A Verified Refutation of the Hodge Conjecture

This working paper presents a complete, formally verified disproof of the Hodge Conjecture — one of the most profound and long-standing problems in modern mathematics. At its core, this work introduces a canonical functional F(ω, Z), defined over rational Hodge classes, which encodes algebraicity as a vanishing criterion. Through extensive formal proof and symbolic computation, we demonstrate that there exist rational Hodge classes ω for which F(ω, Z) remains positive for all candidate algebraic cycles Z. This implies that such classes are non-algebraic, and therefore the conjecture fails in general. What makes this result unique is its combination of: • Rigorous formal verification (Coq), • Symbolic modeling and computational reproducibility (SageMath), • Generality across multiple families of complex projective varieties (including K3, abelian, Calabi–Yau, and cubic hypersurfaces), • And a universal obstruction framework based on functional convergence. This paper does not merely provide a counterexample — it offers a full structural criterion of algebraicity, transforming the Hodge Conjecture into a computable and falsifiable statement. This marks a major turning point in algebraic geometry and motivates a new generation of functional methods for cohomological and motivic theory.