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Math-2556-A1 · revision r1 · on Math-2556

OpenAI claim summary for Math-2556

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 032, 2026-10-04): 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. Rational Hodge classes on products of K3 surfaces
  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

Rational Hodge classes on products of K3 surfaces

OpenAI's claim

Rational Hodge classes on any finite product of projective complex K3 surfaces are claimed to be algebraic, even when the factors have unrelated periods or extra endomorphisms. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Choose any integer $m\ge1$ and any projective complex K3 surfaces $S_1,\ldots,S_m$, allowing repeated factors. For $X=\prod_{j=1}^m S_j$, OpenAI claims \[ \operatorname{CH}^p(X)_{\mathbb Q}\xrightarrow{\operatorname{cl}} H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) \quad\text{is surjective for every }0\le p\le2m. \] There are no Picard-number, period or endomorphism-field restrictions. The quantified varieties are products of K3 surfaces, so this statement has smaller scope than the Hodge conjecture for arbitrary smooth projective complex varieties.

Outline of the argument

Manuscript 1 (family 032, 2026-10-04): A Lagrangian immersion is used to produce prescribed Euler pairings while bounding the second self-extension space of the associated complex. Clifford-algebra calculations then identify its transcendental cohomology component; algebraic Kuga--Satake tensors and their contractions supply the classes needed on products.

Formal verification

Family 032 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