{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/","md":"https://postcutoff.com/e/2026-08-05-sendov-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":"2026-08-05-sendov-conjecture-proved","date":"2026-08-05","date_precision":"day","short_title":"Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro","deck":"Tao simplifies and formalises it","takeaway":"Lech Mazur posted a computer-assisted proof, generated with GPT-5.6 Pro, of Sendov's conjecture for all degrees: if every root of a polynomial lies in the closed unit disk, each root is within distance 1 of a critical point.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Lech Mazur: Sendov conjecture proof (PDF)","url":"https://www.proofatlas.ai/papers/sendov-conjecture/SENDOV_CONJECTURE_PROOF_AUGUST_5_2026.pdf","type":"paper","group":"primary","domain":"proofatlas.ai"},{"n":2,"title":"Terence Tao: A digestion of the proof of Sendov's conjecture","url":"https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"}],"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"],"title":"Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro; Tao simplifies and formalises it","summary":"Lech Mazur posted a computer-assisted proof, generated with GPT-5.6 Pro, of Sendov's conjecture for all degrees: if every root of a polynomial lies in the closed unit disk, each root is within distance 1 of a critical point. Terence Tao called it 'remarkably elementary', simplified it, and used AI agents to shrink the Lean proof from ~90k to ~15k lines.","key_facts":["Conjecture from 1958; previously known for degree < 9 (Brown–Xiang) and for sufficiently large degree (Tao, 2020)","Mazur's preprint 5 Aug 2026 (some lists say 3 Aug); Tao's digestion 12 Aug 2026","Tao extended the method to the Phelps–Rodriguez conjecture"],"key_numbers":[],"tags":["math","complex-analysis","polynomials","lean"],"science":{"field":"mathematics","subfield":"complex analysis / geometry of polynomials","problem":"Sendov's conjecture","result":"Proof of Sendov's conjecture for all degrees, formally verified in Lean.","open_since":"1958","ai_system":["GPT-5.6 Pro"],"human_role":"Human orchestrated (Mazur); Tao simplified and formalised with AI agents","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean; expert-checked by Tao"},"status":"confirmed","shock":"A 68-year-old conjecture that Tao himself had only proved for large degrees fell to an elementary AI-found argument."},"body_md":"## What happened\n\nA non-academic used GPT-5.6 Pro to generate a computer-assisted proof covering the remaining degrees. Tao then digested it into a short argument based on the fundamental theorem of algebra and the Maclaurin inequality.\n\n## Why it matters\n\nIt is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.","disputed":[],"related":[{"id":"2026-09-06-collatz-positive-density-log-time-mazur-codex","url":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/","date":"2026-09-06","date_precision":"day","short_title":"Collatz positive-density convergence, AI-developed, Lean-checked","deck":null,"takeaway":"The first linear lower bound for Collatz-convergent integers (earlier bounds were X^0.84, then X^0.90): an explicit positive proportion of n < X reach 1 within 10.46 ln n steps. Developed mainly by AI agents, Lean-checked twice, not yet peer reviewed. The Collatz conjecture itself stays open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-19","orgs":["ProofAtlas","OpenAI","Axiom Math"]},{"id":"2026-07-27-crouzeix-conjecture-proved","url":"https://postcutoff.com/e/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":"Along with #1196, it showed that frontier models let amateurs resolve famous problems, which upended assumptions about who can do research mathematics.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":1,"filed":"2026-09-29","updated":"2026-09-30","orgs":["OpenAI"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[{"id":"2026-08-12-tao-sendov-conjecture-digestion","title":"A digestion of the proof of Sendov's conjecture","url":"https://postcutoff.com/p/2026-08-12-tao-sendov-conjecture-digestion/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Related: back-link to Lech Mazur's Collatz result (same author and ProofAtlas harness)"},{"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":97,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":36,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":127,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":66,"in_training_data":false}],"short_url":"https://postcutoff.com/s/sendov-1958-conjecture"}