Three claims, three levels of evidence

Start with what can actually be looked at, because the three results are not in the same position. The mathematics and computer science packages are public: a manuscript and a Lean formalization for each, on OpenAI's GitHub repository. The physics package is not. Its shepherd group, led by David Gross, decided to hold the artifacts back and write the work up itself. Until that writeup exists, the physics claim rests entirely on OpenAI's description.

For mathematics, OpenAI reused the approach of [its October 6 release of 722 model-written manuscripts](https://alfw.ai/article/openai-722-math-proofs/): publish the output and let researchers organise their own examination of it. OpenAI is unusually plain about this. It calls the manuscripts raw research data that still has to be converted into scientific understanding. Any coverage that reports three solved problems has gone past its source.

What a Lean check settles, and what it leaves open

Lean formalization is the strongest part of the evidence, and it needs its limits stated precisely. A Lean proof certifies that a specific formal statement follows, inside a specific formal environment. It does not certify that the formal statement is the famous conjecture, in full, as the field understands it. That matching work is human, and it is where the gaps sit.

The public scope documents — the repository's formalization catalogue and the formal statements themselves — show two qualifications, both first flagged by a launch-day analysis of those documents and checked against them here. The first is the ETH bound. The revised manuscript's main theorem (Theorem 1.1, revised October 25) gives deterministic exact counting for every fixed k of at least 2 in time (n + m)^O(1) · 2^O(n log log n / √log n), in polynomial space. The statement OpenAI formalized in Lean bounds the running time by C · (|F| + 1)^d · 2^(A · n (log log n)^2 / √log n): the same square-root-log scale with one extra log log factor in the exponent. The formalized statement still covers every fixed k of at least 2, exact counting of satisfying assignments for canonically encoded k-CNF formulas, and polynomial space at every step. The weaker bound is still subexponential, so it would still be enough for the central claim if the formalization and its interpretation check out, but the manuscript's sharper numbers do not carry the same verification status.

The second qualification is the local smoothing range. The formalized half-wave estimates hold in every dimension n of at least 2, for p of at least 2, with an arbitrarily small positive regularity loss; zero loss appears only at p = 2. The restriction, Bochner-Riesz and Stein square-function conclusions are stated in the formalization's own docstrings as the "strict" or "conjectured strict" ranges: strict inequalities that cover the interior of the conjectured ranges, not every endpoint. That narrows what the formalization covers. It does not by itself falsify the claim, because the open question is how the formalization, the manuscript and the asserted implications fit together. None of this is hidden. It is in the scope documents, which is why they, not headlines, are the unit to read.

Two practical corrections follow. A false ETH would not settle P versus NP: a subexponential algorithm can still be far slower than any polynomial one, and OpenAI itself says practical improvements will require significant further research. And Yang-Mills existence is, in OpenAI's own words, part of the Millennium Prize problem, which also requires a mass gap, so existence alone is not the whole problem. Anyone selling October 26 as a million-dollar solution is selling something the announcement does not contain. Neither the claimed ETH result nor the reported physics result establishes that current encryption has been broken.

The examiners are not a verdict

The examiners are the best in their fields, which is why their disclaimer matters more than their presence. Impagliazzo, Paturi and Williams wrote that their involvement should not be taken as certification of the claim, and that a result of this significance requires careful, independent scrutiny. They signed no nondisclosure agreements and are unpaid, terms OpenAI confirms, and they extracted two commitments first: all raw data behind the ETH result made public, and funding for a conference on what it means for the field.

There is also a live argument about whether named shepherds should exist at all. On October 25, the Advisory Group on Mathematics and Artificial Intelligence (AGMAI) published a statement recommending that labs avoid designating mathematicians to steward AI releases, arguing that unequal access, reduced community agency, opaque selection and reputational lending outweigh the benefits (via the Kingy AI analysis). It was not a response to OpenAI's October 26 announcement, and the two arrangements should not be assumed identical.

What happens next is easy to state and slow to arrive. The computer science group organises an open examination. The physics group writes its result up, or does not. Everyone else starts from the public repository. If the ETH claim survives, an assumption underneath hundreds of papers has to be retired, and the field will say so itself. Until then, the accurate headline is the examiners' own: a claimed refutation, under examination.

Sources

  1. [1] OpenAI, "Working with academic communities to build scientific understanding", October 26, 2026 (primary announcement; all three claims, dates, shepherd groups, release strategies, no-NDA and unpaid terms, raw-data and conference commitments, $5 million Simons contribution)Read source
  2. [2] OpenAI, openai/math GitHub repository (public manuscripts and Lean formalizations; formalization catalogue; the SubexponentialSAT formal statement; the LocalSmoothing comparator statements; "The Exponential Time Hypothesis is False" manuscript, dated October 17, revised October 25, 2026)Read source
  3. [3] Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, "Message to the TCS Community on ETH" (open letter, October 26, 2026; the non-certification statement, read first-hand)Read source
  4. [4] Kingy AI, "OpenAI's Science Claims: Kakeya, ETH and Quantum Fields Explained", October 26, 2026 (independent reading of the public manuscripts and Lean scope documents, which first flagged the weaker ETH Lean bound and the local smoothing scope limits, plus the mass-gap requirement and the October 25 AGMAI statement; the article states it did not rebuild the Lean proofs or certify the theorems)Read source
  5. [5] Unite.AI, "OpenAI Says Its Models Produced Three Proofs in Math, CS, and Physics", October 26, 2026 (formalization details, including the ETH Lean scope: fixed k of at least 2, exact counting for canonically encoded k-CNF formulas, polynomial space)Read source