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Made of Maths

CRR Simulation Architecture

Markovian agents operating within non-Markovian coherence fields

Computational Substrate as Coherence Field

CRR simulations aim to represent the idea of autonomous agents responding to local environmental conditions, operating within computational substrates that accumulate interaction histories. The simulation environment can be thought of as functioning as a 'non-Markovian' coherence field, where temporal memory is encoded through persistent data structures that maintain states across computational cycles. Here we define 'non-markovian' to mean a temporal integral over past states, rather than a linear recurrence. In this sense, system dynamics are influenced by the accumulated history of interactions, not only the immediately preceding state. These simulations approximate the full CRR as best as possible, for efficient computation in small file sizes

Mathematical Conversion: Canonical to Computational

The core theoretical formalism requires mathematical conversion for computational realisation, depending on the simulation, but always based on the same first principles. Pure temporal integrals must be discretised and regularised for numerical stability, whilst preserving the essential mathematical structure of coherence integration, rupture detection, and reconstruction ('Regeneration').

Canonical Coherence Integration
$$C(x) = \int L(x,\tau) \, d\tau$$
Regularised with Temporal Weighting
$$C(x,t) = \int_{-\infty}^{t} L(x,\tau) \cdot e^{-\lambda(t-\tau)} \, d\tau$$
Discrete Implementation
$$C(t) = \sum_{i=0}^{n} L(\tau_i) \cdot e^{-\lambda(t-\tau_i)} \cdot \Delta\tau$$

Exponential temporal weightings prevent unbounded growth ('blow-up'), whilst implementing realistic memory decay. This conversion from continuous to discrete form enables computational tractability whilst preserving "non-Markovian" temporal coupling dynamics.

Coherence Integration

Organisms accumulate coherence through temporal integration of 'memory entanglement density', with exponential decay weighting. This creates memory-dependent dynamics that influence developmental trajectories and emergent behaviours beyond simple reactive responses. To forgive the metaphor, 'memory entanglement density' could be thought of as meaning, or 'qualia' in human phenomenology.

$$C(t) = \sum_{i=0}^{n} L(\tau_i) \cdot e^{-\lambda(t-\tau_i)} \cdot \Delta\tau$$
Discrete temporal integration with exponential decay preserves mathematical structure
Actual Implementation (CRR Dynamics):
updateCoherence(deltaTime) {
    this.L = Math.min(0.4, this.L + this.calculateMemoryDensity() 
        * deltaTime * this.coherence_build_rate);
    
    const coherenceGrowth = this.L * Math.exp(-this.lambda * deltaTime) 
        * deltaTime * 2.0;
    this.C += coherenceGrowth;
    this.C *= (1 - this.lambda * deltaTime * 1.5);
    
    this.C = Math.max(0, Math.min(this.saturation_limit, this.C));
    
    this.memory_trace.push({
        time: time, 
        coherence: this.C, 
        density: Math.min(0.3, this.L)
    });
    if (this.memory_trace.length > 30) this.memory_trace.shift();
}

Rupture Detection

System ruptures are detected through threshold monitoring of coherence states. When coherence drops below critical levels, organisms transition to rupture states that prepare developmental systems for reconstruction through Regeneration processes. Rupture, represented by a Dirac delta, is scale-invariant. By analogy, such events may be compared to Hamiltonian transitions in quantum systems, where coupling terms reconfigure system dynamics.

$$\delta(t-t_0) = \begin{cases} 1 & \text{if } C(t) < C_{\text{threshold}} \\ 0 & \text{otherwise} \end{cases}$$
Binary rupture detection triggers discrete state transitions at threshold crossing
Rupture Implementation:
detectRupture() {
    if (time - this.last_rupture < 4.5) return false;
    
    const recent = this.memory_trace.slice(-3);
    if (recent.length < 2) return false;
    
    const gradient = Math.abs(recent[recent.length-1].coherence - 
        recent[0].coherence);
    const coherenceThreshold = systemParams.ruptureThreshold + 
        Math.random() * 0.1;
    
    return gradient > 0.25 && this.C > coherenceThreshold;
}

Regeneration

The Regeneration operator reconstructs states through exponentially-weighted integration of historical field states. This implements the theoretical framework with causality constraints, thermal coupling parameters, and finite integration windows for efficient computational implementation.

Canonical Regeneration
$$R[\chi](x,t) = \int \phi(x,\tau) \cdot e^{C(x,\tau)/\Omega} \cdot \Theta(t-\tau) \, d\tau$$
Finite Integration Window
$$R(t) = \int_{0}^{t} \phi(\tau) \cdot e^{C(\tau)/\Omega} \cdot H(t-\tau) \, d\tau$$
Discrete Implementation
$$R(t) = \sum_{i: \tau_i \leq t} \phi(\tau_i) \cdot e^{C(\tau_i)/\Omega} \cdot \Delta\tau$$
Regeneration Implementation:
calculateRegeneration() {
    let regeneration = 0;
    const maxTraces = Math.min(15, this.memory_trace.length);
    
    for (let i = this.memory_trace.length - maxTraces; 
         i < this.memory_trace.length; i++) {
        if (i < 0) continue;
        const trace = this.memory_trace[i];
        const tau = time - trace.time;
        
        if (tau > 0) {  // Heaviside function H(t-τ)
            const phi_term = Math.min(0.25, trace.density);
            const coherence_normalized = trace.coherence / this.omega;
            const exp_term = Math.exp(Math.min(1.5, coherence_normalized));
            const temporal_decay = Math.exp(-tau * 0.15);
            
            regeneration += phi_term * exp_term * temporal_decay * 0.01;
        }
    }
    
    return Math.min(0.08, regeneration) * systemParams.regenerationPower;
}

Non-Markovian Field Dynamics

The computational substrate maintains temporal coupling fields that preserve interaction histories across simulation steps. Agentic mathematical organisms 'experience' these fields as environmental memory that influences their developmental trajectories, creating an approximate of non-Markovian dynamics within the simulation space through persistent data structures and temporal integration algorithms.

Memory Density Calculation:
calculateMemoryDensity() {
    let density = 0.08 + Math.sin(time * 0.2) * 0.03;
    
    if (tree) {
        const treeInfluence = Math.min(0.15, tree.getCoherenceInfluence());
        density += treeInfluence * 0.05;
    }
    
    return Math.min(0.12, density) * systemParams.memoryDensity;
}

The simulation's gradients accumulate shared temporal states in persistent data structures, creating memory-dependent dynamics that extend beyond individual agent cognition. The mathematical relationships between coherence, rupture, and Regeneration are concretely realised through discrete temporal integration algorithms that preserve the essential structure of the theoretical formalism, whilst enabling computational tractability.