{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean/","md":"https://postcutoff.com/e/2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean/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":"2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean","date":"2026-11-04","date_precision":"day","short_title":"2×2 matrix mortality claimed undecidable (AI-assisted, Lean)","deck":null,"takeaway":"Paterson showed in 1970 that mortality is undecidable for 3×3 integer matrices; the 2×2 case was the one dimension left open. A Waterloo preprint claims it is undecidable for families of 5,799 2×2 integer matrices, with AI used for research, proofs and a Lean formalization of the reduction.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Quartz, Shevchenko, De Sterck, Liu: Mortality Is Undecidable for 2×2 Integer Matrices (arXiv 2611.06562)","url":"https://arxiv.org/abs/2611.06562","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Undecidable problems: a sampler (lists 2×2 mortality as open, arXiv 1204.0299)","url":"https://arxiv.org/abs/1204.0299","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Bell, Potapov, Semukhin: On the mortality problem (upper-triangular 2×2 decidable)","url":"https://livrepository.liverpool.ac.uk/3142958/","type":"paper","group":"primary","domain":"livrepository.liverpool.ac.uk"}],"official":3,"filed":"2026-11-08","updated":"2026-11-08","orgs":["University of Waterloo","OpenAI","Anthropic"],"title":"Matrix mortality claimed undecidable for 2×2 integer matrices, the last open dimension since 1970: Waterloo team with Codex, Claude Code and GPT-5.6 Sol, Lean-checked reduction","summary":"arXiv 2611.06562 (Thanin Quartz, Ivan O. Shevchenko, Hans De Sterck, Jun Liu; University of Waterloo, announced in the 4 Nov listing) claims that the matrix mortality problem (can products of a finite family of matrices reach the zero matrix?) is undecidable for 2×2 integer matrices, the only dimension left open since 1970. The authors used OpenAI Codex and Claude Code for research, proof development and a Lean 4 formalization of the reduction. Unrefereed preprint; no expert reaction found yet.","key_facts":["Claim: mortality is undecidable for families of 5,799 distinct 2×2 integer matrices; all but one are fixed and nonsingular with determinants ±p for one fixed prime p, and the input affects only a single rank-one matrix (abstract)","History (abstract): trivially decidable in dimension 1, undecidable for integer matrices in dimension ≥ 3 since 1970 (Paterson); 'only the problem in dimension 2 is left open'. Known partial 2×2 results: decidable for two rational matrices (Bournez–Branicky) and for upper-triangular rational families (Bell–Potapov–Semukhin)","Method (abstract): a counter machine is encoded so that a prime p enters the denominator of the encoded configuration where no later matrix can remove it; the rank-one matrix turns halting into reaching zero. 'A companion formalization in the Lean proof assistant verifies the complete reduction'","AI disclosure (PDF, 'AI methodology'): 'We used OpenAI Codex and Claude Code for mathematical research, proof development, manuscript preparation, and Lean 4 formalization. The investigation began in July 2026, when the authors used GPT-5.6-Sol to systematically investigate this problem … did not resolve the problem. The final proof architecture was selected by the authors and developed iteratively with AI assistance. OpenAI Codex with GPT-5.6-Sol was also used to generate a Lean formalization'","Verification status: preprint with a Lean formalization of the reduction (the authors 'reviewed the correspondence between the formalization and the mathematical statements'); not refereed; no comment from the decidability community found on 8 Nov (web search)"],"key_numbers":[],"tags":["math","computer-science","decidability","matrices","codex","claude-code","gpt-5-6-sol","lean"],"science":{"field":"computer-science","subfield":"decidability / matrix semigroups","problem":"Matrix mortality problem in dimension 2 (open since Paterson 1970)","result":"Claimed undecidable for finite families (5,799 matrices) of 2×2 integer matrices","open_since":"1970","ai_system":["OpenAI Codex","GPT-5.6 Sol","Claude Code"],"human_role":"AI-assisted: authors chose the final proof architecture and reworked the exposition; AI used for research, proofs and Lean code","verification":{"key":"lean","label":"Lean-verified","text":"Preprint; Lean formalization of the reduction"},"status":"pending","shock":""},"body_md":"## What happened\n\nIn the arXiv listing of 4 Nov 2026, four University of Waterloo researchers posted a proof that matrix mortality is undecidable\nfor 2×2 integer matrices. Mortality asks whether some product of matrices from a finite family is the zero matrix. Paterson\nshowed in 1970 that it is undecidable for 3×3 integer matrices, and the 2×2 case has been the open one since. The construction\nuses 5,799 matrices, all but one fixed, and the paper says a Lean formalization checks the complete reduction.\n\nThe AI statement says the work began in July 2026 with GPT-5.6 Sol. That first attempt produced partial results but no proof.\nThe final route was chosen by the authors and developed with Codex and Claude Code, and Codex generated the Lean code.\n\n## Why it matters\n\nIt is one of the basic open questions about matrix products, and it sits in a line of AI-assisted decidability results this\nautumn. The Lean check covers the reduction, but the paper has not been refereed and no outside expert has commented yet.","disputed":[{"label":"Verification","text":"Preprint; Lean formalization of the reduction","sources":[]}],"related":[],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-08","type":"filed","text":"Created (science agent; abstract and PDF AI statement read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-08","run":null,"sources_read":"Science agent; abstract and PDF AI statement read","updated":"2026-11-08","human_review":null,"version":{"date":"2026-11-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":188,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":127,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":218,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":157,"in_training_data":false}],"short_url":"https://postcutoff.com/s/2-2-matrix-mortality"}