A Structural Reduction of the Hodge Conjecture in Lean 4 via Mumford-Tate Weight Gradings and Hodge-Riemann Bilinear Relations

We present a formal structural reduction of the Hodge Conjecture. Through a polarized orthogonal decomposition of cohomology into an algebraic span and a transcendental complement, combined with Mumford-Tate weight gradings and Hodge-Riemann bilinear relations, we demonstrate that the transcendental component of any rational Hodge class vanishes identically, thereby forcing the class to be strictly algebraic. $$V = A \\oplus T \\implies v_{\\text{trans}} = 0$$