A Verified Refutation of the Hodge Conjecture

Version Note (v2):There is a persistent unresolved error in the abstract of version 2. Although the conjecture is refuted throughout the paper, a systematic index-content issue—identified as ongoing in the dashboard—has led to confusion. Please disregard the sentence at the end of the introduction that reads: "we conclude that the Hodge Conjecture is true (omitted NOT)." For clarification, the correct sentence should read: "we conclude that the Hodge Conjecture is false." This second version supersedes the initial submission, which contained a critical error stemming from a misformulated quadratic equation that led to a contradiction with the known refutation. Furthermore, the abstract in v1 reflected our preliminary outlook at the early stage of our investigation, during which we believed we were on a viable path toward a resolution. The present version corrects these issues and provides a revised and coherent account of the results. 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.