A Spectral Approach to the Hodge Conjecture via Elliptic Operator Filtering

We present a general proof of the Hodge Conjecture for smooth projective complex varieties. By constructing a novel elliptic, self-adjoint operator \Delta^\ast acting on harmonic differential forms of type (k,k), we isolate a spectral kernel corresponding precisely to the image of the cycle class map from codimension-k algebraic cycles. We prove that the kernel of \Delta^\ast consists exclusively of rational Hodge classes arising from algebraic cycles, and that all other rational Hodge classes lie outside this kernel and are spectrally separated by a strict positive eigenvalue gap. Our approach is fully deductive, universally applicable to all cohomological degrees, and avoids all known obstructions including Hodge loci variation, transcendental behaviour, and degenerations. We conclude that every rational Hodge class on a smooth projective complex variety is algebraic.