Topological Phase Transition of Algebraic Cycles: A Reformulation of the Hodge Correspondence via Ternary Total Integral Embedding

The classical Hodge Conjecture asserts that every rational Hodge (p, p)-class on a smoothcomplex projective variety arises from an algebraic cycle. This paper resolves the conjec-ture by embedding the geometric structure into a ternary metric manifold via the P1→3mapping. We propose that algebraic cycles are not isolated geometric invariants, but stableconfigurations arising from a topological phase transition. By introducing the asymmetricoperator [1−tanh(Φ)] and the Ternary Total Integral, we demonstrate that rational (p,p)-classes are the necessary physical precipitates of manifold cooling in the ∅-state. A detailedderivation for complex tori is provided, establishing that algebraic cycles represent the onlydimensionally consistent resolution of cohomology in ternary space.