The ordinary double point volume gap

The ordinary double point volume gap is a conjecture about the normalized volume of klt singularities, an invariant introduced by Chi Li. In dimension n the largest normalized volume, nⁿ, is attained exactly at smooth points; the conjecture predicts that every singular klt germ has normalized volume at most 2(n − 1)ⁿ, with equality exactly at an ordinary double point. Two manuscripts dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim the conjecture in dimension four and then in every dimension. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

As recounted in the introduction of the every-dimension manuscript:
- For a closed point x of a normal complex variety X that is klt near x with zero boundary, the normalized volume is the infimum of A_X(v)ⁿ vol(v) over valuations v centered at x. Its maximum nⁿ is attained exactly at smooth points (Liu–Xu).
- The conjecture arose from noncollapsed Kähler–Einstein limits. Spotti and Sun proposed that the ordinary double point has the largest density among singular Ricci-flat Kähler cones with isolated vertex, and formulated the normalized-volume conjecture for arbitrary singular klt germs (Conjectures 1.2 and 5.5 of their 2017 paper).
- Earlier cases: surfaces, from Li–Liu's formula 4/|G| for quotient singularities; threefolds, by Liu and Xu; local complete intersections in every dimension, by Liu (2022). For singular toric germs, Moraga and Süß proved the different sharp bound (16/27)nⁿ for n ≥ 3.
- The gap gives global restrictions through Liu's local-to-global volume inequality for K-semistable Fano varieties. For smooth K-semistable Fano manifolds other than projective space, Li and Miao proved the bound 2nⁿ with equality exactly for the quadric and P¹ × Pⁿ⁻¹.

The original results

The normalized-volume gap in dimension four (24 September 2026). Theorem 1.1 of the manuscript states that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality exactly for an analytic ordinary double point. Source: the manuscript The normalized-volume gap in dimension four.
The ordinary double point gap in every dimension (24 September 2026). Theorem 1.1 of the manuscript states that, for every n ≥ 2, every singular closed point x of a normal complex algebraic n-fold X with K_X Q-Cartier and X klt near x (zero boundary) has normalized volume at most 2(n − 1)ⁿ, with equality if and only if the germ is analytically isomorphic to the cone z₀² + … + z_n² = 0. The singular point need not be isolated. The manuscript states that the fourfold theorem is its only companion input.
- A corollary on global volume bounds states that a singular n-dimensional K-semistable Q-Fano variety has (−K)ⁿ ≤ 2((n² − 1)/n)ⁿ < 2nⁿ, and that a K-semistable Q-Fano variety other than Pⁿ has (−K)ⁿ ≤ 2nⁿ, with equality exactly for a smooth quadric or P¹ × Pⁿ⁻¹. The manuscript makes no claim of sharpness for the singular bound.
- Method, as described: induction on the dimension from the cases n ≤ 4. Stable degeneration replaces a germ of large density by a K-polystable cone, which is then smooth away from its vertex and Gorenstein. Two cases follow, according to the stabilizers of an integral grading approximating the minimizing Reeb vector. With small stabilizers, a minimal orbifold rational curve and a weight bound from the Ricci-flat cone metric lead to the Kobayashi–Ochiai characterization of quadrics and projective space. With large stabilizers, a semigroup of leading Taylor exponents, adjoint vanishing and two filtrations produce a same-dimensional singular slice, and a supremum argument closes the induction.
- Source: the manuscript The ordinary-double-point gap in every dimension.
Formalization. The collection has no Lean scope note (lean/docs/) or Comparator challenge for these manuscripts.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 18 October 2026)
by 2026-10-14	10m-u/open-problems-catalog, a GitHub catalogue of open problems from doctoral theses and recent papers, list of excluded problems	Catalogue	The catalogue excludes its problem Q1113 (the ordinary-double-point normalized volume gap) from its open list as "claimed in openai/math", linking the every-dimension manuscript.	The entry describes the result as an "upstream claim, not independently verified" (repository commit `ca8d4d4`).
2026-10-16	J. Jardine, Lattice24, referee review of The ordinary-double-point gap in every dimension in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The referee review reports no gap after reading all eight sections and re-deriving each step of the induction for n ≥ 5. Its two minor notes say that the small-stabilizer case depends on an existence result for an orbifold rational curve cited from a preprint of Li and Zhou, which the review could not check, and that correctness depends on the fourfold companion as the base case and on standard but deep cited results.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.
2026-10-16	J. Jardine, Lattice24, referee review of The normalized-volume gap in dimension four in the same audit	Review (AI-assisted audit)	The referee review reports no gap after checking every step of the main argument. Its two minor notes say that Liu's local-complete-intersection theorem and a bound for nonisolated singularities are used as black boxes, and that a local subadjunction result is cited from an unpublished preprint version of Fujino and Gongyo, a statement the review describes as standard in the local setting.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.