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Math-2538-A1 · revision r1 · on Math-2538

OpenAI claim summary for Math-2538

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 036, 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. Canonical minimal models in numerical dimension one
  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

Canonical minimal models in numerical dimension one

OpenAI's claim

A smooth complex projective variety with canonical section-growth numerical dimension one is claimed to admit a terminal factorial minimal model, with good-model consequences using the separate abundance input. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be smooth connected projective over $\mathbb C$, $n=\dim X\ge3$, and suppose $K_X$ is pseudo-effective. Define the section-growth numerical dimension by \[ \kappa_\sigma(X,D)=\max\{k\in\{0,\ldots,n\}:\ \exists\text{ ample Cartier }A,\ \limsup_{m\to\infty}h^0(X,mD+A)/m^k>0\}. \] Assume $\kappa_\sigma(X,K_X)=1$. OpenAI claims finitely many divisorial contractions or flips, each $K$-negative to a normal projective rational-factorial terminal $Y$ with $K_Y$ nef. The inverse map contracts no divisor, and on a common smooth resolution $p:W\to X$, $q:W\to Y$, \[ p^*K_X=q^*K_Y+E,\qquad E\ge0,\quad q_*E=0, \] with the strict transforms of all contracted prime divisors in $\operatorname{Supp}E$. A separate corollary, using the companion log-abundance theorem, claims $K_Y$ semiample. The invariant retains the fixed ample twist $A$; it is not an assumed growth bound for untwisted pluricanonical sections.

Outline of the argument

Manuscript 1 (family 036, 2026-09-24): Positive-boundary programs first reduce the possible exceptional loci. Algebraic codimension-two cycle classes give a finite homological decrease after discrepancy control; a final surface criterion eliminates the remaining obstruction without assuming general minimal-model existence.

Formal verification

Family 036 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