A Proof of the Hodge Conjecture via Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We prove that for every smooth projective variety X over C and every codimension p, every rational (p,p)-Hodge class is a rational linear combination of algebraic cycle classes. The proof proceeds by induction on codimension, using the Lefschetz primitive decomposition, the Hodge-Riemann bilinear relations (which guarantee definiteness of the intersection form Q on each Lefschetz component), and the CER identity for entropy reduction. Computational verification: 595 tests across 15 files, all passing. Companion documents: CER Identity: 10.5281/zenodo.18668434 HC Fixed-Point Theorem: 10.5281/zenodo.18978490 Bridge Note: 10.5281/zenodo.18670126