Full Linear Perturbation Analysis Beyond the Quasi-Static Limit: Constraint Reduction and the Cross Term in the Disformal Galactic Sector (RCD Paper XVIII)

Papers XVI–XVII of this series established a subluminality bound for the scalar sound speed in the disformal RCD galactic sector within the quasi-static (QS) dust limit. A potential concern is whether terms dropped by the QS approximation — specifically, time-derivative and mixing contributions — could modify the sound speed at the disformal knee, where the matter correction Delta-Q vanishes. We perform the full constraint reduction using Brown–Kuchar canonical dust in the ADM formalism on a static spherically symmetric background. We prove three results: (i) the first-order disformal mixing D0 C0' dt(delta-C) in the Jordan-frame shift is fully absorbed into the Einstein-frame shift via the momentum constraint; (ii) the residual cross term at second order is Xi = 2 rho-tilde D(1+nD) C0'/(1+DX), which vanishes at the knee (nD = -1) by the algebraic identity d(DX)/dX = 0; (iii) away from the knee, the mixing correction Xi2/(4Q2) ~ cs,QS 2/epsilon is negligible (epsilon ~ 3 x 105 in the solar neighborhood). The QS proxy is therefore exact at the knee, and the 14 pc superluminal window of Paper XVII is a genuine feature at the full linear perturbative level.