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Math-2549-A1 · revision r1 · on Math-2549

OpenAI claim summary for Math-2549

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 063, 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 generalized Mukai pseudoindex assertion
  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 generalized Mukai pseudoindex assertion

OpenAI's claim

For a smooth complex Fano $n$-fold, the generalized Mukai bound $\rho(\iota-1)\le n$ is claimed, with equality exactly for the corresponding product of projective spaces. 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 smooth connected projective complex Fano manifold $X$ of dimension $n>0$, let $\rho_X$ be its Picard number and $\iota_X=\min\{-K_X\cdot C:C\text{ an irreducible rational curve}\}$ its pseudoindex. OpenAI claims \[ \rho_X(\iota_X-1)\le n, \qquad\text{equality}\ \Longleftrightarrow\ X\simeq(\mathbb P^{\iota_X-1})^{\rho_X}. \] The pseudoindex is a minimum curve degree, not divisibility of the anticanonical line bundle. There is no dimension restriction.

Outline of the argument

Manuscript 1 (family 063, 2026-09-24): A weighted-projective evaluation construction gives positive lower bounds for point descendants of free rational curves. Basis-invariant alternating traces detect independent curve degrees in quantum correspondences; minimal proper covering families enter in the equality case and identify the projective-space factors.

Formal verification

Family 063 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