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Math-2582-A1 · revision r1 · on Math-2582

OpenAI claim summary for Math-2582

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.

Manuscript 2 (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. Removing a fixed pseudo-effective anticanonical error
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Removing a fixed pseudo-effective anticanonical error

OpenAI's claim

Manuscript 1 (family 068, 2026-09-26): A globally strict anticanonical metric with controlled boundary integrability is claimed to make every pseudo-effective rational divisor effective after a correction on that boundary, and to remove fixed errors from unbounded anticanonical twists. Manuscript 2 (family 068, 2026-09-26): 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.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Y$ be a complex projective variety that is smooth and has positive dimension and $C\ge0$ integral. Assume $-K_Y+C$ carries a singular Hermitian metric $h$ with curvature at least a Kähler form and whose multiplier ideal satisfies $\mathcal J(h)\supset\mathcal O_Y(-C)$. OpenAI claims \[ \forall\text{ pseudo-effective rational }M\ \exists C_0\ge0\text{ supported on }C:\quad M+C_0\sim_{\mathbb Q}E\ge0. \] The multiplier condition means local squared integrability of the canonical section of $C$ in the metric. A separate conversion theorem says that if $X$ is smooth connected projective complex, $L=-K_X$ smoothly semipositive and $P$ pseudo-effective Cartier, then effective integral $N_j\sim m_jL-P$ for unbounded positive $m_j$ imply $H^0(X,mL)\ne0$ for some $m>0$.
Mathematical claim 2.
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): Singular adjoint direct-image positivity descends a strict metric with an exceptional integral boundary and trivial total-space multiplier ideal. Controlled section rings and a single-Iitaka conversion return the corrected sections upstairs; a supported Fano-type base model supplies the required effectivity without losing the disappearing-prime condition.

Manuscript 2 (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