Formulas

C = (1/N) Σᵢ<ⱼ cos(φᵢ - φⱼ)

Phase alignment of N oscillators. C = 1: perfect synchrony, C = 0: random phases.

Λ(x) ≡ Λ₀ ln C(x)

Scalar coherence field. Λ₀ ≈ 10⁻³⁵ J (calibration constant). Units: Joules.

W = |⟨ψ|Ô|ψ⟩| / ‖Ô‖

Normalized observation strength. W ∈ [0, 1].

dΛ/dt + ∇·J_Λ = σ_Λ - γ_Λ

Conservation law for coherence field. J_Λ: coherence flux, σ_Λ: source, γ_Λ: dissipation.

P(outcome) = |ψ|²

Emerges from microstate statistics at equilibrium. Not a fundamental axiom in FRC.

δP ∈ [10⁻⁴, 10⁻³]

Measurable under resonant driving. Falsifiable prediction distinguishing FRC from QM.

dS + k* d ln C = 0 ⟹ S + k* ln C = const

Entropy and coherence are conjugate. k* = 1 (information) or k_B (thermodynamic).

ΔG = −k*T Δln C

Connects coherence to thermodynamic free energy. Isothermal projection.

∂_t ln C = −∇·J_C + S_C, J_C = −D_C ∇ln C

Well-posed diffusion-reaction form. D_C > 0 (diffusion coefficient).

σ(t) ≡ k* D_C ∫ ‖∇ ln C‖² dV ≥ 0

Non-negative dissipation under Neumann/Dirichlet boundary conditions.

RER(p→q) = C[q]/C[p] = exp[−D_KL(p∥q)/k*]

Coherence ratio from KL divergence.

C_XY = exp[−I(X;Y)/k*]

Joint coherence from mutual information. I(X;Y) = D_KL(p_XY ∥ p_X p_Y).