Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics
Awaiting review
Importance: major (4 of 5)The 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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 7 October 2026
Your AI and this story
- GPT-6 Astra111 days after its cutoff
- Claude Opus 5.550 days after its cutoff
- Gemini 4 Hydrogen141 days after its cutoff
- Grok 4.780 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 50 days before it.
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
What happened
A 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.
Why it matters
The 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.
What is disputed or not yet verified
| Verification | Preprint only; not formalised; not peer-reviewed |
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Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Science block: short label for the open-problems table
- Filed