{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/","md":"https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-08-30-bounded-prime-gaps-186","date":"2026-08-30","date_precision":"day","short_title":"GPT-6 Astra lowers the bounded prime gaps record from 246 to 186","deck":null,"takeaway":"An OpenAI preprint (30 Aug 2026) claims lim inf (p_{n+1} − p_n) ≤ 186, improving Polymath8b's bound of 246, which had stood since 2014.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: short gaps between primes (PDF)","url":"https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf","type":"paper","group":"primary","domain":"cdn.openai.com"},{"n":2,"title":"Julia Stadlmann: Bounded gaps between primes (arXiv 2608.31126; human-only, bound 240)","url":"https://arxiv.org/abs/2608.31126","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Terence Tao on Mathstodon: Stadlmann shaves 246 to 240 without modern AI tools","url":"https://mathstodon.xyz/@tao/117197525544971208","type":"discussion","group":"reactions","domain":"mathstodon.xyz"},{"n":4,"title":"Weijie Su on X (Lean formalisation)","url":"https://x.com/weijie444/status/2095600108956262911","type":"discussion","group":"reactions","domain":"x.com"}],"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"],"title":"GPT-6 Astra lowers the bounded prime gaps record from 246 to 186","summary":"An OpenAI preprint (30 Aug 2026) claims lim inf (p_{n+1} − p_n) ≤ 186, improving Polymath8b's bound of 246, which had stood since 2014. It uses 'triply densely divisible' conditions feeding a multidimensional Selberg sieve and was announced with a Lean formalisation. Julia Stadlmann independently reached 240 at about the same time.","key_facts":["Previous record: 246 (Polymath8b, 2014), building on Zhang (2013) and Maynard (2013)","New claimed bound: 186","Lean formalisation announced (Weijie Su); independent human verification not complete","Human counterpart: Julia Stadlmann (UIUC), arXiv 2608.31126 (submitted 31 Aug 2026), proves 240 alone, 'with the assistance of traditional numerical computation, but not modern AI tools' (Tao); key idea: Motohashi–Pintz–Zhang estimates for only 'partly smooth' moduli"],"key_numbers":[],"tags":["math","number-theory","primes","astra","lean"],"science":{"field":"mathematics","subfield":"analytic number theory","problem":"Bounded gaps between primes (toward the twin prime conjecture)","result":"Claimed proof that infinitely many pairs of primes differ by at most 186.","open_since":"2014","ai_system":["GPT-6 Astra"],"human_role":"Largely AI-generated per OpenAI","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed","text":"Formal proof in Lean (announced); not yet independently peer-reviewed"},"status":"pending","shock":"A record that a large Polymath collaboration of top number theorists could not push for 12 years moved by 60 in one AI result."},"body_md":"## What happened\n\nOpenAI's model found a refinement of the Maynard–Tao sieve set-up that substantially improves the gap bound.\n\n## Why it matters\n\nBounded prime gaps were one of the celebrated stories of 2013–14. An AI improving the collaborative record is a striking, if still pending, result.","disputed":[{"label":"Verification","text":"Formal proof in Lean (announced); not yet independently peer-reviewed","sources":[]}],"related":[{"id":"2026-09-03-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"OpenAI releases GPT-6 Astra, its first GPT-6 model","deck":null,"takeaway":"Astra is the first GPT-6-generation model and the first frontier release after the Hugging Face sandbox-escape incident and OpenAI's August training pause.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":20,"official":11,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-08-01-openai-astra-ten-advances","url":"https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/","date":"2026-08-01","date_precision":"day","short_title":"OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs","deck":null,"takeaway":"It moved the frontier from individual AI-assisted results to a lab producing batches of significant theorems.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":5,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[{"id":"2026-09-03-tao-open-problems-non-renewable-resource","title":"Tao: open problems have become a non-renewable resource","url":"https://postcutoff.com/p/2026-09-03-tao-open-problems-non-renewable-resource/"},{"id":"x-weijie444-2095600108956262911","title":"“Announcing that GPT-6 Astra has pushed the prime gap to 186, with Lean formalization! I…”","url":"https://postcutoff.com/p/x-weijie444-2095600108956262911/"}],"videos":[],"models":[],"changes":[{"date":"2026-09-29","type":"edit","text":"Corrected arXiv 2608.31126 label (it is Stadlmann's human paper, not OpenAI's); added Tao's Mathstodon post on it"},{"date":"2026-09-29","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-09-29","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":122,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":61,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":152,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":91,"in_training_data":false}],"short_url":"https://postcutoff.com/s/gpt-6-astra-lowers-bounded"}