{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-22-odlyzko-poonen-conjecture-proved/","md":"https://postcutoff.com/e/2026-09-22-odlyzko-poonen-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":"2026-09-22-odlyzko-poonen-conjecture-proved","date":"2026-09-22","date_precision":"day","short_title":"Odlyzko–Poonen conjecture (1993) proved unconditionally","deck":"'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation","takeaway":"Constantin Kogler posted an unconditional proof of the Odlyzko–Poonen conjecture: a random monic 0/1 polynomial with constant term 1 is irreducible over Q with probability tending to 1 (arXiv 2609.26771, 22 Sep 2026).","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2609.26771: The Odlyzko–Poonen Conjecture on Irreducibility of Random Polynomials (Kogler)","url":"https://arxiv.org/abs/2609.26771","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: ckkogler/odlyzko-poonen-lean","url":"https://github.com/ckkogler/odlyzko-poonen-lean","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Constantin Kogler","OpenAI"],"title":"Odlyzko–Poonen conjecture (1993) proved unconditionally: 'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation","summary":"Constantin Kogler posted an unconditional proof of the Odlyzko–Poonen conjecture: a random monic 0/1 polynomial with constant term 1 is irreducible over Q with probability tending to 1 (arXiv 2609.26771, 22 Sep 2026). Breuillard and Varjú had proved this in 2019 only under the Generalized Riemann Hypothesis. The paper says the proofs are due to GPT-6 Astra. The model also formalised everything in Lean (about 22,000 lines in about 30 hours), and Emmanuel Breuillard checked the proof.","key_facts":["Conjecture: Odlyzko and Poonen (1993); Konyagin (1999) gave a c/log n lower bound; Breuillard–Varjú (2019) proved it under GRH; Bary-Soroker–Koukoulopoulos–Kozma (2023) handled coefficients from at least 35 consecutive integers","New: P(P_n irreducible) → 1 unconditionally, with P(reducible) = sqrt(2/(πn)) + O(1/n)","Machine Contribution Statement: 'The proofs are due to GPT-6 Astra. The author rewrote and checked the arguments'; first found on 14 Sep 2026, after the model had proved a related result on Bernoulli convolutions (first under GRH, then unconditionally)","'GPT-6 Astra formalized all results from this paper and all necessary results from previous work. The formalization took around 30 hours and around 22'000 lines of code were written' (repository ckkogler/odlyzko-poonen-lean)","Kogler thanks Emmanuel Breuillard 'for checking the proof', and thanks Breuillard and Péter Varjú for help with the writing"],"key_numbers":[],"tags":["math","number-theory","random-polynomials","lean","astra"],"science":{"field":"mathematics","subfield":"number theory / probability","problem":"Odlyzko–Poonen conjecture on irreducibility of random 0/1 polynomials","result":"Unconditional proof that random 0/1 polynomials are irreducible with probability tending to one, with the sharp asymptotic for the reducible probability.","open_since":"1993","ai_system":["GPT-6 Astra"],"human_role":"AI-generated proofs and formalisation; the author directed the work, rewrote and checked it; an expert (Breuillard) checked the proof","verification":{"key":"lean","label":"Lean-verified","text":"Formal proof in Lean 4, plus expert check; preprint, not peer-reviewed"},"status":"confirmed","shock":"","short":"Odlyzko–Poonen conjecture"},"body_md":"## What happened\n\nWhile studying Bernoulli convolutions with GPT-6 Astra, Kogler saw the model remove a GRH assumption from a related result. He then asked it to try the Odlyzko–Poonen conjecture. The model produced a proof and formalised it together with all the prior results it needed.\n\n## Why it matters\n\nThe unconditional case had stayed open after Breuillard and Varjú's conditional proof in 2019. The paper assigns the mathematics fully to the model and includes both a machine-checked proof and a check by a leading expert, which is unusually complete verification for an AI result.","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":"2026-09-03-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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-07","type":"edit","text":"Science block: short label for the open-problems table"},{"date":"2026-09-30","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-10-07","human_review":null,"version":{"date":"2026-10-07"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":145,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":84,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":175,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":114,"in_training_data":false}],"short_url":"https://postcutoff.com/s/odlyzko-poonen-conjecture-1993"}