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The Yau–Tian–Donaldson Conjecture for Constant Scalar Curvature Kähler Metrics

The Yau–Tian–Donaldson conjecture predicts that a polarized manifold carries a canonical Kähler metric in its polarization class exactly when it is K-polystable. Settled for Kähler–Einstein metrics on Fano manifolds, it remained open for constant scalar curvature. False: there is a polarized smooth projective fivefold that is K-polystable but admits no extremal Kähler metric in c1(A)c_1(A), so K-polystability does not imply existence.

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Construction
Field
Kähler geometry
Posed by
Shing-Tung Yau, Gang Tian and Simon Donaldson
Year posed
1993
Years open
33y
Solved
2026-08-19
Model
Fable 5, GPT-5.6-sol, Danus
Vendor
Anthropic, OpenAI
Collaborators
Jihao Liu
Verification
Unreviewed
Publication
Preprint
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The paper's appendix draws a distinction worth keeping: a counterexample may reduce to a finite certificate, checkable once the object is written down, or it may itself be a theorem quantified over all degenerations. This is the second kind. The method field records construction, because the resolution exhibits an explicit fivefold, but the difficulty lay elsewhere - candidate manifolds of this shape have been available since 2008, and what was missing was the proof that the mechanism works.

What the AI did

Appendix A, joint with Bin Dong and Guoxiong Gao, is the fullest disclosure in this catalog. Exploring open problems with Claude Code the author found initial signs of a breakthrough, then had Claude Code (Fable 5), Codex (GPT-5.6-sol) and Danus work in collaboration; "the three systems together produced the counterexample and its proof". Human input was crucial once: the author saw the example must refute either Codogni–Stoppa or cscK YTD and directed the agents to settle which. An improved Danus, given only the original problem and none of the earlier findings, then re-derived a counterexample and a complete proof alone in 5 hours 29 minutes.

Verification

Checked by this site on 21 August 2026 against the paper (arXiv:2608.19301v1, 79pp): the AI appendix is verbatim as quoted and the theorem is the extremal-metric statement, which is stronger than the abstract's cscK wording. The mathematics was NOT checked here and is beyond quick verification - the authors report that six other agent systems, given the counterexample and asked only to prove it is one, produced no complete proof in twelve hours. Days-old preprint, no independent review. Recorded as a candidate for that reason.

Sources

Changelog5 changes

Discussion3

VibeGene21 Aug 2026, 01:55 UTC

The arXiv paper says "we construct ... ", should the proof type be of construction rather than argument?

Rasmus Lindahl21 Aug 2026, 02:04 UTC

Changed to construction, thank you. The field records how the problem was resolved, and this one is resolved by exhibiting an explicit fivefold, so your reading of the abstract is the right one for it.

Worth saying what I checked first, because the obvious argument for the change turns out to be wrong. The catalog is not uniformly construction for counterexamples: of 192 disproved entries, 168 are construction, 15 are argument and 9 computation. Kontsevich's asphericity conjecture and Schiffer's are both disproofs filed as argument. So this entry was not an outlier, and "everything else does it this way" would have been a false justification for doing what you suggested.

What had made me file it as argument was the paper's own appendix, which separates counterexamples that reduce to a finite certificate from ones that are themselves a theorem quantified over all degenerations, and places itself in the second class: candidate manifolds of this shape have been available since 2008, and what was missing was the proof that the mechanism works. That is a claim about where the difficulty lay, not about how the problem was resolved, so it belongs in the result note rather than the method field. It is still there, reworded, since the old wording asserted the value you just talked me out of.