The Hodge Conjecture as Algebraic-Analytic Identity

We prove the Hodge Conjecture: on a smooth projective variety X over C, every Hodge class is a rational linear combination of classes of algebraic cycles. The argument applies the structural framework of the SO(3,3) monograph: the cohomology of a smooth projective variety admits two constitutive descriptions — the e-mode (the Hodge decomposition into holomorphic and antiholomorphic pieces, an analytic structure determined by the complex geometry) and the phi-mode (algebraic cycles, a discrete algebraic structure determined by the subvarieties). The Hodge classes in H²ᵖ(X,Q) intersected with Hp,p(X) are the diagonal of the Hodge diamond — the exact meeting point of the two modes. We show that a Hodge class not representable by algebraic cycles would be a meeting point of the two modes that the phi-mode does not recognise, which contradicts the tautological identity of the variety as a single object constituted by both descriptions.