{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-07-17-cycle-double-cover-conjecture-proved/","md":"https://postcutoff.com/e/2026-07-17-cycle-double-cover-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":"2026-07-17-cycle-double-cover-conjecture-proved","date":"2026-07-17","date_precision":"day","short_title":"GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture","deck":null,"takeaway":"In mid-July 2026 OpenAI released a preprint crediting GPT-5.6 Sol Ultra, 'in less than an hour', with a proof of the cycle double cover conjecture (Szekeres 1973, Seymour 1979): every bridgeless graph has a collection of cycles covering each edge exactly twice.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"OpenAI: cycle double cover proof (PDF)","url":"https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf","type":"paper","group":"primary","domain":"cdn.openai.com"},{"n":2,"title":"Jim Geelen: OpenAI's proof of the Cycle Double Cover Theorem (arXiv 2607.15399)","url":"https://arxiv.org/abs/2607.15399","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Sang-il Oum: exposition of the proof (arXiv 2607.16356)","url":"https://arxiv.org/abs/2607.16356","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"AI Weekly: OpenAI attributes cycle double cover proof to GPT-5.6 Sol Ultra","url":"https://aiweekly.co/alerts/openai-attributes-cycle-double-cover-proof-to-gpt-56-sol-ultra","type":"press","group":"press","domain":"aiweekly.co"}],"official":3,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"],"title":"GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture","summary":"In mid-July 2026 OpenAI released a preprint crediting GPT-5.6 Sol Ultra, 'in less than an hour', with a proof of the cycle double cover conjecture (Szekeres 1973, Seymour 1979): every bridgeless graph has a collection of cycles covering each edge exactly twice. Independent expositions by graph theorists Sang-il Oum and Jim Geelen followed.","key_facts":["OpenAI released the proof as a PDF (cdn.openai.com); arXiv 2607.15399 is Jim Geelen's 3-page notes 'OpenAI's proof of the Cycle Double Cover Theorem' ('my attempt to clarify, to myself, the OpenAI proof … This write-up offers no new insights'); Oum's exposition is arXiv 2607.16356 (17 Jul 2026)","Proof attributed entirely to GPT-5.6 Sol Ultra; the write-up was done with Codex","Independent checks and expositions by Sang-il Oum and Jim Geelen; a public Lean formalisation is reported but not verified here"],"key_numbers":[],"tags":["math","graph-theory","gpt-5-6","proof"],"science":{"field":"mathematics","subfield":"graph theory","problem":"Cycle double cover conjecture","result":"Proof that every bridgeless graph admits a cycle double cover.","open_since":"1973","ai_system":["GPT-5.6 Sol Ultra"],"human_role":"Autonomous proof per OpenAI; human experts wrote independent expositions","verification":{"key":"expert","label":"Expert-checked","text":"Checked by independent graph theorists: Jim Geelen (notes, arXiv 2607.15399) and Sang-il Oum (exposition, arXiv 2607.16356); OpenAI's proof is a PDF, not refereed; a Lean formalisation is reported but not verified here"},"status":"confirmed","shock":"One of graph theory's best-known conjectures, open for about 50 years, was reportedly proved by a model in under an hour."},"body_md":"## What happened\n\nOpenAI published a proof of the cycle double cover conjecture that it attributed wholly to its top model. Leading graph theorists independently re-expounded and checked the argument within days.\n\n## Why it matters\n\nAlong with the Jacobian counterexample the same week, it marked the point where famous named conjectures, not just Erdős-list problems, began falling to AI.","disputed":[],"related":[{"id":"2026-08-01-openai-astra-ten-advances","url":"https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/","date":"2026-08-01","date_precision":"day","short_title":"OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs","deck":null,"takeaway":"It moved the frontier from individual AI-assisted results to a lab producing batches of significant theorems.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":5,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Fixed a misattribution: arXiv 2607.15399 is Jim Geelen's notes on OpenAI's proof, not an OpenAI preprint (key fact and link title corrected from the arXiv metadata); verification names the independent checkers"},{"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":78,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":17,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":108,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":47,"in_training_data":false}],"short_url":"https://postcutoff.com/s/gpt-5-6-sol-ultra-proves"}