A Verified Refutation of the Hodge Conjecture

This version (v3) incorporates all necessary corrections to the errata identified in the previous releases. The issues present in earlier versions have now been fully resolved, ensuring that the text reflects the intended precision and coherence of the original work. No further modifications have been made beyond these corrections. This version should be considered the definitive and final form of the paper. This 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.