The ordinary-double-point gap conjecture for normalized volumes of klt singularities
For a klt germ of dimension with zero boundary, the normalized volume over valuations centred at is at most , with equality exactly at smooth points (Liu-Xu). Spotti and Sun, motivated by volume densities of tangent cones of Kahler-Einstein limits, conjectured that the ordinary double point comes next. It was known for surfaces (Li-Liu), threefolds (Liu-Xu) and local complete intersections (Liu), with a different sharp bound for toric germs (Moraga-Suss). Does every singular -dimensional klt germ satisfy , with equality exactly for an ordinary double point?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry, K-stability
- Posed by
- Cristiano Spotti and Song Sun
- Year posed
- 2017
- Years open
- 9y
- 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
- 32 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for and a singular closed point of a normal complex algebraic -fold with -Cartier, klt with zero boundary, , with equality iff the germ is analytically an ordinary double point; the point need not be isolated. The companion proves dimension four (bound 162), the base case. Corollary 1.2 gives a strict anticanonical volume bound below for singular K-semistable -Fano -folds, so off projective space only for the smooth quadric or . Not shown: nonzero boundary, the next gaps, or sharpness of the singular global bound.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts, both dated September 24, 2026; the every-dimension paper uses the dimension-four paper as its base case.
Verification
No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both manuscripts were read against the conjecture as cited. The proofs (cone reduction through stable degeneration, an orbifold rational-curve argument for small stabilizers, jet cones and adjoint jets, induction on dimension from the fourfold case) were not refereed. They rely on the published theory of normalized-volume minimizers (Blum, Xu, Xu-Zhuang, Li-Wang-Xu) as inputs. No Lean formalization accompanies this family.