OpenAI said an internal model refuted a 25-year-old computer science assumption and cracked two other landmark problems in one announcement. Hours later, Anthropic said Claude Opus 5.5 found the same computer science result on its own, without seeing OpenAI's work.
On October 26, OpenAI said one of its unreleased internal models had produced proofs for three separate landmark problems: a refutation of the Exponential Time Hypothesis, a resolution of the Kakeya conjecture and Stein's restriction conjecture across all dimensions, and an existence proof for pure Yang-Mills theory, the gauge theory underpinning quantum chromodynamics. Any one of those would be a career-defining result for a human mathematician. OpenAI claimed three in a single release.
Then Anthropic said its own frontier model had beaten OpenAI to part of it, quietly, days earlier. According to reporting from Officechai, an internal Claude research model independently rederived the Exponential Time Hypothesis refutation on October 16, ten days before OpenAI's public announcement, using roughly 14 million output tokens and no human steering of the proof itself. Two labs, working apart, landed on the same answer to one of theoretical computer science's open questions.
That's the detail that turned a dense computer science result into a viral moment. A post from the X account AI Notkilleveryoneism Memes quoted Claude Opus 5.5's own assessment of the cluster of results: "If these hold up, this is the biggest single day in the history of math and theoretical computer science." The line spread fast, in part because it wasn't a marketing claim from Anthropic or OpenAI. It was the model talking about the field it had just helped move.
The Exponential Time Hypothesis has sat underneath hundreds of computer science papers for more than two decades. It's the working assumption that certain hard problems, Boolean satisfiability chief among them, can't be solved meaningfully faster than brute force. If that assumption is false, as the new proof claims, it reopens questions about how hard a large class of problems actually is. That's roughly Turing Award territory, and it's the specific claim both OpenAI's model and Anthropic's Claude model reached independently.
Claude Opus 5.5 is beating OpenAI's Astra on coding benchmarks and price Terminal-Bench 4.0 scores put Opus 5.5 at 66.4% against Astra's 57.9%, while Anthropic prices it at $4/$20 per million tokens versus OpenAI's $10/$50, fueling a real-time developer debate on Reddit. - claude opus 5.5 vs openai astra coding benchmarks - best ai model for coding terminal tasks cheaper
The Kakeya and restriction result is arguably the bigger mathematical prize. The three-dimensional case of the Kakeya conjecture alone was only settled in 2025, a proof the field treated as a once-in-a-century event. OpenAI's claimed result extends that to every dimension and simultaneously resolves Stein's restriction conjecture, a connected problem in harmonic analysis. Mathematicians have called work at this level Fields Medal-caliber when it comes from a human.
The third claim touches physics directly: an existence proof for pure Yang-Mills theory, a piece of the Yang-Mills mass gap problem that the Clay Mathematics Institute lists among its seven Millennium Prize Problems, each carrying a $1 million prize for a verified solution. According to Scientific American's reporting on the release, the review process includes Nobel laureate David Gross and Arthur Jaffe, who co-authored the original Clay Institute problem statement. Both researchers were careful to draw a line in public: their involvement, Gross and Jaffe said, "should not be taken as certification of the claim."
That caveat matters more than the headline. Lean, the formal proof-checking software used to verify the work, can confirm that a proof is logically airtight relative to a formal statement. It cannot confirm that the formal statement actually captures the conjecture mathematicians have argued about for decades. That translation step, checking that the formalized problem is the real problem, is exactly where independent experts will spend the coming weeks, and it's the step that separates a Lean-verified claim from a result the field accepts.
Skepticism isn't limited to technical caution, either. The Association for Human Mathematics reportedly called the release "a demonstration of power" and urged mathematicians to stop collaborating with the companies behind it, and several Fields medalists have separately raised alarms about AI's growing role in pure mathematics. Whatever the proofs turn out to be worth, the reaction shows the math community is not simply cheering.
Why Anthropic needed a win like this #
The timing helps explain why Anthropic's side of the story traveled as far as it did. Anthropic filed a draft S-1 with the SEC in June, is reportedly eyeing a listing with investors pricing the company near $965 billion after a $65 billion Series H-1 round in May, and faces real doubt about whether it clears the roughly $2 trillion some backers are hoping for. Prediction markets have given long odds on Anthropic closing above even $1.25 trillion at IPO, and the S-1 is expected to list AI backlash as a named risk factor.
Against that backdrop, a frontier model independently rediscovering a Turing Award-level result, without being shown OpenAI's path first, is a cleaner story for Anthropic than another benchmark chart. Opus 5.5 launched on September 22, built for the kind of long-horizon technical and research work this result represents rather than chat polish. A model quietly doing serious theoretical work in the background is a sharper answer to "does Anthropic have a technical edge" than anything in a pitch deck.
None of this is settled science yet. What's confirmed is that two labs produced overlapping claims on a problem that has resisted humans for 25 years, that the physics claim is now in front of two people who understand the Millennium Prize problem better than almost anyone alive, and that the verification fight ahead will take weeks, not headlines. If the Yang-Mills claim actually holds, it's worth noting what's still missing: the full mass gap problem, not just existence, remains open, and the Clay Institute's $1 million prize isn't paid out on existence proofs alone.
Scott Aaronson Says AI Labs Are Quietly Testing Models Against Encryption Anthropic's Claude Mythos Preview already cut HAWK's effective key strength in half and cracked a reduced AES variant up to 800 times faster than known methods, spending roughly $100,000 per attack. - AI companies testing models against internet encryption security - how AI labs secretly evaluating cryptographic algorithm vulnerabilities
Also read: ASM International Raises 2027 Sales Target Three Years Ahead of Schedule • Google DeepMind unveils Genie 4, the world model that once tanked gaming stocks • Nvidia is in talks to lead a $20 million round for Simplismart
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