CRR-FEP Inner Screen Simulator

Control Parameters

Ω (Omega) - Rigidity-Liquidity Parameter: 2.0

Low Ω: Rigid, peaked at high-coherence | High Ω: Liquid, uniform history access
Note: Ω < 0.3 approaches computational instability

C (Coherence) - Accumulated Success: 5.0

C(t) = ∫₀ᵗ L(τ)dτ where L = -F (negative variational free energy)
Effect: Higher coherence = stronger memory amplification via exp(C/Ω)
Accumulated coherence amplifies ALL template accessibility exponentially

Derived FEP Quantities

Markov Blanket Thickness

55.0

A_B = κ/Ω + A_min

Thickness inversely proportional to Ω

Precision (π)

1.741

π(Ω) = αΩ^-β

High Ω → Low precision (loose constraints)

Coherence Amplification

148.41

exp(C/Ω)

Memory weighting multiplier from accumulated coherence

Mathematical Framework

Regeneration Operator (Full Implementation)

R[χ](x,t) = ∫₀ᵗ φ(x,τ)·exp(C(τ)/Ω)·Θ(t-τ) dτ

Templates φ weighted by exp(C/Ω) - exponential amplification controlled by Ω
Implementation: weight(shape) = exp(C_global/Ω) × exp(C_shape/Ω)

Memory Weight Function (As Implemented)

w(τ) = exp(C_accumulated/Ω) × exp(C_shape/Ω)

Low Ω: Exponentially peaked at recent high-C | High Ω: Approximately uniform across history
Global coherence C amplifies all template accessibility multiplicatively

Effective Memory Depth

τ_eff = ∫₀ᵗ exp(C(τ)/Ω) dτ ≈ exp(C/Ω) for constant C

Grows exponentially with accumulated coherence at high Ω
Higher coherence = deeper temporal integration

Coherence Accumulation Dynamics

dC/dt = -F = -(D_KL[q(η|μ)||p(η|b)] - log p(b))

Successful prediction (low VFE) increases coherence
Coherence measures cumulative inferential success over system lifetime

Boundary Dissolution Limit

Ω → Ω_crit ⟹ A_B → A_min ⟹ |U⟩ ≠ |S⟩|E⟩

At critical Ω, separability breaks down and FEP formalism becomes ill-defined
Safeguard: Ω_min = 0.3 prevents numerical instability

Phenomenological Interpretation

Daily Experience (Low Ω, Medium C): Clear, stable representations. Recent high-coherence patterns dominate. Thick Markov blanket maintains separation. Templates sharply peaked in time.
Dream State (High Ω, Variable C): Fluid, permeable representations. Deep-time patterns accessible. Thin blanket allows novel combinations. Uniform temporal weighting creates non-linear associations.
High Coherence Training Effect: Accumulated successful prediction (high C) amplifies memory accessibility exponentially. System has "learned" patterns deeply. All templates become more accessible - represents stable, well-trained cognitive state.
Low Coherence: Little accumulated success. Templates weakly accessible. Represents naive/untrained system or high-uncertainty state. Regeneration draws from limited memory.
Boundary Dissolution (Ω→Ω_crit): Direct experience of constraint structure itself. Timeless, ineffable, unitary. Extreme coherence at high Ω creates explosive memory amplification.