{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-05-12-gaussian-completely-monotone-disproved/","md":"https://postcutoff.com/e/2026-05-12-gaussian-completely-monotone-disproved/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":"2026-05-12-gaussian-completely-monotone-disproved","date":"2026-05-12","date_precision":"day","short_title":"GPT-5.5 Pro finds counterexample disproving McKean's 1966 conjecture and the Gaussian completely monotone conjecture","deck":null,"takeaway":"Gu and Sellke (arXiv 2605.11656) presented an explicit probability measure, found by GPT-5.5 Pro, for which the 5th time-derivative of entropy along the heat flow is positive.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"Gu & Sellke: counterexample to the GCM conjecture (arXiv 2605.11656)","url":"https://arxiv.org/abs/2605.11656","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Follow-up: multidimensional counterexample (arXiv 2605.18081)","url":"https://arxiv.org/abs/2605.18081","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Suvrit Sra: GPT, the Counterexample Machine (arXiv 2608.29595)","url":"https://arxiv.org/abs/2608.29595","type":"discussion","group":"reactions","domain":"arxiv.org"}],"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"],"title":"GPT-5.5 Pro finds counterexample disproving McKean's 1966 conjecture and the Gaussian completely monotone conjecture","summary":"Gu and Sellke (arXiv 2605.11656) presented an explicit probability measure, found by GPT-5.5 Pro, for which the 5th time-derivative of entropy along the heat flow is positive. This disproves the Gaussian completely monotone conjecture, McKean's 1966 Gaussian-optimality conjecture (1-D) and Toscani's 2015 entropy power conjecture.","key_facts":["arXiv 2605.11656 (12 May 2026)","Counterexample found by GPT-5.5 Pro; proof written by the human authors","Follow-ups: a hexagonal multidimensional counterexample (arXiv 2605.18081) and log-concave families (2608.30275)","AI disclosure (Gu–Sellke abstract, arXiv 2605.11656): 'The explicit counterexample was found by GPT-5.5 Pro.'","AI disclosure (Zou–Fan–Gao–Wang abstract, arXiv 2605.18081, which disproves the Cheng–Geng log-convexity conjecture in dimension ≥ 2): 'The explicit two-dimensional counterexample was found by GPT-5.5 Pro.'"],"key_numbers":[],"tags":["math","probability","information-theory","counterexample"],"science":{"field":"mathematics","subfield":"probability / information theory","problem":"McKean's conjecture on entropy along the heat flow; Gaussian completely monotone (GCM) conjecture","result":"Explicit measure on R with positive 5th derivative of entropy under heat flow, refuting three related conjectures.","open_since":"1966","ai_system":["GPT-5.5 Pro"],"human_role":"AI found the counterexample; humans verified and wrote the proof","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint (Gu & Sellke, arXiv 2605.11656) with a human-written proof; the multidimensional disproof by Zou, Fan, Gao and Wang (arXiv 2605.18081) is also an unrefereed preprint; no formal proof"},"status":"pending","shock":""},"body_md":"## What happened\n\nOpenAI researcher Mark Sellke and a co-author used GPT-5.5 Pro to search for a distribution violating a 60-year-old monotonicity conjecture, and it produced one.\n\n## Why it matters\n\nIt is part of 2026's striking pattern of AI counterexamples: models proved especially good at finding objects that break long-believed conjectures. Suvrit Sra documented 15+ such counterexamples in \"GPT, the Counterexample Machine\".","disputed":[{"label":"Verification","text":"Unrefereed preprint (Gu & Sellke, arXiv 2605.11656) with a human-written proof; the multidimensional disproof by Zou, Fan, Gao and Wang (arXiv 2605.18081) is also an unrefereed preprint; no formal proof","sources":[]}],"related":[{"id":"2026-05-20-ai-disproves-erdos-unit-distance-conjecture","url":"https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/","date":"2026-05-20","date_precision":"day","short_title":"OpenAI model disproves Erdős's 80-year-old unit distance conjecture","deck":null,"takeaway":"This is the moment AI crossed from solving competition problems to settling a famous open research conjecture, reshaping debate about AI's role in mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":3,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending (unrefereed preprints, no independent check found); added both papers' AI disclosures verbatim from the arXiv abstracts"},{"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":12,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":true},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":42,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":true}],"short_url":null}