An Anatomy of Illusions: A Pedagogical Taxonomy of Failed Resolutions to the Hodge Conjecture

We present a chronological taxonomy of synthetic mathematical frameworks attemptingto resolve the Hodge Conjecture. By constructing a theoretical pipeline—from the Lelong-Zariski Transform to the Virtual Super-Connection Isomorphism—we systematically exposethe foundational incompatibilities between differential geometry, geometric measure theory,non-commutative algebra, and algebraic K-theory. This paper serves as a pedagogical autopsy,demonstrating why continuous analytic methods consistently fail to capture rigidalgebraic cycles without violating their own structural axioms.