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A.1 Core invariants

At every scale the FRC framework rests on two equations:

*\\text{(1) Entropy–Coherence Reciprocity:}\\qquad dS \+ k\_\\\,d\\ln C \= 0,

\\text{(2) Adaptive Resonance Formula:}\\qquad \\Delta S \= RΨC. Equation (1) expresses conservation of total transformation potential. Equation (2) describes instantaneous entropy change as a function of receptivity, potential, and coherence.
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A.2 From reciprocity to dynamical law

Start from dS \= \-k\_\\\,d\\ln C. Differentiate with respect to time: \\dot S \= \-k\_\\,\\frac{\\dot C}{C}. Split the total entropy rate into internal and external components: \\dot S \= \\dot S\_\\text{int} \+ \\dot S\_\\text{ext}. Define F \= \-\\dot S\_\\text{ext} (entropy flux into the system) and S\_C \= \\dot S\_\\text{int} (entropy exported). Then, \\dot C \= \\frac{α}{k\_}(F \- S\_C). Absorbing constants into α yields the coherence-pump equation: \\boxed{\\dot C \= α(F \- S\_C).}
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A.3 Coupled-level derivation
For adjacent μ-domains, the coherence exchange term T\_{μ,μ+1} arises from gradient coupling in the field potential V(φ): T\_{μ,μ+1} \= \-\\frac{∂V}{∂φ\_{μ}} \\approx T\_0 e^{-|μ-μ+1|/λ} e^{iφ\_{μμ+1}}. The general N-level system becomes: \\frac{dC\_{μ}}{dt} \= α\_{μ}(F\_{μ} \- S\_{C,μ}) \+ \\sum\_{j\\ne μ} T\_{μ,j}(C\_{j} \- C\_{μ}). Integration over all μ yields the global invariant: \\sum\_{μ} (dS\_{μ} \+ k\_\^{(μ)}\\,d\\ln C\_{μ}) \= 0\.
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A.4 Deriving ARF from local reciprocity

Let J\_C \= \\dot C/C and define receptivity R \= ∂J\_C/∂Ψ. Then internal entropy rate \\dot S\_\\text{int} \= \-k\_\RΨ. Integrating over one resonance cycle: \\Delta S \= RΨC. This defines the Adaptive Resonance Formula (ARF) used throughout the text.
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A.5 Stability criterion
Linearizing \\dot C \= α(F \- S\_C) around equilibrium C=C\_0: \\dot C \= \-αβ(C \- C\_0), with β \= ∂S\_C/∂C. Solution: C(t)=C\_0 \+ (C\_i \- C\_0)e^{-αβt}. Stability demands αβ \> 0\. Since β \> 0 (entropy increases with order), stable regimes have α \> 0 small or α≈0 — the flow zone.
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A.6 Oscillatory feedback solution
When α varies with coherence, \\dot α \= γ(C \- C\_\\text{opt}), we obtain: \\ddot C \+ βγC \= βγC\_\\text{opt} \+ α\_0β(F \- S\_0), a damped harmonic oscillator with natural frequency ω\_0 \= √(βγ). Oscillation period: T \= 2π/√(βγ).
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A.7 The resonance constant k\_\

Units:* S (entropy): J·K⁻¹ ln C: dimensionless Thus k\_ has units J·K⁻¹, analogous to Boltzmann’s constant. Different μ-levels have scaled constants: k\_^{(μ)} \= κ\\,λ\_{μ}^d\\,k\_B, where λ\_μ is correlation scale, d effective dimension, κ a normalization factor. Empirical measurement of k\_\ per domain defines the “fractal spectrum” of coherence.
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A.8 The energy interpretation

Multiply (1) by temperature T: TdS \+ Tk\_\\,d\\ln C \= 0\. Then TdS \= −dE\_coh, identifying E\_\\text{coh}=Tk\_\\ln C as stored coherence energy. Thus the reciprocity law is a conservation of energy between heat and structured information.
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A.9 The phase-space formulation

Define generalized coordinates x\_i and conjugate momenta p\_i. Entropy S(x,p), coherence C(x,p). Hamiltonian form: \\frac{dC}{dt} \= \\frac{∂H}{∂S}, \\qquad \\frac{dS}{dt} \= \-\\frac{∂H}{∂C}. Choosing H \= k\_S\\ln C reproduces dS \+ k\_\\,d\\ln C \= 0\. Hence the reciprocity law is the Hamiltonian generator of the FRC dynamics.
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A.10 Summary

Symbol	Meaning	Equation
S	entropy	disorder measure
C	coherence	correlation measure
α	adaptive gain	system responsiveness
R, Ψ	receptivity & potential	ARF parameters
k\_\	coherence constant	scale factor linking energy & information

Symbol Meaning Equation
S entropy disorder measure
C coherence correlation measure
α adaptive gain system responsiveness
R, Ψ receptivity & potential ARF parameters
k\_\ coherence constant scale factor linking energy & information
Together they form a closed algebra under the transformation group preserving Σ \= S \+ k\_\ \\ln C.**

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