Generative-AI-Assisted Counterexample Disproves the cscK Yau–Tian–Donaldson Conjecture
Jihao Liu constructs a smooth polarized projective fivefold claimed to be K-polystable for every normal ample test configuration but to admit no constant scalar curvature Kähler metric, disproving the classical cscK Yau–Tian–Donaldson conjecture for general polarizations while leaving the log Fano, uniform, and completed variants unaffected; the paper credits Fable 5, GPT-5.6-sol, and Danus with producing the counterexample and proof.
In a manuscript dated August 24, 2026, Jihao Liu constructs a smooth polarized projective fivefold that is K-polystable with respect to every normal ample algebraic test configuration of every positive exponent, but whose polarization class contains no extremal—and therefore no constant scalar curvature Kähler—metric. The claimed example disproves the classical cscK form of the Yau–Tian–Donaldson conjecture for general polarized varieties.
The construction is a projective-line bundle over a product of four curves with explicitly specified genera and line-bundle degrees. The paper proves global nonnegativity of the Donaldson–Futaki invariant and classifies the zero-invariant test configurations as products. It emphasizes that the counterexample does not affect the established Yau–Tian–Donaldson correspondence for log Fano varieties or the stronger uniform and completed K-stability formulations; whether the classical statement survives when the connected automorphism group is trivial or finite also remains open.
Liu reports that Claude Code using Fable 5, Codex using GPT-5.6-sol, and Danus jointly produced the counterexample and proof, followed by human polishing and checking. An improved Danus system was then run from the original problem without the earlier findings and reportedly reached a complete solution independently. The detailed AI-use appendix is joint with Bin Dong and Guoxiong Gao.