The four-dimensional normalized-volume gap
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
At a singular point of a normal complex algebraic fourfold with zero boundary and Q-Cartier canonical divisor that is klt nearby, must normalized volume be at most 162, with equality exactly for the analytic ordinary quadratic double point in $C^{5}$?Short English statement
At a singular point of a normal complex algebraic fourfold with zero boundary and Q-Cartier canonical divisor that is klt nearby, must normalized volume be at most 162, with equality exactly for the analytic ordinary quadratic double point in $C^{5}$?Sources
- S1 Original question source or mathematical context. https://aimath.org/WWN/singularvariety/problems_final.pdf.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2571
Manuscript 1 (family 037, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.
The four-dimensional normalized-volume gap
OpenAI's claim
The normalized volume of a singular complex klt fourfold germ with zero boundary is claimed to be at most $162$, attained precisely by ordinary double points. The cited manuscript argues for that result; its relation to this problem is specified below.Mathematical statement of the claimed result
Mathematical claim 1.
For a singular boundary-zero klt closed point $x$ of a normal complex algebraic fourfold with rational Cartier $K_X$, OpenAI claims \[ \widehat{\operatorname{vol}}(x,X)\le162, \qquad\text{equality}\ \Longleftrightarrow\ (X,x)_{\mathrm{an}}\simeq(\{z_0^2+\cdots+z_4^2=0\}\subset\mathbb C^5,0). \] Normalized volume is the infimum of $A_X(v)^4\operatorname{vol}(v)$ over valuations centered at $x$, with $\operatorname{vol}(v)$ the asymptotic valuation-ideal colength multiplied by $4!$. The theorem has zero boundary and complex dimension four.Outline of the argument
Manuscript 1 (family 037, 2026-09-24): Density and integral discrepancy estimates reduce the high-volume cone to a terminal Gorenstein hyperplane section. Threefold classification bounds its embedding dimension, making the original cone a hypersurface; the established complete-intersection gap then gives the sharp bound and equality description.Formal verification
Family 037 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.
Status caveat
OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.Sources
- OpenAI, The normalized-volume gap in dimension four, 2026-09-24; citation key
OAI:The-normalized-volume-gap-in-dimension-four-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.