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Math-2571-A1 · revision r1 · on Math-2571

OpenAI claim summary for Math-2571

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. The four-dimensional normalized-volume gap
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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