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The ordinary-double-point volume gap

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Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most \(2(n-1)^n\). Equality holds precisely for an analytic ordinary double point.

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The paper states

Theorem 1.1

In The normalized-volume gap in dimension four, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.1 (The ordinary double point gap) in The ordinary-double-point gap in every dimension () and Corollary 1.2 (Global volume bounds) in The ordinary-double-point gap in every dimension ().

Let x\in X be a singular closed point of a normal complex algebraic variety of dimension four. Suppose that K_X is \Q-Cartier and that X is klt near x, with boundary zero. Then

\nvol(x,X)\le 162.

Equality holds if and only if the analytic germ (X,x) is isomorphic to

\left(\left\{z_0^2+z_1^2+z_2^2+z_3^2+z_4^2=0\right\}\subset\C^5,0\right).
Reported proof verification
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Papers and explanations
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The ordinary-double-point gap in every dimension

24 September 2026, current edition.

Read it herePDF

We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most \(2(n-1)^n\), with equality precisely at an analytic ordinary double point.

The normalized-volume gap in dimension four

24 September 2026, current edition.

Read it herePDF

We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

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