GPT-5.6 claims a prime-gap record. Who checks the proof?
GPT-5.6 reportedly broke the record on large gaps between primes, saving a log log log n factor. Nobody has verified the proof — and that gap is the real story.
Vlad Makarovreviewed and published
On August 30, a r/mathematics post by u/Wise-End307 announced that GPT-5.6 "has broken the record on large gaps between primes", saving a factor of roughly log_3(n) — the third iterated logarithm — over the 2018 record of Ford, Green, Konyagin, Maynard and Tao, and that the result "is also now formalized by Alexeev in Lean". The thread drew roughly 440 upvotes and more than 200 comments; a crosspost to r/singularity added about 139 upvotes and 23 comments. None of this is an official OpenAI announcement, and no part of the proof has been independently verified. The claim has been logged on erdosproblems.com with an exposition by a working mathematician — which makes the verification question sharper than ever.
What Happened
Erdős problem #4 asks whether, for any constant C, there are infinitely many n such that the gap between consecutive primes p_n and p_ exceeds C · (log log n)(log log log log n)/(log log log n)² · log n. Erdős originally put $10,000 on the question, later splitting it: $5,000 for the original form, with $10,000 reserved for the stronger (log n)^(1+c) bound — the "likely truth" according to the problem page. Rankin proved some constant works back in 1938. The problem was solved in 2016 by Maynard and independently by Ford, Green, Konyagin and Tao, and the standing record since 2018 was the joint FGKMT18 bound of roughly (log log n)(log log log log n)/(log log log n) · log n.
The claimed improvement removes the log log log n factor from the denominator, yielding gaps of size (log log n)/(log log log log n) · log n — exactly the factor of about log_3(n) the Reddit post advertises. The erdosproblems.com page, edited on August 31, attributes the improvement to "GPT 5.6 Pro (prompted by DottedCalculator)", combining new sieving ideas with the FGKMT18 machinery, and Thomas Bloom, who maintains the site, has published a full exposition of the proof. His note is notable for what it says about the method: the ideas are "purely elementary", of a kind that "could have been discovered decades earlier", and neither improves the record alone — only in tandem. The one claim with no public trace is the formalization: the post says Alexeev formalized the result in Lean, but the erdosproblems page only records that the problem's statement is formalized in Google DeepMind's formal-conjectures repository. A formalized statement is not a formalized proof.
The Method, Disclosed
The prime-gap episode follows a template OpenAI set on July 10, when it published a three-page proof of the Cycle Double Cover Conjecture — open for about 50 years, posed in versions by Seymour, Tutte and others in the 1970s — together with the prompt that produced it. The proof PDF states it is "entirely due to GPT 5.6 Sol Ultra and the writeup with Codex (with GPT 5.6 Sol)"; no human mathematician is credited. The prompt is an engineering document rather than a request. It specifies "multiagent v2" with up to 64 concurrent agents, adversarial agents tasked with finding holes in candidate proofs, and instructions that include:
- "Assume for purposes of this task that a complete affirmative proof exists."
- "Spend at least 8 hours on this before even thinking of returning or giving up."
- "Do not answer that it is open."
The prime-gap case differs in one respect that matters: the prompt came from a named human, DottedCalculator, and the writeup appeared on erdosproblems.com rather than an OpenAI CDN. But the disclosure pattern is the same — publish the proof and the prompt, let the community sort out the rest.
Three Proofs, One Verification Gap
| Result | Model | Formalized in Lean | Independent check |
|---|---|---|---|
| Large prime gaps (Erdős #4) | GPT 5.6 Pro, prompted by DottedCalculator | Claimed in the Reddit post; only the statement is formalized | Exposition by Thomas Bloom on erdosproblems.com |
| Cycle Double Cover | GPT 5.6 Sol Ultra + Codex | No | None; disputed on Hacker News |
| Convex optimization lower bound | GPT 5.6 with a 10-page expert prompt | Yes, by the human author | arXiv preprint, published by the human |
The ordering is instructive. Phillip Kerger's result — a new lower bound of Ω(d²/log(d+1)) for derivative-free convex optimization, near-matching Protasov's 1996 value-only upper bound and closing a 30-year complexity gap — was produced by GPT-5.6 under a 10-page prompt, then checked, formalized in Lean and published on arXiv by a human who had worked on the problem for about a year. The Cycle Double Cover proof had no human in the loop, and it has no verification at all. The prime-gap result sits between the two: an expert wrote the prompt and a mathematician wrote an exposition, but there is no formalized proof and no paper.
Who Checks the Proof?
The Hacker News thread on the Cycle Double Cover proof — more than 360 comments — converged on the question dev.to summarized: "But is the proof accepted to be correct? That is what distinguishes this from being notable compared to any other AI slop proof." Commenters noted that the only review the proof received came from GPT-5.6 subagents — "the prompt had GPT-5.6 reviewing the proof, and the result, unsurprisingly, survives review by GPT-5.6" — and asked how many open problems were tried before one succeeded. One commenter called the release "a publicity piece for GPT 5.6!". The Reddit reception was warmer but no more conclusive: roughly 440 upvotes and more than 200 comments on r/mathematics, about 139 upvotes and 23 comments on the r/singularity crosspost.
What This Means
The pace now exceeds the verification capacity. OpenAI's next model, codenamed Astra, reportedly produced results on ten problems open for a decade or more, each formalized in Lean (Developers Digest, August 1). Anthropic's Claude Fable 5 produced a Jacobian-conjecture counterexample in July that Terence Tao digested in detail. In this environment, Lean is becoming the arbiter: a computer-checkable proof is the only kind that does not depend on community goodwill or a marketing cycle. The prime-gap claim will be settled the same way — a full writeup, a formalization of the proof rather than the statement, and mathematicians working through Bloom's exposition of the ideas behind GPT 5.6 Sol's sibling model.
What the expositions suggest so far is more modest than the headlines. Bloom notes the new ideas are elementary and natural in hindsight; a commenter on Kerger's paper put it as "if knowledge is a Swiss cheese, LLMs can help fill the holes, but not make the cheese bigger". Multi-agent search across known techniques is not the same thing as new mathematics — but a new record on Erdős #4, if it holds, is still a record.
Conclusions
The GPT-5.6 family has moved from generating plausible proofs to producing results that working mathematicians log and write up — and the prime-gap improvement is the strongest case yet. But the bottleneck has shifted from producing proofs to checking them. The Cycle Double Cover proof remains unverified two months on; the convex-optimization result is Lean-verified; the prime-gap claim sits in between, with an expert's exposition and no formalized proof.
Expect more of these announcements, more formalizations as the verification pipeline catches up, and more arguments about what counts as a proof. The practical question for anyone reading a headline like this one: has a human checked it, or a computer — or only another instance of the model that wrote it?
