Covariant Completion of Radial Coherential Dynamics: From No-Hair on Fixed Backgrounds to Static Solutions with Backreaction Paper F in the RCD Research Program

We investigate static, spherically symmetric solutions of Radial Coherential Dynamics (RCD) coupled to gravity, systematically escalating from test-field on fixed backgrounds to full gravitational backreaction with non-minimal coupling. At each level, we determine whether the coherence field C (r) admits non-trivial regular solutions that could support gravitational wave echoes from compact objects. At Level 1 (fixed Schwarzschild, no backreaction), three independent numerical solvers confirm that no static hair exists: all non-trivial profiles violate the physical domain C > 0. At Level 2 (self-gravitating backreaction with minimal coupling), the coupled Einstein-RCD system produces either domain violations or horizon formation for all tested initial conditions. At Level 3 (full non-minimal coupling with analytically correct regular-center expansion), a comprehensive sweep over 200 configurations spanning 8 values of the coupling constant and 25 values of the core coherence yields zero solutions satisfying simultaneously: regularity, domain preservation, absence of horizons, and asymptotic Schwarzschild decay. These results establish a structural boundary of the RCD black hole sector: the action S = integral of sqrt (-g) R/16piG + (1-C) (dC) ²/2 - Vcore (C) + xiC²R does not admit static spherically symmetric solutions describing regular compact objects without horizons. This delimitation is foundational rather than fatal: it specifies precisely what the theory cannot do in its current formulation, while leaving open alternative kinetic structures, time-dependent collapse scenarios, and extended gravitational couplings as future directions.