Pixton's conjecture: do Pixton's relations give all tautological relations on the moduli of stable curves?
The tautological ring is the image of the strata algebra of decorated stable graphs in the Chow ring. In 2012 Pixton proposed an explicit system of graph relations extending the Faber-Zagier relations; Pandharipande-Pixton-Zvonkine proved them valid in cohomology and Janda in Chow. Pixton conjectured completeness: that the span of his relations (with the operations it is closed under) is the whole kernel of the realization map, in Chow and in cohomology (recorded as Pixton's conjecture in Canning 2025). Is for every stable pair ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Algebraic geometry; moduli of curves; tautological rings
- Posed by
- Aaron Pixton, Conjectural relations in the tautological ring of the moduli of stable curves, arXiv:1207.1918 (2012), Conjecture 2
- Year posed
- 2012
- Years open
- 14y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 38 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims that Pixton's original relations are not complete in rational Chow or rational cohomology: on with and an alternating product of corrected small diagonals vanishes but lies outside the span. It does NOT give a small counterexample (the parameters are not claimed minimal), does NOT address enlarged relation systems, and is distinct from the Gorenstein question already settled negatively (Petersen-Tommasi, Canning).
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The single manuscript defines the counterexample class explicitly and proves nonmembership and vanishing by separate arguments.
Verification
No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 were read against Pixton's Conjecture 2 as fixed in Section 2 of the paper: for and an explicit formal class maps to zero in rational Chow (hence in rational cohomology) but is not in Pixton's original relation span. The vanishing argument relies on derived Quot-scheme virtual classes and O'Sullivan's tensor reconstruction, cited inputs not checked here. No Lean formalization. The paper limits itself to the original relation system as it defines it.