Machine learning systems claim credit for a cascade of mathematical proofs

A newly compiled index of 100 modern mathematical milestones attributes 87 percent of its recent breakthroughs to automated systems, raising questions about verification across the mathematical sciences.

For centuries, mathematical truth advanced through a steady, deliberate sequence of human reasoning. A mathematician identified an intractable open conjecture, introduced geometric or analytical tools to untangle its structural obstacles, and circulated an intricate manuscript for years of meticulous peer review. When a proof survived that trial, the result settled into the collective foundation of the discipline. That slow verification rhythm is now colliding with an unprecedented flood of automated assertions, as machine learning models generate purported solutions to classical open problems faster than human referees can read them.

The tension matters because mathematics provides the foundational certainty beneath modern cryptography, physical dynamics, and computation. Unlike empirical laboratory experiments, which can tolerate probabilistic error margins, a mathematical proof requires absolute logical validity across every deductive transition. If flawed automated arguments enter the literature unnoticed, downstream theorems built on those foundations risk sudden collapse.

Understanding this transition requires tracing how modern theorem proving operates. A conjecture begins as an unproven mathematical statement about numbers, geometric shapes, or physical equations. To establish its truth, a researcher must build an unbroken logical bridge from fundamental axioms to the final claim. Automated reasoning systems attempt to cross this bridge by combining large statistical language models, which suggest intuitive proof steps, with interactive proof assistants such as Lean or Coq. The proof assistant functions as an unforgiving referee, verifying that every symbolic deduction strictly satisfies formal grammatical and logical rules before accepting the step. In principle, this combination allows an algorithm to explore vast combinatorial spaces of potential proofs while keeping deduction rigorous.

In October 2026, an online catalogue compiled and shared by Will Depue presented a striking snapshot of how rapidly this landscape is shifting.¹ The compilation, presented under the title of a ranking comparing Fable 5.1 and GPT-6 Pro, catalogued 100 major mathematical and scientific milestones. According to the release posted by Depue, 87 percent of the listed advances were attributed to artificial intelligence systems, while human efforts accounted for only 13 percent of the breakthroughs.¹ Furthermore, Depue reported that 59 percent of the solutions across the list were released on that single day in October 2026.¹

Which classical problems are automated systems claiming to solve?

The catalogue attributes dozens of long-standing mathematical problems to automated systems developed by major artificial intelligence laboratories. Among the most prominent claims in the compilation is a reported Lean-formalized resolution of the Navier–Stokes existence and smoothness problem, specifically the Clay Mathematics Institute statements C and D concerning finite-time blowup for three-dimensional incompressible fluid equations with smooth forcing, credited to an internal multi-agent model from OpenAI.¹ In fluid mechanics, establishing whether smooth solutions to the Navier–Stokes equations can break down in finite time has remained one of the seven Millennium Prize Problems since 2000.

The listing also credits automated systems with disproving classical algebraic conjectures. According to the record shared by Depue, models labeled openai/math families 196 and 197 disproved Kaplansky's zero-divisor conjecture, while producing explicit counterexamples to direct-finiteness and Gottschalk surjunctivity.¹ In number theory, the catalogue lists a resolution of the rational Hodge conjecture for CM abelian varieties by openai/math family 032, and an evaluation of the irrationality exponent of pi as exactly 2 by openai/math family 017.¹ Other claimed resolutions include the Unique Games Conjecture by model 102, the undecidability of Hilbert's tenth problem over the rational numbers by model 004, and a non-sofic group resolution credited to OpenAI Astra and certified in the Lean formal proof system.¹

The compilation similarly credits systems developed by Anthropic, noting a Lean-formalized increase in the proportion of Riemann zeta zeros on the critical line from 41.6 percent to 67.2 percent by an unreleased Claude model.¹ Another listed result reports an explicit counterexample in dimension three or higher to the Jacobian conjecture, attributed to Alpöge working alongside Claude Fable 5.¹

How do human mathematical breakthroughs compare to automated lists?

Human mathematicians continue to achieve profound structural breakthroughs that require deep conceptual frameworks rather than raw symbolic search. In February 2025, mathematicians Hong Wang and Joshua Zahl submitted a preprint establishing the Kakeya set conjecture in three dimensions.²¹ A Kakeya set is a geometric point set containing a unit line segment in every spatial direction, and mathematicians had long asked how small the volume and dimension of such a set could possibly be.

Wang and Zahl studied families of slender delta tubes in three-dimensional space, focusing on configurations where only a limited number of tubes can lie inside any common convex set.² Through these geometric constraints, Wang and Zahl proved that the union of tubes from such collections must possess almost maximal volume.² As a direct consequence of their volume estimates, Wang and Zahl proved that every Kakeya set in three-dimensional Euclidean space has Minkowski and Hausdorff dimension equal to 3.² Their preprint spanned 127 pages and 14 figures, demonstrating the scale and analytical complexity required to resolve a central problem in metric geometry and classical analysis.²

Similarly, in August 2024, researchers Yu Deng, Zaher Hani, and Xiao Ma submitted a 192-page manuscript resolving a major branch of Hilbert's sixth problem.³¹ The sixth problem, proposed by David Hilbert in 1900, asks for an axiomatic derivation of macroscopic fluid equations from the fundamental microscopic laws of classical mechanics. Deng, Hani, and Ma provided a rigorous derivation of Boltzmann's kinetic transport equation from a system of hard spheres modeling a rarefied gas, establishing validity for arbitrarily long times as long as the solution to the Boltzmann equation exists.³

The work of Deng, Hani, and Ma extended Oscar Lanford's landmark 1975 theorem, which had justified the derivation of the Boltzmann equation only for sufficiently short time intervals.³ Deng and his coauthors based their long-time derivation on propagating a cumulant ansatz that preserves memory of the complete collision history of particles, proving the smallness of these cumulants through an elaborate cutting algorithm applied to molecular collision diagrams.³ While Depue's 2026 catalogue includes both the Wang–Zahl theorem and the Deng–Hani–Ma derivation among its 13 percent human contributions, it sets them directly beside newly posted automated claims across adjacent fields.

What can an automated catalogue of mathematical claims show?

The catalogue shared by Depue is an observational inventory of claims rather than a verified record of established mathematical truth. A catalogue entry stating that an artificial intelligence model has resolved an open conjecture does not constitute independent mathematical verification. While several items on the list note formalization or certification in the Lean proof assistant, the underlying code repositories, formal theorem files, and complete logical libraries were not detailed in the shared document. Translating mathematical concepts into Lean requires defining definitions and axioms accurately; a subtle flaw in formal definitions can permit a machine learning system to produce a valid machine check for a trivial or unintended proposition.

Furthermore, the compilation presents automated announcements alongside human preprints that underwent extensive development over many years. The manuscripts by Wang and Zahl, as well as Deng, Hani, and Ma, were submitted to the arXiv preprint repository, where they remain publicly available for exhaustive inspection by specialist mathematicians.³² In contrast, many of the automated accomplishments listed by Depue cite unreleased proprietary models and internal corporate families that outside academic departments cannot independently query or inspect.

What happens next as proof generation accelerates?

The immediate challenge facing the mathematical community is constructing reliable infrastructure for evaluating machine-generated arguments. When human authors submit a 192-page manuscript such as the derivation produced by Deng, Hani, and Ma, mathematical journals assemble teams of expert referees who spend months analyzing the underlying lemmas. If commercial artificial intelligence laboratories continue to announce dozens of proofs simultaneously, the human peer review system will rapidly become overwhelmed.

To manage this volume, the discipline is shifting toward automated verification pipelines. Formal interactive proof assistants such as Lean and Coq provide a mechanism to verify steps mechanically, but human mathematicians must still audit whether the formal theorem statements correspond faithfully to the genuine open conjectures. Until automated proofs are accompanied by public, open-source formal code and verified by independent researchers, catalogues of artificial intelligence breakthroughs will remain ambitious claims rather than settled science.

What this rests on

19 sentences trace to 3 sources.

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1. 1 H https://x.com/willdepue Document · x.com · 6 Oct 2026 Fable 5.1 and GPT-6 Pro Ranking Claims 81% of Recent Discoveries Were Released Today See the source
2. 2 A arxiv.org Paper · arxiv.org Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions Preprint · may not have been peer reviewed See the source
3. 3 A arxiv.org Paper · arxiv.org Long time derivation of the Boltzmann equation from hard sphere dynamics Preprint · may not have been peer reviewed See the source