{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2025-08-20-gpt-5-pro-convex-optimization-proof/","md":"https://postcutoff.com/e/2025-08-20-gpt-5-pro-convex-optimization-proof/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":"2025-08-20-gpt-5-pro-convex-optimization-proof","date":"2025-08-20","date_precision":"day","short_title":"GPT-5 Pro proves an improved convex-optimisation bound, which humans had already surpassed","deck":null,"takeaway":"OpenAI's Sébastien Bubeck reported that GPT-5 Pro, in about 17 minutes, proved that gradient descent on L-smooth convex functions yields a convex sequence of function values for step sizes up to 1.5/L.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Sébastien Bubeck on X","url":"https://x.com/SebastienBubeck/status/1958198661139009862","type":"official","group":"primary","domain":"x.com"},{"n":2,"title":"What does GPT-5's new math claim actually mean?","url":"https://allthings.how/what-does-gpt-5s-new-math-claim-actually-mean/","type":"press","group":"press","domain":"allthings.how"},{"n":3,"title":"whataifound.org: GPT-5 convex bound","url":"https://whataifound.org/finding/2025-08-gpt5-convex-bound","type":"discussion","group":"reactions","domain":"whataifound.org"}],"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"],"title":"GPT-5 Pro proves an improved convex-optimisation bound, which humans had already surpassed","summary":"OpenAI's Sébastien Bubeck reported that GPT-5 Pro, in about 17 minutes, proved that gradient descent on L-smooth convex functions yields a convex sequence of function values for step sizes up to 1.5/L. The paper's v1 had proved it for 1/L. However, the authors' own v2 had already proved the tight 1.75/L bound.","key_facts":["Problem: for which step sizes η is the optimisation curve of gradient descent convex? v1 proved η ≤ 1/L and gave a counterexample above 1.75/L","GPT-5 Pro proved η ≤ 1.5/L by a different argument; Bubeck checked it","The human authors' updated version had already closed the gap at 1.75/L","Bubeck: 'Claim: gpt-5-pro can prove new interesting mathematics.'"],"key_numbers":[],"tags":["math","optimization","gpt-5","proof"],"science":{"field":"mathematics","subfield":"convex optimisation","problem":"Convexity of the gradient-descent optimisation curve for L-smooth convex functions","result":"A correct, novel proof of the bound η ≤ 1.5/L, improving v1's 1/L but weaker than the humans' tight 1.75/L result already posted.","open_since":"","ai_system":["GPT-5 Pro"],"human_role":"Human posed the problem and verified the proof","verification":{"key":"expert","label":"Expert-checked","text":"Informal proof checked by Sébastien Bubeck (OpenAI), who posed the question; not written up as a paper or refereed; the human authors' revised paper already had the tight 1.75/L bound"},"status":"pending","shock":"A general chatbot produced a correct, non-trivial research-level proof in minutes, although the result was already superseded."},"body_md":"## What happened\n\nBubeck gave GPT-5 Pro the open question from v1 of a paper; the model produced a valid proof of an intermediate bound.\n\n## Why it matters\n\nIt was one of the first widely discussed cases of an LLM producing correct new research-level mathematics. The fact that humans had already done better also foreshadowed later disputes over novelty.","disputed":[{"label":"Verification","text":"Informal proof checked by Sébastien Bubeck (OpenAI), who posed the question; not written up as a paper or refereed; the human authors' revised paper already had the tight 1.75/L bound","sources":[]}],"related":[{"id":"2025-10-17-gpt-5-erdos-problems-controversy","url":"https://postcutoff.com/e/2025-10-17-gpt-5-erdos-problems-controversy/","date":"2025-10-17","date_precision":"day","short_title":"OpenAI researchers claim GPT-5 'solved' 10 Erdős problems","deck":"The solutions were already in the literature","takeaway":"The episode set the standard of scepticism for later AI maths claims, and led to Tao's public wiki tracking exactly what AI contributed to each Erdős problem.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"retracted","labels":["Retracted"]},"sources":3,"official":0,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},{"id":"2025-08-07-gpt-5","url":"https://postcutoff.com/e/2025-08-07-gpt-5/","date":"2025-08-07","date_precision":"day","short_title":"OpenAI launches GPT-5","deck":null,"takeaway":"Brought reasoning-model capability to hundreds of millions of free users by default.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":2,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]}],"people":[{"id":"sebastien-bubeck","name":"Sébastien Bubeck","url":"https://postcutoff.com/person/sebastien-bubeck/"}],"posts":[{"id":"x-sebastienbubeck-1958198661139009862","title":"“Claim: gpt-5-pro can prove new interesting mathematics. Proof: I took a convex…”","url":"https://postcutoff.com/p/x-sebastienbubeck-1958198661139009862/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending (the only check is by the OpenAI researcher who posed the question; no paper); verification made precise"},{"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":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":null,"in_training_data":true},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":true},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":null,"in_training_data":true},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":true}],"short_url":null}