Paper XIV-A: Dynamics and Stability of the Coherence Field — Mathematical Foundations for an Effective Protection Operator

We present a rigorous mathematical framework for the Coherence Field (C-field) as an effective protection operator within Radial Coherential Dynamics (RCD). Unlike standard scalar field approaches (quintessence, k-essence, scalar-tensor theories), the C-field is defined not as an additional energy-density contributor but as a modulator of effective coupling structures. We derive stability conditions, identify emergent scales, and establish precise criteria distinguishing "absorbable" dynamics (reproducible by standard parametrizations) from "non-absorbable" signatures (requiring genuine protection mechanisms). This framework provides falsifiable triggers for domain-specific applications (Paper XIV-B) and impossibility theorems (Paper XIV-C). This is Paper XIV-A of the Radial Coherential Dynamics Series, establishing the mathematical foundations for the XIV trilogy. Part of the XIV Trilogy:- XIV-A: Structure (this paper)- XIV-B: Application (cosmological implications)- XIV-C: Necessity (impossibility theorem)