ZFC-Constructible Realization of Rational Hodge Classes via Generic Grassmannian Pullbacks

This archive contains a formally structured and ZFC-constructible derivation of the Hodge Conjecture for smooth projective varieties defined over the field of rational numbers Q\mathbb{Q}Q. The main result establishes that every rational Hodge class of type (p,p)(p,p)(p,p) can be realized as the pullback of a universal algebraic class on a suitable Grassmannian variety via a Zariski-generic morphism. The derivation builds on a constructive adaptation of Totaro’s pullback formalism, combined with a systematic enumeration of Schubert classes and a complexity-controlled search mechanism. All steps are compatible with primitive-recursive constructions definable within the axiomatic framework of Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). The document includes explicit examples, formal lemma statements, and computational estimates. A brief verification sketch in the style of Lean 4 is included to illustrate compatibility with formal proof assistants, although no complete Lean formalization is provided at this stage.