GPT-6-Astra Proves Prime Gaps ≤186 in Lean as Records Fall in Days

On Sep 4, OpenAI open-sourced several Lean-formalized proof repos on GitHub, most notably openai/PrimeGaps186: a Lean 4 proof by a model codenamed GPT-6-Astra showing infinitely many adjacent primes differ by at most 186, i.e. liminf(p{n+1}-pn) ≤ 186, corresponding to the conditional result DHL[40,2]. It is the first research output after GPT-6 Astra's release, arriving days after a fresh human record, which was then broken three times in quick succession.

Confirmed

• The repo is open-sourced; the proof was produced by GPT-6-Astra and verified in Lean 4, deriving DHL[40,2] from given inputs; per 机器之心, the result was announced as a paper by OpenAI researcher and UPenn professor Su Weijie, titled "Improved Short Ga…" (truncated in reports)
• Per @NoFaithlessness951, OpenAI's GitHub simultaneously showed new repos including openai/LongGapsBetweenPrimes and openai/ten-proo, read by the community as paving the way for Astra
• Per @scaling01 and @johnowhitaker, the previous best bound of 240 was published by Julia Stadlmann on Aug 31 and was immediately surpassed
• @teortaxesTex's timeline: the human record was 246 from the 2014 Polymath project; Stadlmann then proved 240, AxiomMath gave 212, and GPT-6-Astra reached 186 — three record breaks in two days
• Per @新智元 and 量子位's history: Yitang Zhang proved gaps below 70 million in 2013, Maynard and others improved it, and Polymath's 246 (with Terence Tao involved) stood 12 years before three breaks in three days
• @AcerFur (cited by @scaling01) explains the result is conditional on 186 because it relies on Deligne-type conditions
• Per @burnytech, weeks earlier Jack Kleeman and Claude had formalized Goldston's result as precursor work to 186
• Per @raphagl and @ChrSzegedy relaying the launch event, mathematician Mehtaab Sawhney announced the companion long-gap result, improving the upper bound by roughly a log log n factor — called the first improvement of this order since the 1930s
• Per math-community comments relayed by @teortaxesTex, the long-gaps proof is extremely elementary yet seemingly missed by all analytic number theorists; its Lean formalization is only 4500 lines, is current SOTA, and improves the record on Erdős problem 4
• Per @scaling01 citing Noam Brown: among GPT-6 Astra's uses he is most excited about scientific discovery, adding OpenAI has not maxed out the model's math and science capabilities

Unconfirmed

• GPT-6-Astra's exact identity, capability limits and official release timing remain unexplained by OpenAI; "paving the way for Astra" is community interpretation
• @新智元 additionally reported OpenAI released GPT-6 codenamed Astra, with Greg Brockman saying "history will prove it's this model," a blog post that 404'd, and limited user access; these details are highly unusual and unverified
• The "first improvement since the 1930s" claim for the long-gap result comes from relayed posts and awaits primary-source confirmation

Why it matters

• This is a number-theory result independently produced by a frontier model and rigorously formalized, in the same direction as Zhang's 2013 bounded prime gaps breakthrough, demonstrating AI capability in formal mathematics
• The 12-year-old 246 record fell to 240, then 212, then 186 within days — an extraordinary pace; if the companion long-gap result is confirmed as the first order-of-magnitude improvement since the 1930s, its significance is even greater
• The proof is Lean-formalized and open-sourced for community verification and reuse; Noam Brown's remarks suggest OpenAI is prioritizing scientific discovery for the model

2026-09-04 ~ 2026-09-04 · 22 related posts

• [source] OpenAI repo claims GPT-6-Astra Lean-formalized a prime-gaps bound of 186 — scaling01 · 2026-09-04
• GPT-6-Astra Lean proof cuts prime gap bound from 240 to 186, days after previous record — scaling01 · 2026-09-04
• [source] OpenAI open-sources PrimeGaps186, a Lean formalization of a prime-gap bound of 186 — johnowhitaker · 2026-09-04
• OpenAI quietly publishes Lean proof repos, seen as warm-up for Astra release — NoFaithlessness951 · 2026-09-04
• Prime gap record falls to 186 as GPT 6 Astra 'speedruns' mathematics — teortaxesTex · 2026-09-04
• Elementary proof with 4.5k-line Lean formalization beats SOTA on Erdős problem 4 — teortaxesTex · 2026-09-04
• Noam Brown: GPT-6 Astra Lean proof bounds consecutive prime gaps, touts scientific discovery — scaling01 · 2026-09-04
• OpenAI says GPT-6 Astra pushed the prime gap from 246 to 186 with Lean formalization — weijie444 · 2026-09-04
• Report: OpenAI unveils GPT-6 Astra, claims prime gap breakthrough toppling 90-year-old bounds — 新智元 · 2026-09-04

13 near-duplicate retellings: scaling01 · scaling01 · scaling01 · burny_tech · weijie444 · rapha_gl · weijie444 · Singularitarian · khademinori · 量子位 · 新智元 · ChrSzegedy · 机器之心