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OpenAI and Anthropic’s AI Proof Claims: What They Announced—and What Remains Unverified
On October 26, 2026, OpenAI announced three AI generated research claims; Anthropic soon claimed a separate refutation of ETH. OpenAI released mathematical manuscripts with Lean formalizations, while its physics materials were held back for an outside group’s write up.
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Key takeaways
- On October 26, 2026, OpenAI announced three AI generated research claims; Anthropic soon claimed a separate refutation of ETH.
- OpenAI released mathematical manuscripts with Lean formalizations, while its physics materials were held back for an outside group’s write up.
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openai.comCreate a landscape editorial hero image for this Studio Global article: What did OpenAI and Anthropic claim their internal AI models achieved on October 26, 2026, and how do these unreviewed claims, their signifi. Article summary: On October 26, 2026, OpenAI claimed that its internal models had produced breakthroughs in harmonic analysis, computational complexity, and quantum field theory; Anthropic announced a competing claimed refutation of the . Topic tags: general, general web. Style: premium digital editorial illustration, source-backed research mood, clean composition, high detail, modern web publication hero. Use reference image context only for broad subject, composition, and topical grounding; do not copy the exact image. Avoid: logos, brand marks, copyrighted characters, real person likenesses, fake screenshots, UI text, readable text, watermarks, charts with fake numbers, clic
On October 26, 2026, OpenAI said its internal models had produced results in harmonic analysis, computational complexity and quantum field theory. Hours later, Anthropic announced a competing claim about the Exponential Time Hypothesis (ETH). The announcements attracted attention because the claimed results would matter well beyond AI—but reporting at the time said they had not been independently confirmed. 2
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What OpenAI claimed
A proof of local smoothing. OpenAI’s public repository records a manuscript and Lean formalization for Local smoothing for the Euclidean wave equation. It says the proof establishes Sogge’s local smoothing conjecture for waves in every spatial dimension of at least two. The repository also says the result yields the Kakeya maximal and dimension conjectures, as well as Stein’s spherical restriction estimates within the interior of the conjectured range. 17 A separate report dates Sogge’s conjecture to 1991.
6 That stated scope should not be inflated into a claim that every version of Stein’s restriction conjecture has been resolved.
A disproof of ETH. OpenAI said its model found a disproof of ETH, a conjecture about the computational difficulty of Boolean satisfiability problems, and described ETH as an assumption used in many conditional results in computer science. 13 Reporting says the model produced the result on October 16 and that it was formalized in Lean the following day.
32 OpenAI’s public repository lists a manuscript titled The Exponential Time Hypothesis is False.
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If correct, a disproof could change the status of results that rely on ETH. But a subexponential algorithm is not automatically a polynomial-time algorithm: for example, 2^√n is subexponential in n but still grows faster than any fixed polynomial. So even a valid 2^o(n) algorithm for 3-SAT would refute ETH without, by itself, proving P = NP. The distinction matters because the two questions are related, but not interchangeable. 13
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An existence result in quantum field theory. OpenAI also claimed a mathematically rigorous existence proof for certain interacting four-dimensional quantum field theories, including quantum chromodynamics (QCD). 1 Reports describe the claim as concerning existence; they do not establish that the broader Yang–Mills Millennium Prize problem, including its mass-gap question, has been solved.
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Anthropic’s competing ETH claim
Anthropic soon presented its own claimed ETH refutation. A report identified the paper as A 2^o(n) Time Algorithm for 3-SAT and credited Claude; it also reported that Anthropic mathematician Levent Alpöge said the core algorithm had been found on October 16 “with no human input,” with researchers later working with Claude on the paper’s presentation. 9
33 Those details are reported claims about the work’s provenance, not independent confirmation of the proof.
The close timing is notable: reports say both labs dated their core ETH results to October 16. 9
32 OpenAI’s planned review was to involve Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, according to reporting on its outside review group.
20 But the central question is not which lab announced first; it is whether independent experts can verify the reasoning and its implications. The reports available here said the claims had not yet completed independent scrutiny.
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Different release plans—and transparency questions
OpenAI described its AI-generated manuscripts as “raw research data” that outside academics would help turn into scientific understanding. 2
11 It posted the local-smoothing and ETH manuscripts with Lean formalizations in its public repository.
17 By contrast, reporting said the physics group held back the AI-produced materials while preparing its own account; David Gross was named among the outside experts associated with that work.
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There were also transparency concerns around earlier OpenAI releases. Reporting summarized by AI Edge Briefing said that 10 of 719 released manuscripts included model chain-of-thought output, and that requested machine-readable metadata linking natural-language manuscripts to formal artifacts was missing. 18 That is relevant context for evaluating release practices, but it does not, on its own, establish every objection raised about the October 26 announcements.
The useful distinction is between a striking research claim and a verified result. OpenAI made two mathematical claims inspectable through public manuscripts and Lean artifacts, while the physics materials remained unpublished in the reported review process. Anthropic offered a competing ETH claim with a reported account of how its model contributed. Until outside experts can examine the proofs and confirm what they establish, neither lab’s announcement should be treated as a settled scientific discovery. 3
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On October 26, 2026, OpenAI announced three AI generated research claims; Anthropic soon claimed a separate refutation of ETH.
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On October 26, 2026, OpenAI announced three AI generated research claims; Anthropic soon claimed a separate refutation of ETH. OpenAI released mathematical manuscripts with Lean formalizations, while its physics materials were held back for an outside group’s write up.