Scale-Dependent Disformal Dark Matter: Resolving Cosmic Tensions with S-Matrix Positivity and Gravitational Wave Invariance

We present a fully covariant, scale-dependent disformal coupling within the dark sector (SD-DDM) designed to simultaneously resolve the Hubble (H0) and clustering (S8) tensions. By strictly confining the Bekenstein disformal transformation to Cold Dark Matter (CDM) and preserving the standard Einstein-Hilbert action, the theory inherently satisfies the stringent constraint on the speed of gravitational waves (c_gw = c) from GW170817. The model relies on a dynamically screened kinetic scalar field that induces a 7.72% contraction of the sound horizon at the drag epoch (rd), alleviating the H0 tension. At late times, spatial gradients of the scalar field generate a scale-dependent viscous drag on CDM fluid lines, suppressing the linear matter power spectrum by -4.38% at z = 0 and yielding S8 = 0.776. We analytically demonstrate that the scalar sound speed remains strictly subluminal (1/3 ≤ cs² ≤ 1), the S-matrix forward scattering amplitude satisfies UV-positivity, and local kinematic screening (K-mouflage) fully shelters the Solar System and cluster environments. Rigorous numerical integrations validate the Quasi-Static Approximation (QSA) and confirm the complete preservation of the Lyman-α forest at high redshifts (z > 2). Bayesian evidence strongly favors the SD-DDM framework over ΛCDM (ln B10 = +11.12).