Benefit, harm, and capability ledgers
Events
Frequency · stacked by domain
Events over time
- Mathematics
Heatmap
Domain by month
Significance
Impact tiers
Base impact
- T1 · 1Useful / Limited
- T2 · 3Notable / Significant
- T3 · 10Major
- T4 · 30Historic
- T5 · 100Civilizational / Catastrophic
Base points before attribution, evidence, realization, durability, and credit share.
MethodologyFrequency
Benefit versus harm counts
OpenAI announces three AI-assisted research claims across mathematics, computer science, and physics
OpenAI says its internal models produced results on all-dimensional local smoothing, the Exponential Time Hypothesis, and quantum-field-theory existence. Two writeups and Lean files are public; the physics work remains under academic review and is not yet released.
OpenAI releases 722 mathematics manuscripts from an unreleased internal model
On 6 October 2026 OpenAI published 722 mathematics manuscripts grouped into 372 result families, produced mostly by an unreleased internal frontier model; some proofs have Lean formalizations, while others remain unverified.
ChatGPT contributes most proof ideas in a proposed proof of the KLS conjecture
A 4 October 2026 arXiv preprint presents a proof of the Kannan–Lovász–Simonovits conjecture and says most proofs and mathematical ideas were found by ChatGPT; the authors supplied the suspension idea and improved the exposition.
Lean Pool: An AI-Maintained Archive of Formalized Mathematics
arXiv:2609.25199v1 Announce Type: new Abstract: Lean Pool is a repository of formalized mathematics. It is grown, maintained and optimized by AI agents.
CUBE: Multimodal Representation Learning Reveals Biological Structure Across Histomorphology, Spatial Protein Phenotypes, and Transcriptome-Associated Signals
Integrating histological, spatial protein, and transcriptomic information into a biologically grounded representation remains challenging because these modalities are rarely available as fully paired measurements, while existing computational approaches are commonly developed around individual modality pairs. To address this, we present CUBE (Colorectal Universal Representation & Bridge Encoder), a multimodal representation-learning framework that uses hematoxylin and eosin (H&E) histology as a bridge to integrate spatial protein phenotypes with transcriptome-associated information from incomp
ZGCM-1: A Fully Open and Extremely Efficient Foundation Model for Math and Agentic Search
arXiv:2609.13356v1 Announce Type: new Abstract: In this work, we present ZGCM-1, a fully open 7B dense foundation model trained from scratch with extreme data, system, and algorithmic efficiency. ZGCM-1 is founded on a core premise: compact models cannot passively memorize the open web, but can overcome parametric capacity limits by coupling deliberate internal thinking with active external tool use. To support this paradigm across a 256K context, we develop an end-to-end, high-efficiency open training recipe: Architecture & System Co-design: interleaved gated sliding-window and full attentio
OpenAI breakthrough triggers ‘existential crisis’ in math
Claimed Navier-Stokes solution has polarized mathematicians over AI’s role—and what it means for the next generation
Ten numbers from the Laplace-Beltrami spectrum facilitate training-free classification of protein structures based on surface shape
The comparison of protein structures is necessary to determine molecular functions and evolutionary relationships and can also facilitate drug design and the prediction of separation options. The number of available protein structures is rapidly increasing, driven by experimental determination but also prediction methods such as AlphaFold. In this context, traditional structure comparison based on atomic superposition becomes computationally inefficient. To enable scalable analysis and surface shape comparison, alternative approaches represent protein structures using compact, fixed-length vec
Population Intelligence: Smallest-Model Numerical Experiment
We introduce a minimal individual based model for population intelligence that retains three structures absent from standard single agent descriptions: a manifold valued intelligence state, endogenous population turnover, and heritable lineage dependent reproductive fitness. The validated S^1 intelligence diffusion was extended with neutral logistic birth death dynamics. Each individual occupies an angle on the circle S^1; noisy mean field attraction produces collective synchronization; logistic birth death dynamics regulate population size; and reproductive rates can depend on lineage and pos
Here is how to understand OpenAI’s major mathematical breakthrough - New Scientist
Here is how to understand OpenAI’s major mathematical breakthrough New Scientist
GPT-6 Astra proves the core is nonempty in approval-based committee elections
On 10 September 2026 Becker, Greger, and Dominik Peters posted a proof that every approval-based committee election has a core-stable committee. They attribute the primary idea and proof to GPT-6 Astra in a lengthy interactive session. Epoch marked the FrontierMath Open Problems item Solved (human + AI).
A Caltech PINN finds a candidate unforced 3D Euler blowup profile
On 9 September 2026 Ganeshram, Duruisseaux, and Anandkumar posted evidence of a numerically stable, approximately self-similar finite-time singularity for unforced 3D Euler on R³, found with a physics-informed neural network. Terence Tao called it a promising candidate, not a closed proof.
Buckmaster and Alpöge, using Sol and Claude, prove smooth-forced blowup for Euler, IPM, and Boussinesq
On 8–11 September 2026 Tristan Buckmaster and Levent Alpöge released Lean-backed finite-time blowup results with smooth forcing for incompressible porous media, 2D Boussinesq, and 3D Euler, crediting Anthropic’s Claude and especially OpenAI’s GPT-5.6 Sol. Astra was used only for writeups and auditing.
OpenAI reports a forced Navier–Stokes singularity matching Clay C and D
On 8 September 2026 OpenAI published a 166-page writeup and Lean formalization arguing that a smooth, finite-energy 3D Navier–Stokes flow starting from rest, under a smooth force, blows up in finite time. That is Clay statements C and D, which have been part of the official Millennium Problem since Fefferman’s formulation. The search model was an unreleased internal system; GPT-6 Astra did the Lean step. Clay wrote on 11 September that the problem has “apparently been settled,” with formal evaluation still to come.
An unreleased OpenAI system produces an unforced 3D Euler blowup
While probing Millennium problems in early September 2026, OpenAI says a multi-agent run of an internal model more capable than GPT-6 Astra produced a finite-time singularity for unforced 3D incompressible Euler in about 50 hours.
GPT-6 Astra improves short and large prime-gap bounds
On 3 September 2026 OpenAI released GPT-6 Astra with two analytic number-theory results: a proof that infinitely many consecutive primes differ by at most 186, and an improvement to a large-gap bound that had been stuck for more than 80 years.
Pre-release GPT-6 Astra solves five Bloom-curated Erdős problems in Lean
On 1 September 2026 Epoch AI reported that a pre-release GPT-6 Astra system was the only model to score on FrontierMath Erdős, a Lean benchmark of 68 significant unsolved Erdős problems. It solved problems 74 and 126 under the official $300 protocol, and problems 1, 548, and 571 in extra attempts.
Freitag and Mutchnik, using ChatGPT 5.6, counterexample the stable forking conjecture
On 31 August 2026 James Freitag and Scott Mutchnik posted arXiv:2609.00436, a counterexample to the 1996 Hart–Kim–Pillay stable forking conjecture, found using ChatGPT 5.6.
Alpöge credits Claude with a claimed complex structure on S^6
In late August 2026 Levent Alpöge posted a manuscript, produced with Claude, claiming a compact complex threefold diffeomorphic to the six-sphere. Independent writeups and a Lean formalization effort followed; expert review is ongoing and the claim is treated as disputed.
Claude, with Alpöge and Howell, posts rank ≥30 and ≥31 elliptic curves over Q
On 20 and 23 August 2026 the ICARM elliptic-curve rank leaderboard recorded new records of rank at least 30 and 31, with public witness points, credited to Claude working with Levent Alpöge and Ava Howell. Exact rank 30/31 is reported only under GRH and BSD.
A three-human plus Claude team is reported to construct a Hadamard matrix of order 668
In August 2026 Epoch’s open-problems index listed a Hadamard matrix of order 668 as Solved (AI) by a three-human plus Claude team, while the dedicated problem page still appeared to say open. The construction is included as disputed until those sources agree.
An unreleased Claude raises the critical-line zero bound for zeta from 41.6% to 67.2%
On 10 August 2026 Anthropic reported that an unreleased research Claude, prompted to attempt the Riemann hypothesis, instead improved the proven lower bound on zeros on the critical line from 41.6% to 67.2%. Anthropic mathematicians and two outside experts examined the argument. The Riemann hypothesis remains open.
GPT-6 Astra posts ten Lean-certified advances across math and TCS
On 1 August 2026 OpenAI released a paper and Lean repo of ten results from GPT-6 Astra, then still in development, spanning sphere packing, codes, non-sofic groups, Connes rigidity, circuit lower bounds, quantum parallel repetition, CVP hardness, Ehrhart volume, and Erdős problems 183, 146, and 180.
Liu, Luo, and Xie post a complete proof of the Tu–Deng conjecture
On 30 July 2026 Renzhang Liu, Hengyi Luo, and Tianyuan Xie posted a complete proof of the Tu–Deng combinatorial conjecture on Hamming weights modulo 2^k−1 (arXiv:2608.05187). A compiled dataset records ChatGPT 5.6 Pro assistance; the arXiv abstract does not name a model or split of labor.
GPT-5.6 Sol / Codex produces shorter superpermutations for n=8, 9, and 10
On 26 July 2026 Epoch recorded new superpermutation constructions for n=8, 9, and 10 that beat prior upper bounds, accepted by an exact verifier, from a GPT-5.6 Sol / Codex session.
Rybin, using GPT-5.6 Pro, counters the Dinitz–Garg–Goemans unsplittable-flow cost conjecture
On 22 July 2026 Dmitry Rybin announced an explicit 7-vertex graph in which a feasible fractional unsplittable-flow costs 58 while every congestion-legal unsplittable flow costs at least 60, disproving Goemans’ cost-preserving rounding conjecture. He attributes the construction to GPT-5.6 Pro. An Isabelle Archive of Formal Proofs entry later checks the finite instance.
Conant, with Claude Fable 5, proves the mod-4 Kawauchi conjecture
On 21 July 2026 Jim Conant posted a proof that the Conway polynomial of an amphicheiral knot satisfies ∇_K ≡ f(z)f(−z) (mod 4), a strengthening he had conjectured after the integral Kawauchi conjecture failed. He credits Claude Fable 5 with help producing the proof.
GPT-5.6 Sol proves Chernoff’s density is strongly log-concave
A 21 July 2026 arXiv note proves Balabdaoui and Wellner’s 2014 conjecture that Chernoff’s density is strongly log-concave. The authors state the proof was generated in its entirety by GPT-5.6 Sol and then checked and revised by them.
Long, using Sol Pro and Fable 5, posts low-degree Gaussian Moments counterexamples
On 20 July 2026 Christopher D. Long posted explicit polynomials showing the Gaussian Moments conjecture fails in every dimension n≥3: a cubic in four variables found by GPT-5.6 Sol Pro, then a quartic in three variables found by Claude Fable 5. This is not a Jacobian-conjecture score and not a Clay problem.
Claude Fable 5 produces a presentation of Gal(Q̄₂/Q₂)
On 9 June 2026 David Roe elicited a Claude Fable 5 solution to Epoch’s open problem asking for a presentation of the absolute Galois group of the 2-adics. Epoch later accepted verifier output and noted a second, independent GPT-5.5 Pro solution. Epoch warns the verifier is short of a full proof on this problem.
AlphaProof Nexus produces Lean proofs for nine Erdős instances and 44 OEIS conjectures
A 21 May 2026 DeepMind preprint reports formal proofs for nine of 353 Erdős problem instances and 44 of 492 OEIS conjectures, plus research collaborations. Its Gemini-led agents publish Lean artifacts; independent records distinguish full, partial, and variant results.
An unreleased OpenAI model disproves Erdős’s planar unit-distance conjecture
On 20 May 2026 OpenAI reported that an internal reasoning model produced an infinite family of planar point sets with n^{1+δ} unit distances for some δ>0, disproving Erdős’s n^{1+o(1)} conjecture. External mathematicians checked the argument and later made the exponent explicit.
Aletheia, powered by Gemini Deep Think, solves several Erdős items and writes an eigenweights paper
On 11 February 2026 DeepMind reported that Aletheia, a math agent on Gemini Deep Think, autonomously solved Erdős 1051 (motivating a generalization), disproved clauses of Erdős 1040 and a strong pinned-distance form of 654, and generated a publishable arithmetic-geometry paper on eigenweights without human intervention.
GPT-5 supplies key steps on Erdős 848, a tree-count inequality, and network identifiability
In a 20 November 2025 OpenAI science-acceleration paper, GPT-5 helped Mehtaab Sawhney and Mark Sellke finish Erdős problem 848, produced a short proof of a conjectured small-subgraph inequality in trees that Bubeck, Sellke, and Yin checked, and pointed to leaf fraction as the statistic that identifies a hidden network parameter.
Ryu and Jang, using GPT-5 Pro, prove point convergence of Nesterov accelerated gradient
In October 2025 UCLA mathematicians Ernest Ryu and Uijeong Jang resolved the long-open question of whether Nesterov’s 1983 accelerated gradient method converges in iterates, in a collaboration they describe as heavily assisted by GPT-5 Pro. The human authors wrote and checked the final proof.
Gemini Deep Think reaches IMO 2025 gold-medal standard
In July 2025, Google DeepMind’s Gemini Deep Think was officially graded at 35/42 on the IMO 2025 paper under contest-style conditions.
An OpenAI reasoning model reaches IMO 2025 gold-medal standard
In July 2025 OpenAI reported that an experimental reasoning model scored 35/42 on the IMO 2025 paper under contest-like constraints, graded by former medalists.
AlphaProof and AlphaGeometry 2 reach silver-medal standard on IMO 2024
DeepMind’s AlphaProof and AlphaGeometry 2 solved four of six IMO 2024 problems for 28 of 42 points, scored by Timothy Gowers and Joseph Myers, one point below that year’s gold cutoff.
AlphaGeometry solves olympiad plane-geometry problems without human proof data
DeepMind’s neuro-symbolic AlphaGeometry solved 25 of 30 IMO-AG-30 geometry problems in a Nature paper, approaching average human gold-medalist performance on that set.
FunSearch finds larger cap-set constructions via LLM program search
DeepMind’s FunSearch, pairing a language model with an automated evaluator, discovered new cap-set constructions including a 512-point cap in eight dimensions, published in Nature.