{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/","md":"https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/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":"2026-08-19-yau-tian-donaldson-cscK-disproof-claim","date":"2026-08-19","date_precision":"day","short_title":"Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics","deck":null,"takeaway":"On 19 Aug 2026 Jihao Liu (Peking University) posted a 79-page preprint describing a polarized smooth projective fivefold that is K-polystable but has no constant scalar curvature Kähler (cscK) metric.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2608.19301: Disproof of the Yau–Tian–Donaldson conjecture (Liu)","url":"https://arxiv.org/abs/2608.19301","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2608.01040: The equality case of Ehrhart's volume conjecture (Liu; same AI setup)","url":"https://arxiv.org/abs/2608.01040","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Peking University","Jihao Liu","Anthropic","OpenAI"],"title":"Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics","summary":"On 19 Aug 2026 Jihao Liu (Peking University) posted a 79-page preprint describing a polarized smooth projective fivefold that is K-polystable but has no constant scalar curvature Kähler (cscK) metric. That would disprove the cscK form of the Yau–Tian–Donaldson conjecture. The paper says 'the main result of this paper was obtained using generative AI', namely Claude Code (Fable 5), Codex (GPT-5.6 Sol) and the Danus research agent. A later solo Danus run re-derived counterexample and proof in 5 h 29 min.","key_facts":["arXiv 2608.19301 (19 Aug 2026), 79 pages, single author, with an appendix on AI use written jointly with Bin Dong and Guoxiong Gao (Peking University)","Claim: a K-polystable polarized smooth projective fivefold with no cscK metric. The Kähler–Einstein (Fano) case of Yau–Tian–Donaldson was proved by Chen–Donaldson–Sun in 2015; the cscK form was open","Process (appendix): exploring open problems with Claude Code, the author saw 'initial signs of a possible breakthrough', then had Claude Code (Fable 5), Codex (GPT-5.6 Sol) and Danus work on it together; 'the three systems together produced the counterexample and its proof'","Human input 'was crucial at one point': the author noticed the example must refute either the Codogni–Stoppa conjecture or the cscK YTD conjecture and told the agents to settle which","Replication: an improved Danus working alone from the bare problem 'settled the problem 5 hours and 29 minutes into its run … and produced a counterexample and a complete proof'","Comparison (per the appendix): QED, ProofCouncil, MechMath, Codex (GPT-5.6 Sol), Claude Code (Fable 5) and GPT-5.6 Sol Pro all failed in 12 hours, even when given the counterexample and asked only to prove it","Danus is described as a specialised research agent built on the Rethlas system (Liang Xiao and Bin Dong's group); the same author used it for the equality case of Ehrhart's volume conjecture (arXiv 2608.01040)","Status: preprint; no formal verification; no independent expert confirmation found as of 30 Sep 2026"],"key_numbers":[],"tags":["math","kahler-geometry","k-stability","counterexample","agents","danus","claude-fable-5","gpt-5-6"],"science":{"field":"mathematics","subfield":"Kähler geometry / K-stability","problem":"Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics","result":"Claimed explicit K-polystable polarized smooth projective fivefold without a cscK metric.","open_since":"","ai_system":["Danus (Rethlas-based agent)","Claude Fable 5 (Claude Code)","GPT-5.6 Sol (Codex)"],"human_role":"AI-generated main result with one key human redirection; the author checked and polished the text","verification":{"key":"preprint","label":"Preprint only","text":"Preprint only; not formalised; not peer-reviewed"},"status":"pending","shock":"An agent system is reported to have refuted a flagship conjecture of Kähler geometry, and to have reproduced the result alone in under six hours.","short":"Yau–Tian–Donaldson conjecture (cscK metrics)"},"body_md":"## What happened\n\nA Peking University algebraic geometer released a counterexample to the cscK form of the Yau–Tian–Donaldson conjecture. He credits a combination of Claude Code, Codex and his group's Danus agent for the example and its proof. The appendix also records a control experiment: a stronger Danus reproduced the result alone, while six other agent systems could not solve it or even verify the given counterexample.\n\n## Why it matters\n\nThe Yau–Tian–Donaldson programme is central to modern complex geometry, and the paper is explicit that its main result is AI-generated. The appendix argues that such a counterexample is itself a proof, with no finite certificate, because it asserts something about every possible degeneration. If the result is confirmed, it would be one of the deepest AI-derived results of 2026.","disputed":[{"label":"Verification","text":"Preprint only; not formalised; not peer-reviewed","sources":[]}],"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":"2026-08-01-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"]},{"id":"2026-06-09-claude-fable-5-mythos-5","url":"https://postcutoff.com/e/2026-06-09-claude-fable-5-mythos-5/","date":"2026-06-09","date_precision":"day","short_title":"Anthropic releases Claude Fable 5 and Claude Mythos 5","deck":"First generally available Mythos-class model","takeaway":"This was the first time the class of model Anthropic had withheld in April (Mythos Preview) became available to the public.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":7,"official":5,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Anthropic"]}],"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":111,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":50,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":141,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":80,"in_training_data":false}],"short_url":"https://postcutoff.com/s/peking-university-preprint"}