Derivation of the ECSM Coherence Parameter from Compressible Medium Dynamics

This paper derives the ECSM coherence parameter from the linear response of a compressible finite-response medium. Starting from a density-phase order parameter, the medium is shown to support coherent long-wavelength propagation with a characteristic propagation speed and healing length. The finite time required for the medium to reorganise over this coherence length defines an intrinsic response time, tau_resp. When the medium is driven on a timescale comparable to or shorter than this response time, the response amplitude is suppressed by the transfer function of a finite-relaxation system. This yields the ECSM coherence law: chi = 1 / (1 + (tau_resp / tau_drive)^2) showing that chi is not an arbitrary phenomenological modifier, but the effective response efficiency of a finite-relaxation medium. The derivation provides a bridge between the microscopic ECSM scaffold and the macroscopic coherence parameter used in gravitational, optical, inertial, localisation, and cosmological applications. In the coherent limit, chi approaches 1 and standard propagation behaviour is recovered. In the response-limited regime, chi becomes small and the medium may exhibit lag, saturation, localisation, confinement-like behaviour, or coherence-dependent propagation corrections. This paper therefore provides a core derivational basis for the ECSM programme by linking its central control parameter to compressible medium dynamics rather than treating it as a free phenomenological insertion.