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Math-2535-A1 · revision r1 · on Math-2535

OpenAI claim summary for Math-2535

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.

The source treats projective complex log canonical pairs, with a rational Cartier divisor in the pullback hypothesis. With those conventions, its finite coefficient set, dimensions $1\le d\le4$ and positive-dimensional base agree with this question. It asserts the exact rational-function presentation with uniform $p_0\mid p$, effective discriminant in a rational DCC set, and b-nef moduli with $p$ times it b-Cartier. It also retains the generalized lc conclusion and generalized klt conclusion for klt inputs.

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

Manuscript

Contents
  1. Bounded denominators in low-dimensional log Calabi–Yau fibrations
  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

Bounded denominators in low-dimensional log Calabi–Yau fibrations

OpenAI's claim

Uniform moduli denominators, trivializing degrees, and effective rounded systems are claimed for the stated log Calabi--Yau fibrations of total dimension at most four. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a finite $\Phi\subset[0,1]\cap\mathbb Q$ and $1\le d\le4$. For every projective complex log canonical $d$-fold pair $(X,B)$ with coefficients in $\Phi$ and a contraction $f:X\to Z$, $\dim Z>0$, with $K_X+B\sim_{\mathbb Q}f^*D$ for a rational Cartier divisor $D$ on $Z$, OpenAI claims uniform positive integers $p_0\mid p$ and a rational descending-chain-condition set $\mathcal B\subset[0,1]$, depending only on $(d,\Phi)$, such that \[ K_X+B+p_0^{-1}\operatorname{Div}(\psi)=f^*D_Z,\qquad D_Z\sim_{\mathbb Q}D,\quad D_Z=K_Z+B_Z+M_Z \] for some rational function $\psi$. The integer $p_0$ clears $\Phi$, $B_Z\ge0$ has coefficients in $\mathcal B$, and the moduli b-divisor $\mathbf M$ is b-nef with $p\mathbf M$ b-Cartier. The resulting generalized pair is log canonical, and klt when the original pair is klt. This is the denominator result in dimensions at most four, not a bound in unrestricted dimensions.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): Reduction to curves measures monodromy characters using the index of the entire reduced special fiber. Bounded generalized pairs, smooth-locus topology, Kummer theory, and finitely generated divisor classes provide a common torsion exponent, after which positive base divisors yield effective field-generating systems.

Formal verification

Family 034 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