{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-07-27-crouzeix-conjecture-proved/","md":"https://postcutoff.com/e/2026-07-27-crouzeix-conjecture-proved/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":"2026-07-27-crouzeix-conjecture-proved","date":"2026-07-27","date_precision":"day","short_title":"Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run","deck":null,"takeaway":"A preprint posted 27 Jul 2026 proves Crouzeix's conjecture (2004): for every square matrix A and polynomial f, ‖f(A)‖ ≤ 2·max over the numerical range W(A) of |f|.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2608.03841: A solution to Crouzeix's conjecture (Lorist, Schwenninger)","url":"https://arxiv.org/abs/2608.03841","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"SCMP: Chinese doctor stuns maths world cracking decades-old problem using ChatGPT","url":"https://www.scmp.com/tech/tech-trends/article/3363966/chinese-doctor-stuns-maths-world-cracking-decades-old-problem-using-chatgpt","type":"press","group":"press","domain":"scmp.com"},{"n":3,"title":"Alex Townsend: SIAM News essay on the Crouzeix conjecture (PDF)","url":"https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf","type":"discussion","group":"reactions","domain":"alextownsend.net"}],"official":1,"filed":"2026-09-29","updated":"2026-09-30","orgs":["OpenAI"],"title":"Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run","summary":"A preprint posted 27 Jul 2026 proves Crouzeix's conjecture (2004): for every square matrix A and polynomial f, ‖f(A)‖ ≤ 2·max over the numerical range W(A) of |f|. The proof came from one uninterrupted 16-hour autonomous GPT-5.6 Sol run prompted by Shanmu Jin, a self-taught neurosurgery resident. Michel Crouzeix, Anne Greenbaum and Alex Townsend checked it.","key_facts":["Previously best known constant: 1+√2 (Crouzeix–Palencia 2017); conjectured optimal constant 2","Single 16-hour autonomous run of GPT-5.6 Sol","Checked by Crouzeix himself, Anne Greenbaum and Alex Townsend (SIAM News essay)","Independent proof: Emiel Lorist and Felix Schwenninger, 'A solution to Crouzeix's conjecture' (arXiv 2608.03841, 4 Aug 2026); their AI disclosure says GPT-5.6 Sol Pro in ChatGPT 'was used to explore proof strategies for this note'"],"key_numbers":[],"tags":["math","matrix-analysis","gpt-5-6","amateur"],"science":{"field":"mathematics","subfield":"matrix analysis / operator theory","problem":"Crouzeix's conjecture","result":"Proof that the numerical range W(A) is a 2-spectral set for every matrix A.","open_since":"2004","ai_system":["GPT-5.6 Sol"],"human_role":"Autonomous: non-specialist prompted; experts verified","verification":{"key":"expert","label":"Expert-checked","text":"Expert-checked (Crouzeix, Greenbaum, Townsend); preprint"},"status":"confirmed","shock":"A doctor with no formal maths training settled a well-known conjecture in numerical analysis by letting a model run overnight."},"body_md":"## What happened\n\nA non-mathematician set GPT-5.6 Sol on the problem. The model produced a complete proof in one long run, which the conjecture's originator and other specialists confirmed.\n\n## Why it matters\n\nAlong with #1196, it showed that frontier models let amateurs resolve famous problems, which upended assumptions about who can do research mathematics.","disputed":[],"related":[{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-07-09-gpt-5-6-sol-terra-luna","url":"https://postcutoff.com/e/2026-07-09-gpt-5-6-sol-terra-luna/","date":"2026-07-09","date_precision":"day","short_title":"OpenAI broadly releases GPT-5.6 after government-gated preview","deck":null,"takeaway":"First frontier model whose public release was explicitly gated by US government review, and the model family involved in the July 2026 sandbox-escape incident.","category":"model-release","category_label":"Model releases","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":7,"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},{"id":"2026-05-03-erdos-1196-primitive-sets","url":"https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/","date":"2026-05-03","date_precision":"day","short_title":"Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets","deck":"Tao co-authors the paper","takeaway":"It was the first AI solution to a well-known, decades-old Erdős conjecture that specialists had actively worked on, not just an obscure entry.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-09-30","type":"edit","text":"Added the independent Lorist–Schwenninger proof (arXiv 2608.03841)"},{"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":"2026-09-30","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":88,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":27,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":118,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":57,"in_training_data":false}],"short_url":"https://postcutoff.com/s/neurosurgery-resident-uses"}