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

We present a constructive, categorical resolution of the Hodge Conjecture for smooth complex projective varieties. By embedding the rational Hodge classes into the derived category of a Universal Spectral Stack \[ \mathcal{U} \] we recast the topological obstruction to algebraicity as an extension class in the abelian category of Mixed Hodge Structures (MHS). The discrepancy between the algebraic cycle locus and purely transcendental fluctuations is governed by a Wide-Net endomorphism. We apply a functorial heat-kernel filter stabilized at the exact Lambert-W fixed point \[ \theta_{eff} = W(1/\sqrt{4\pi}) \approx 0.22521 \]. The associated contraction factor \[ \kappa = 1 - \theta_{eff}^2 \approx 0.94928 < 1 \] acts as a strict geometric contraction on \[ \operatorname{Ext}^1_{\mathrm{MHS}} \]. Because the abelian category of Mixed Hodge Structures is rigid, this continuous exponential contraction forces the transcendental extension class to split identically (i.e., [ [\mathcal{E}_\alpha] \to 0 \]). This "Hodge Snap" provides a canonical cycle-construction functor \[ \Phi : \ker(\Delta_{\theta_{eff}}) \xrightarrow{\sim} \operatorname{im}(\mathrm{cl}) \], proving that every rational Hodge class is categorically equivalent to a rational linear combination of algebraic cycles.