The Kramers-Kronig Dispersion Bridge: Unifying Amplitude Drag, Phase Velocity, and Tensor Elasticity in the Cosserat Continuum

**Preprint | Continuum Field Entropy Empirical Validation Series** Standard ΛCDM cosmology evaluates cosmological expansion and photon attenuation as independent phenomena, requiring Dark Energy (Λ) to balance high-redshift luminosity deficits. In this paper, we unify geometric expansion and optical dimming using the Continuum Field Entropy (CFE) framework. By modeling the Primordial Field as a dispersive Cosserat continuum, we prove that the macroscopic geometric wave speed (ceff) and the localized thermodynamic amplitude drag (ΓFlux) are inextricably linked via the Kramers-Kronig relations. We derive the Redshift-Coupled Dispersion Integral to demonstrate that high-frequency photons in the early universe face a thermodynamic “Double Penalty.” To empirically validate this dispersive bridge, we subject the CFE integral to 1,590 deep-space Type Ia Supernovae, utilizing the full 1701 × 1701 systematic covariance matrix. The CFE framework outperforms standard Dark Energy (ΛCDM) by a massive Δχ² ≈ 408.9. Furthermore, by evaluating high-redshift X-ray quasars from the Lusso (2020) catalog, we independently isolate the continuum’s Spectral Drag Index at n ≈ 0.285. When subjected to strict high-precision instrumental constraints using the Exact Kramers-Kronig Integral anchored to the primordial saturation limit (Γ₀ = 0.721), the primordial Spectral Index refines to n = 0.2836 ± 0.0001. Finally, by dynamically testing the theoretical cube-root limit ($n = 0.36$), we uncover the CFE-Maxwell Elastic Tensor: the field elastically trades Base Drag (Γ₀ = 0.3929) to perfectly conserve a 4.71-magnitude X-ray attenuation penalty. This provides decisive statistical proof that deep-space optical dimming is not geometric acceleration, but the rigid thermodynamic elasticity of a relaxing vacuum. **Project Integration:**This document is a standalone validation report. The underlying universal field equations, foundational axioms, and the complete multi-disciplinary validation framework can be found in the primary master manuscript (DOI: 10.5281/zenodo.20631794).