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Math-2567-A1 · revision r1 · on Math-2567

OpenAI claim summary for Math-2567

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 068, 2026-09-26): 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. Polynomiality of the natural invariant anticanonical index
  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

Polynomiality of the natural invariant anticanonical index

OpenAI's claim

The paper claims a polynomial invariant Euler index on divisible anticanonical exponents and a conversion that removes a fixed pseudo-effective error from unbounded effective anticanonical twists. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Consider a holomorphic action by a compact real torus $T$ on a compact complex manifold $F$, with natural differential linearization on $L_F=-K_F$. Define \[ I_T(m)=\sum_{q\ge0}(-1)^q\dim H^q(F,L_F^m)^T. \] OpenAI claims some integer $a>0$ and $P\in\mathbb Q[t]$ with $I_T(m)=P(m)$ for every positive multiple of $a$, and $P(0)=I_T(0)$. Its independent conversion theorem takes smooth connected projective complex $S$, smoothly semipositive $L=-K_S$ and pseudo-effective Cartier $D$; sections $H^0(S,m_jL-D)\ne0$ for a strictly increasing positive sequence $m_j$ are claimed to imply $H^0(S,kL)\ne0$ for some $k>0$. This conversion statement has no group-action hypothesis.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): Lefschetz denominators become weighted lattice counts with selected faces deleted, and a polyhedral retraction computes their value at exponent zero. For conversion, a maximal rational base makes section ratios descend to base functions and forces affine horizontal vanishing orders, permitting descent of forms before the fixed error is removed.

Formal verification

Family 068 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