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Math-2526-A1 · revision r1 · on Math-2526

OpenAI claim summary for Math-2526

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, 2026-10-03): 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. Algebraicity of the prescribed Kuga–Satake map
  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

Algebraicity of the prescribed Kuga–Satake map

OpenAI's claim

For every projective complex K3 surface, the precisely normalized Kuga--Satake embedding into the full even-Clifford abelian target is claimed to come from an algebraic cycle. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $S$ be any smooth projective complex K3 surface and let $T(S)=\operatorname{NS}(S)_{\mathbb Q}^{\perp}$ in $H^2(S,\mathbb Q)$. Fix the full even-Clifford Kuga--Satake construction, including the rational quadratic polarization, parity identification and rational weight-one polarization, with associated abelian variety $A_S$. These choices specify $\kappa_S:T(S)\to H^2(A_S^2,\mathbb Q)$. OpenAI claims \[ \exists\Gamma_S\in\operatorname{CH}^2(S\times A_S^2)_{\mathbb Q}\quad (p_{A_S^2})_*\left(\operatorname{cl}(\Gamma_S)\smile p_S^*x\right)=\kappa_S(x) \quad\text{for all }x\in T(S). \] Here $\operatorname{CH}^2$ is the group of rational codimension-two cycles modulo rational equivalence and $p_S,p_{A_S^2}$ are the projections. The assertion includes every isogenous realization with the prescribed map transported by its rational Hodge isomorphism.

Outline of the argument

Manuscript 1 (family 032, 2026-10-03): The construction engineers a Lagrangian whose second cohomology has a nineteen-dimensional deficit, yielding a corresponding bound on self-extensions after mirror realization. The resulting characteristic class is transported through K3 moduli by semiregularity and specialization, retaining the prescribed Clifford tensor.

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