The Validated Late-Time RCD Kernel is Strictly Parabolic: Principal Symbol Analysis and Absence of Causal Cone in Regime III

We analyze the linearized dynamics of the validated late-time coherence-field kernel of Radial Coherential Dynamics (RCD), the bistable Allen–Cahn / Wright–Fisher reaction–diffusion equation whose coarsening behavior in expanding Friedmann–Robertson–Walker (FRW) backgrounds was established in prior work [1]. Computing the principal symbol of the operator governing small perturbations δC about a homogeneous background, we find it strictly parabolic: the dispersion relation is purely imaginary in frequency, ω = −i[(D_C/a²)k² − f′(C₀)], with no real part at any wavenumber. Consequently the kernel admits no finite propagation speed and no finite-speed causal cone within the validated regime. The nonlinear reaction term does not alter this classification; it only shifts the spinodal threshold that drives coarsening. A direct numerical check confirms Re(ω) = 0 across the full wavenumber range and locates the spinodal boundary. This result delimits the regime in which RCD has been validated: any emergent causal structure in RCD must originate elsewhere — in the quantum-to-classical transition or in an inertial extension, both outside the validated kernel and identified as separate open problems. Removing the late-time dissipative kernel as a candidate origin of an emergent cone is itself a result: it relocates the question to where it actually lives.