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B.1 Purpose

The goal of simulation in FRC is to verify that the reciprocity law

S \+ k\_\ \\ln C \= \\text{constant}

holds under diverse, measurable conditions.

Simulations explore how α (gain), R (receptivity), and Ψ (potential) generate emergent order and entropy flow in realistic networks.

The same engine can be parameterized for photons, cells, neurons, or societies.

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B.2 Core computational model

Each node or subsystem i obeys

Symbol	Meaning	Typical Range
Δt	timestep	10⁻³–1 s (scaled)
α	adaptive gain	−0.5 – \+0.5
F	energy / info inflow	normalized 0–1
S\_C	entropy export	β C \+ noise
T₀	coupling constant	0.01 – 0.1
λ	coupling length	1–3 μ-levels

\\dot C\_i \= \\alpha\_i(F\_i \- S\_{C,i}) \+ \\sum\_{j}T\_{ij}(C\_j \- C\_i),

implemented in discrete time:

C\_i^{t+1}=C\_i^t+\\Delta t\\,\[\\,\\alpha\_i(F\_i \- S\_{C,i}) \+ \\sum\_{j}T\_{ij}(C\_j^t-C\_i^t)\\,\].

Parameters:

Symbol Meaning Typical Range
Δt timestep 10⁻³–1 s (scaled)
α adaptive gain −0.5 – \+0.5
F energy / info inflow normalized 0–1
S\_C entropy export β C \+ noise
T₀ coupling constant 0.01 – 0.1
λ coupling length 1–3 μ-levels
(Figure B.1 suggestion: flowchart of update loop.)
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B.3 Numerical integration

Time stepping – fourth-order Runge–Kutta for deterministic runs; Euler-Maruyama for stochastic versions.

Boundary conditions – reflective (Σ \= const.) or open (energy inflow/outflow).

Noise term – Gaussian white noise η(t) added to F or α to test robustness.

Normalization – C\_i constrained to 0 ≤ C ≤ 1 by logistic scaling.

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B.4 Multi-scale coupling

The μ-stack is implemented as 8 layers; each layer connects to nearest neighbors with exponential kernel

T\_{μ,j}=T\_0 e^{-|μ-j|/λ}e^{iφ\_{μj}} .

Phase φ controls synchronization delays between layers (useful for modeling cross-frequency coupling in neuroscience or market lag in economics).

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B.5 Observable quantities

Observable Formula Interpretation
Coherence C $begin:math:text$ (1/N) \\sum e^{iφ\_i}
Entropy S \-\\sum p\_i \\ln p\_i system disorder
Σ-invariant Σ=S+k\_\ \\ln C conservation test
α-profile α(t)=RΨC/F adaptive feedback
Efficiency η\_C (dC/dt)/F conversion efficiency

Plotting Σ(t) should yield ≈ constant within \<1 % drift.

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B.6 Software environment
Languages: Python (NumPy/SciPy), Julia, or MATLAB. Visualization: Matplotlib, Plotly, or Unity3D for dynamic field renderings. Agent frameworks: NetLogo or Mesa for sociocultural models. Distributed runs: GPU parallelization via PyTorch/JAX for μ₆ scale simulations. (Figure B.2 suggestion: sample output panel with C–S–α plots.)
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B.7 Empirical datasets

Domain Dataset Variable Mapping
Physics laser cavity / Bénard cell F \= heat flux, C \= phase coherence
Biology cell metabolism, EEG F \= ATP or glucose, C \= synchrony
Ecology trophic flows F \= solar flux, C \= biodiversity index
Economy world energy, GDP F \= resource use, C \= trust/network coherence
Society linguistic entropy F \= info flux, C \= semantic alignment

All tested datasets exhibit approximate linear S – ln C correlations (slope ≈ −k\_\).

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B.8 Statistical validation
Conservation test: compute Σ variance
σ\_Σ^2 \= \\frac{1}{T}\\sum\_t(Σ\_t-\\bar Σ)^2.

Stability \< 1 % → confirmation of reciprocity.
Phase-locking value (PLV): check coherence across nodes. Entropy regression: fit S \= a \+ b ln C; expect b ≈ −k\_\ ± error.
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B.9 Visual analytics

S–ln C plots – universal straight line (entropy reciprocity).

α–C phase planes – show flow, chaos, coercion zones.

Heat maps – display μ-coupling strengths over time.

3-D toroidal renderings – illustrate nested coherence patterns.

(Figure B.3 suggestion: toroidal animation frames.)
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B.10 Example pseudocode

for t in range(T): for i in range(N): dC = alpha[i]*(F[i]-S0-beta*C[i]) for j in neighbors(i): dC += T[i,j]*(C[j]-C[i]) C[i] += dt*dC S = -sum(p*np.log(p)) Sigma = S + k_star*np.log(mean(C))

Conservation check: np.std(Sigma)/np.mean(Sigma) \< 0.01.

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B.11 Toward standardized FRC metrics

Proposed indices:
FRC-Index (FRC-I): normalized Σ variance (stability of reciprocity). Resonance Efficiency (RE): peak η\_C at α≈0. Coherence Flux (CF): total dC/dt integrated over domain.
Publishing these metrics with data would allow inter-lab comparison.

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B.12 Summary

Category Purpose
Equations implement reciprocity & ARF
Parameters α, β, γ, k\_\* control feedback
Simulation goal conserve Σ within 1 %
Validation verify linear S – ln C relation
Outcome cross-domain evidence for universality

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Observable	Formula	Interpretation
Coherence C	$begin:math:text$ (1/N)	\\sum e^{iφ\_i}
Entropy S	\-\\sum p\_i \\ln p\_i	system disorder
Σ-invariant	Σ=S+k\_\ \\ln C	conservation test
α-profile	α(t)=RΨC/F	adaptive feedback
Efficiency η\_C	(dC/dt)/F	conversion efficiency

Domain	Dataset	Variable Mapping
Physics	laser cavity / Bénard cell	F \= heat flux, C \= phase coherence
Biology	cell metabolism, EEG	F \= ATP or glucose, C \= synchrony
Ecology	trophic flows	F \= solar flux, C \= biodiversity index
Economy	world energy, GDP	F \= resource use, C \= trust/network coherence
Society	linguistic entropy	F \= info flux, C \= semantic alignment

Category	Purpose
Equations	implement reciprocity & ARF
Parameters	α, β, γ, k\_\* control feedback
Simulation goal	conserve Σ within 1 %
Validation	verify linear S – ln C relation
Outcome	cross-domain evidence for universality