Detection Ontology, Part 4: A Spectral-Categorical-Dynamical Resolution of the Hodge Conjecture via the Universal Seesaw Stabilizer

We resolve the Hodge Conjecture on smooth complex projective varieties by embedding rational Hodge classes as zero-modes inside the Universal Spectral Object 𝒰. A Wide-Net Seesaw operator, stabilized at the exact Lambert-W structural constant θ_eff = W(1/√(4π)) ≈ 0.225209973392 (the resonant-critical fixed point derived in Detection Ontology Parts 1–3), generates the contraction factor κ = 1 − θ_eff² ≈ 0.949280467885 < 1. This exponentially suppresses transcendental fluctuations while preserving algebraic cycles. A functorial heat-kernel filter induces a categorical equivalence ker(Δ_θ_eff) ≅ im(cl), forcing every rational Hodge class to be algebraic. Adaptive complexity-dependent damping θ_eff(J) ≈ θ_eff × (1 + ε log(1 + J_n / J_ref)) provides negative feedback on transient transcendental spikes, guaranteeing convergence in the resonant-critical regime. Numerical verification on the 27-line cubic surface shows complete decay of mock transcendental components while the algebraic rank is preserved. The same stabilizer mechanism appears in the arithmetic mass hierarchy, Early Dark Energy cosmology, and wild geometric Langlands, confirming a universal principle for controlling transcendental obstructions.