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Math-2527-A1 · revision r1 · on Math-2527

OpenAI claim summary for Math-2527

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, 2026-09-18): 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 split Weil-type eightfold classes
  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 split Weil-type eightfold classes

OpenAI's claim

Every rational Weil class of a split eight-dimensional abelian variety is claimed to have an algebraic representative, uniformly over imaginary-quadratic fields and compatible polarizations. 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 $K=\mathbb Q(\sqrt{-d})$ be imaginary quadratic, $A$ a complex abelian variety of dimension eight with a $K$-action, and $\lambda$ a compatible polarization of Weil type, with eigenspace multiplicities $(4,4)$. Put $V=H_1(A,\mathbb Q)$, let $\psi$ be the rational Riemann form and $J$ multiplication by $\sqrt{-d}$. Assume the associated Hermitian form \[ h(x,y)=\psi(Jx,y)+\sqrt{-d}\,\psi(x,y) \] vanishes on a four-dimensional $K$-subspace of $V$ (the split condition). Define the rational Weil space $W_K(A)=\bigwedge_K^8H^1(A,\mathbb Q)\subset H^8(A,\mathbb Q)$ using the two conjugate $K$-eigenspace exterior powers. OpenAI claims \[ W_K(A)\subset\operatorname{im}\left(\operatorname{CH}^4(A)_{\mathbb Q}\xrightarrow{\operatorname{cl}}H^8(A,\mathbb Q)\right). \] The field and compatible polarization type are arbitrary; extra endomorphisms are allowed. The statement concerns these Weil classes, rather than all Hodge classes on all varieties.

Outline of the argument

Manuscript 1 (family 032, 2026-09-18): An embedded spin Lagrangian detects the exceptional middle-cohomology component and makes two homological obstruction maps injective. Finite-cutoff weighted Floer operations are assembled into algebraic module identities; cancellation in alternating specialization then transports the desired Chern character to an algebraic cycle.

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