OpenAI claim summary for Math-2552

Hodge support on a specified K3 quadratic locus

OpenAI's claim

On the specified quadratic-lattice locus of projective K3 surfaces, all rational Hodge classes on every power are claimed to be algebraic; an exact algebraic Kuga--Satake tensor also gives a conditional criterion beyond that locus. 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 projective complex K3 surface $S$, write $T_S=\operatorname{NS}(S)_{\mathbb Q}^{\perp}$ with cup-product quadratic form $q$. Assume a rational isometric embedding \[ (T_S,q)\hookrightarrow\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4, \] where $\mathbb U$ is the hyperbolic plane with Gram matrix $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$. Choose any full even-Clifford realization $H^1(A,\mathbb Q)=C^+(T_S,q)$, any $w\in T_S$ with $q(w)\ne0$, and any rational weight-one polarization. The Kuga--Satake map is formed from $v\mapsto(a\mapsto vaw)$ and this polarization. OpenAI asserts existence of $\Gamma\in\operatorname{CH}^2(S\times A^2)_{\mathbb Q}$ inducing this exact prescribed map on $T_S$, also after transporting it to an isogenous realization. It further claims rational Hodge and generalized Hodge conjectures in all degrees of $S^m$ for every $m\ge1$. The quadratic embedding is a hypothesis, so this theorem alone is restricted to that locus.

Outline of the argument

Manuscript 1 (family 032, 2026-09-30): Clifford multiplication carries exterior powers of the transcendental space into degree-two cohomology of an abelian target. Transposed cycles and a polynomial inverse on the finite-dimensional image recover these summands, so divisor algebraicity supplies the required classes without an inverse Lefschetz correspondence.

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

• OpenAI, Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus, 2026-09-30; citation key OAI:Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: main:theorem.
• OpenAI catalogue entry.