FRC 566.001 — Entropy-Coherence Reciprocity and UCC

1. Introduction

We formalize the FRC 566 reciprocity between entropy S and coherence C and the associated flow equation (UCC). The goal is to provide unit-consistent definitions, thermodynamic projections, and reproducible validations.

2. Definitions and Units

Entropy S is measured in nats (information layers) or J/K (thermodynamic layers). Coherence C is a dimensionless scalar gauge. We adopt two conventions for the coherence constant k:

• Information/cognition layers: k = 1 (nats)
• Thermo/physical layers: k = k_B (Boltzmann constant)

Regularization for numerical work uses C_epsilon = 1/(Z^2 + epsilon) with small epsilon > 0.

3. Reciprocity Law

The core relationship:

dS + k* d ln C = 0  =>  S + k* ln C = const.     (1)

In information form, with a distribution p, define C[p] = exp[-H(p)/k] where H is Shannon entropy (nats).

KL divergence and mutual information yield coherence ratios:

RER(p->q) = C[q]/C[p] = exp[-D_KL(p||q)/k*]      (2)

I(X; Y) = D_KL(p_XY || p_X p_Y) => C_XY = exp[-I/k*]   (3)

Thermodynamic projection (isothermal) gives the free-energy relation:

deltaG = -k*T delta(ln C)                          (4)

4. Universal Coherence Condition (UCC)

Local flow form:

d_t ln C = -div(J_C) + S_C,   J_C = -D_C grad(ln C)    (5)

with D_C > 0.

Energy-like dissipation under suitable boundary conditions:

sigma(t) = k* D_C integral(||grad ln C||^2 dV) >= 0     (6)

5. Connections

• FRC-100-001 — Original coherence definition
• FRC-200-001 — Lambda operator foundations
• FRC-300-001 — Witness magnitude framework

6. Key Results

Entropy and coherence are reciprocal: gaining one costs the other

The UCC provides a local flow equation for coherence dynamics

Free energy couples directly to coherence changes via deltaG = -k*T delta(ln C)

Dissipation is always non-negative (thermodynamic consistency)