led rather than autonomous because the owner directed and reviewed the work.", "registrations": [ { "registry": "palomar", "id": "PALOMAR-2026-10-02-000002", "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-10-02-000002&version=1", "repository": "vltanh/lean4-squares-in-circles", "commit": "621f2340a12f661a51b613719107a75cf40ab03a", "theorems": [ "SquaresInCircles.optimal_radius", "SquaresInCircles.optimal_packings" ], "checked": "2026-10-02", "note": "Registered against Challenge.lean, which restates the definitions of Geometry.lean word for word and states both theorems with sorry. Permitted axioms are propext, Classical.choice and Quot.sound." } ], "sources": [ { "label": "GitHub: vltanh/lean4-squares-in-circles", "url": "https://github.com/vltanh/lean4-squares-in-circles", "kind": "research" }, { "label": "Erich Friedman: Squares in Circles", "url": "https://erich-friedman.github.io/packing/squincir/", "kind": "commentary" } ], "discussion": [ { "label": "Reddit r/math: AI in mathematics, October 3, 2026", "url": "https://www.reddit.com/r/math/comments/1wwqvov/ai_in_mathematics_october_03_2026/pdnjvo8/" } ], "tags": [ "discrete-geometry", "packing", "lean", "formalization", "argument" ], "added": "2026-10-06" }, { "id": "2026-10-01-anisotropic-watson-integral", "title": "Closed form for the anisotropic Watson integral of the cubic lattice", "claim": "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.", "field": "mathematics", "date": "2026-10-01", "lab": "Independent", "model": "Claude (the announcement names Claude Fable 5)", "humans": [ "Noam Elkies", "Thomas W. Grimm", "Matthew D. Schwartz" ], "year_posed": 1939, "verification": "claimed", "autonomy": "ai-assisted", "detail": "The lattice Green's function \\[W(\\alpha_1, \\alpha_2, \\alpha_3; w) = \\frac{1}{\\pi^3}\\int_{[0, \\pi]^3} \\frac{d^3k}{w - \\alpha_1\\cos k_1 - \\alpha_2\\cos k_2 - \\alpha_3\\cos k_3}\\] gives the return probability \\(R = 1 - 1/(w^\\star W)\\) at the band edge \\(w^\\star = \\alpha_1 + \\alpha_2 + \\alpha_3\\). 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 \\(\\mathrm{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.", "novelty_check": "Read the announcement and the authors' BootLoops summary on 2026-10-06 and searched the web for closed forms of the anisotropic simple cubic lattice Green's function. The solved cases are documented: Watson (1939) for equal rates at the band edge, Joyce (1973) for equal rates at every \\(w\\), Glasser and Zucker (1977) in gamma functions, and Delves and Joyce (2001) for two equal rates. Zucker's survey '70+ years of the Watson integrals' (J. Stat. Phys., 2011) called the three-rate case the final problem and judged its difficulties insuperable. No earlier closed form for three unequal rates was found. The paper is not yet public, so the claim could not be checked against its own literature review.", "caveats": "Claimed because there is no paper yet, only a summary that lists it as in preparation, so the proofs of the fifth-order equation, of the absence of an elliptic formula and of the closed form itself are not public. What is public is the formula as runnable code, and agreement to many digits is a numerical check rather than a proof, as the summary itself says; one further statement there, an absolute theta identity, is left as a conjecture. The published script handles generic rates only: on the walls where two or three rates are equal, the cases the classical formulas already cover, it raised an error or, at one equal-rate point tried here, returned a wrong value. That is a limit of the script, not evidence against the theorem, which the summary says holds on the walls too. Autonomy is graded ai-assisted, the weaker defensible reading: the summary says only that the authors used AI tools interactively and that the code was written by Claude under Schwartz's supervision, the essay frames the problem as one Claude found through the harness, and the division of labour on the proofs is not documented. The essay is on Anthropic's blog, Schwartz is a visiting researcher there and the code is copyright Anthropic, although the essay states that BootLoops is not an Anthropic project. The essay's other results are not entered here and need separate grading as their manuscripts appear.", "independent_checks": [ { "who": "whataifound.org (numerical comparison, mpmath)", "url": "https://www.bootloops.ai/files/lgf/WatsonW.py", "outcome": "Evaluated the published closed form and compared it with the classical Bessel-function form \\(\\int_0^\\infty e^{-wt} I_0(\\alpha_1 t) I_0(\\alpha_2 t) I_0(\\alpha_3 t)\\,dt\\), computed separately by quadrature that reproduces Watson's \\(0.5054620197\\) at equal rates. At the summary's points \\((1, 5, 7; 17)\\) and \\((5, 6, 9; 22)\\) the two agree to 24 and 23 digits, and at four further generic rate triples, two of them close to the walls, to at least 15. On the walls the script fails. Numerical agreement only, not a check of the proof." } ], "sources": [ { "label": "Anthropic: Claude-shaped science (guest essay by Matthew Schwartz)", "url": "https://www.anthropic.com/research/claude-shaped-science", "kind": "announcement" }, { "label": "BootLoops: The Watson integral (authors' summary)", "url": "https://www.bootloops.ai/summaries/watson.html", "kind": "announcement" }, { "label": "BootLoops: WatsonW.py, the closed form as code", "url": "https://www.bootloops.ai/files/lgf/WatsonW.py", "kind": "research" } ], "tags": [ "lattice-green-functions", "random-walks", "algebraic-geometry", "argument" ], "added": "2026-10-06" } ]