FRC 100.009: Local Reciprocity Dynamics: Open-System Activation, Adaptive Resonance, and Self-Starting Coherence Fields

This paper develops a local field-level formulation of the Adaptive Resonance Framework (ARF) within Fractal Resonance Cognition (FRC). It addresses a structural gap in the public architecture of FRC: without a local derivation, the adaptive relation ΔS = RΨC can be read as a useful ansatz rather than as the localized activation form of entropy-coherence reciprocity. Starting from the global reciprocity relation established in FRC 566.001 (dS + k* d ln C = 0), we ask how this law should appear in an open, distributed dynamical system with local storage, flux, source, sink, and defect structure. We introduce a local entropy-like density, a coherence field, a reciprocity storage functional, a reciprocity flux, and a local reciprocity defect. We then define ARF as a state-dependent activation or mobility operator that gates transport and transformation according to the local product of receptivity, driving potential, and coherence, ARF(x,t) = R(x,t) Ψ(x,t) C(x,t). In this interpretation, ARF is not a separate conservation law; it is the local permission for dynamics to become effective when reciprocity is open, imbalanced, or driven. The resulting equations reduce to standard drift-diffusion, reaction-diffusion, and chemotaxis-type forms in appropriate limits, while extending them through reciprocity-sensitive activation, open-system exchange, and state-dependent mobility. The central claim is modest and structural: exact homogeneous reciprocity-balanced equilibrium should remain silent, while real activation arises from openness, flux, defect, or environmental exchange. This supplies the missing bridge from global entropy-coherence reciprocity to observable transport, pattern formation, and adaptive transformation.