A Spectral-Categorical Resolution of the Hodge Conjecture via Cohomological Splitting and the Universal Seesaw Stabilizer

We propose a constructive framework that resolves the Hodge Conjecture on smooth complex projective varieties. Rational Hodge classes are treated as zero-modes of a dynamical operator on the Universal Spectral Object 𝒰. A Wide-Net Seesaw operator, stabilized at the exact damping constant θ_eff = W(1/√(4π)) ≈ 0.225209973392 (induced by the fixed-point equation θ e^θ = 1/√(4π)), generates a contraction factor κ = 1 - θ_eff² ≈ 0.949280467885 < 1 that exponentially suppresses transcendental fluctuations. A functorial heat-kernel filter induces a categorical equivalence between the stabilized kernel and the image of the cycle class map, forcing every rational Hodge class to be algebraic. Numerical verification on the exact 27-line cubic surface intersection lattice shows complete decay of mock transcendental components while preserving the algebraic rank. The same stabilizer mechanism appears in taming wild ramification in geometric Langlands, suggesting a common principle for controlling transcendental obstructions.