{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/","md":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/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":"2026-09-10-astra-leanopenproblems-september-results","date":"2026-09-10","date_precision":"day","short_title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results","deck":"Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","takeaway":null,"category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.11500: A Counterexample to an Eternal Domination Conjecture","url":"https://arxiv.org/abs/2609.11500","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: tadamcz/dittert","url":"https://github.com/tadamcz/dittert","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"GitHub: tadamcz/phi-mixing-clt (Ibragimov–Iosifescu)","url":"https://github.com/tadamcz/phi-mixing-clt","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"GitHub: tadamcz/n-conjecture-strong","url":"https://github.com/tadamcz/n-conjecture-strong","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"VibeMathed: Ibragimov–Iosifescu conjecture status","url":"https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture","type":"discussion","group":"reactions","domain":"vibemathed.com"},{"n":6,"title":"Wikipedia: List of mathematical discoveries by artificial intelligence","url":"https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence","type":"discussion","group":"reactions","domain":"en.wikipedia.org"}],"official":4,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"],"title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results: Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","summary":"After the Köthe disproof, the same September 2026 Epoch AI run of pre-release GPT-6 Astra over the Formal Conjectures collection produced more machine-written Lean results, published by Tom Adamczewski: a proof of the full Dittert permanent conjecture, a counterexample to the Ibragimov–Iosifescu φ-mixing CLT conjecture, a disproof of the strong n-conjecture for n=4, and a 243-vertex graph refuting the Gamma–Theta eternal-domination conjecture (arXiv 2609.11500, with William Klostermeyer). Most results have not had independent expert review.","key_facts":["Setting: Epoch AI's LeanOpenProblems harness; pre-release GPT-6 Astra tried each research-open Formal Conjectures statement once, autonomously (see the Köthe entry)","Dittert conjecture: φ(A) ≤ 2 − n!/n^n for nonnegative n×n matrices with entries summing to n, with equality only for the all-1/n matrix. Lean proof passed the Comparator check (repo tadamcz/dittert); the exposition is not independently reviewed. Humans had earlier proved n ≥ 17 (arXiv 2606.01531) and n = 16 (arXiv 2607.19439, GPT-5.6 Sol-assisted)","Ibragimov–Iosifescu conjecture (Ibragimov, 1971): disproved with a strictly stationary φ-mixing counterexample; a 13,047-line Lean proof, 'Lean-checked, statement unaudited', announced 5 Sep 2026 (repo tadamcz/phi-mixing-clt)","Strong n-conjecture, n = 4: disproved in Lean with extra SymPy arithmetic checks (repo tadamcz/n-conjecture-strong)","Eternal domination: 243-vertex graph with γ(G) = γ∞(G) < θ(G), refuting the Gamma–Theta conjecture. Tom Adamczewski & William F. Klostermeyer, arXiv 2609.11500, 10 Sep 2026","The repositories say they were 'machine-written by AI assistants at the direction of Tom Adamczewski'","Same run, Wikipedia collection ('wikipedia-vega-1000usd', all 222 research-open statements, $1,000 and 96 h each): disproof of Smale's mean value conjecture for K = 1 (repo tadamcz/mean-value-problem; own entry 2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra). Epoch's LeanOpenProblems README publishes no overall solved count (checked 16 Oct)"],"key_numbers":[],"tags":["math","lean","astra","epoch-ai","combinatorics","probability","counterexample"],"science":{"field":"mathematics","subfield":"combinatorics / matrix theory / probability / number theory","problem":"Dittert conjecture; Ibragimov–Iosifescu φ-mixing CLT conjecture; strong n-conjecture (n=4); Gamma–Theta eternal domination conjecture","result":"One proof (Dittert, all n) and three disproofs, each with a Lean formalization or an explicit checkable counterexample.","open_since":"","ai_system":["GPT-6 Astra (pre-release)"],"human_role":"Autonomous proof search in Epoch AI's harness; Tom Adamczewski directed packaging; Klostermeyer co-wrote the domination paper","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed","text":"Formal proofs in Lean (mechanically checked). Statements and write-ups mostly not independently audited"},"status":"pending","shock":""},"body_md":"## What happened\n\nEpoch AI ran pre-release GPT-6 Astra once on each research-open statement in the Formal Conjectures collection. Besides Köthe, several more outputs were packaged as Lean repositories by Tom Adamczewski in the first half of September 2026. For the graph-theory counterexample, domination expert William Klostermeyer co-wrote an arXiv paper.\n\n## Why it matters\n\nAutonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines. The bottleneck is shifting to human auditing of whether the formal statements are the intended ones.","disputed":[{"label":"Verification","text":"Formal proofs in Lean (mechanically checked). Statements and write-ups mostly not independently audited","sources":[]}],"related":[{"id":"2026-09-21-openai-100-open-problems-claim","url":"https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/","date":"2026-09-21","date_precision":"day","short_title":"OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)","deck":null,"takeaway":"If substantiated, it would mean open problems are being resolved at industrial scale.","category":"science","category_label":"Science & math","importance":3,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-29","updated":"2026-10-10","orgs":["OpenAI"]},{"id":"2026-09-07-koethe-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/","date":"2026-09-07","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample","deck":null,"takeaway":"If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]},{"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":"2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra autonomously disproves Smale's mean value conjecture in Lean","deck":"HKU mathematicians build on it to show Smale's constant exceeds 1.1018","takeaway":"A Smale problem from 1981 fell to an unattended model run in an Epoch AI benchmark, with a Lean proof; six weeks later two complex analysts turned the construction into a quantitative bound M > 1.1018.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":9,"filed":"2026-10-16","updated":"2026-11-12","orgs":["OpenAI","Epoch AI","University of Hong Kong"]},{"id":"2026-08-30-bounded-prime-gaps-186","url":"https://postcutoff.com/e/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":"Bounded prime gaps were one of the celebrated stories of 2013–14.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Related: back-links to the Smale mean value and Köthe entries"},{"date":"2026-10-16","type":"edit","text":"Linked the Smale mean-value disproof from the same run; noted that Epoch publishes no overall results count"},{"date":"2026-09-29","type":"filed","text":"Created (grouped several September 2026 Astra/Epoch results)"}],"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":133,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":72,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":163,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":102,"in_training_data":false}],"short_url":null}