I. Canonical CRR Formalism
The Coherence-Rupture-Regeneration framework describes systems that maintain identity through discontinuous transformations via three coupled operators:
1. Coherence Integration
C(x,t) = ∫₀ᵗ L(x,τ) dτ
where L(x,τ) is the memory density function measuring how environmental conditions and system state contribute to accumulated temporal structure at time τ.
2. Rupture Detection
δ(t - t₀) when C(x,t) exceeds threshold or degradation detected
Ruptures occur at discrete times {t₀, t₁, t₂, ...} with amplitudes ρᵢ(x) scaled by pre-rupture coherence.
3. Regeneration Operator
R[χ](x,t) = ∫₀ᵗ K(t-τ) · φ(x,τ) · exp(C(τ)/Ω) · Θ(t-τ) dτ
where:
- K(t-τ): Memory kernel (temporal weighting function)
- φ(x,τ): Historical field signal (environmental state at time τ)
- C(τ): Coherence at time τ (from history)
- Ω: System temperature parameter (normalization constant)
- Θ(t-τ): Heaviside function ensuring causality (only past influences future)
II. Implementation in This Simulation
A. Coherence Integration (Coral Polyps)
The memory density L is computed from multiple environmental factors:
L(x,t) = L_nutrition + L_stability + L_flow - L_stress
L_nutrition = nutrientField[x] × 0.5
L_stability = (1 - tempStress) × (1 - oxygenStress) × lightLevel × 0.3
L_flow = min(1, flowSpeed × 5) × 0.2
L_stress = crowding × 0.2
// Discrete-time integration: C(t+Δt) = C(t) + L·Δt
const L = nutritionTerm + stabilityTerm + flowBenefit - stressTerm;
this.coherence += L;
Implementation Detail: The coherence integral is approximated using Euler's method with timestep Δt = 1 frame. This discrete summation converges to the continuous integral as Δt → 0.
B. Historical Memory Storage
To implement the full regeneration operator R[χ], the system maintains history buffers:
this.coherenceHistory = [];
this.fieldHistory = [];
These buffers enable non-Markovian dynamics: future regeneration depends on the entire past trajectory, not just the current state.
C. Rupture Detection
Rupture triggered when: C(x,t) > Ω × ln(10) ≈ 2.3Ω
const RUPTURE_THRESHOLD = OMEGA * Math.log(10);
if (this.coherence > RUPTURE_THRESHOLD &&
this.age - this.lastRuptureTime > 80) {
this.rupture();
}
The refractory period prevents pathological rapid-fire ruptures, implementing a minimum timescale for reorganization.
D. Validated Memory Regeneration
The core CRR principle: regeneration draws on historically validated states (high C(τ)) rather than merely recent states.
R = ∑ τ∈history K(t-τ) × φ_quality(τ) × exp(C(τ)/Ω)
let regenerationPotential = 0;
const tau_memory = 50;
for (let i = 0; i < this.fieldHistory.length; i++) {
const hist = this.fieldHistory[i];
const cohHist = this.coherenceHistory[i];
const dt = this.age - hist.time;
const kernel = Math.exp(-dt / tau_memory);
const fieldQuality = hist.nutrients * (1 - Math.abs(hist.temp - 26) / 10);
const coherenceWeight = Math.exp(cohHist.coherence / this.omega);
regenerationPotential += kernel * fieldQuality * coherenceWeight;
}
regenerationPotential /= this.fieldHistory.length || 1;
const numBranches = regenerationPotential > 1.5 ? 2 :
regenerationPotential > 0.5 ? 1 : 0;
Key Innovation: Child polyps inherit coherence weighted by exp(C(τ)/Ω) from validated historical periods, not just the parent's current state. This implements the "validated memory principle" from the theoretical framework.
E. Memory Preservation Through Rupture
During severe stress (bleaching), the system demonstrates distributed memory encoding:
this.preStressCoherence = this.coherence;
this.coherence *= 0.05;
this.bleached = true;
if (conditions_improve) {
const recoveryRate = Math.exp(this.preStressCoherence / this.omega) * 0.0015;
this.coherence += recoveryRate;
}
This demonstrates how the scalar coherence C(t) can reset locally (enabling plasticity) while the field structure {C(τ)}τ
III. Environmental Field Dynamics
Navier-Stokes Inspired Fluid Flow
∂u/∂t = -u·∇u - ∇p + ν∇²u + f_external
Implemented via operator splitting: advection → diffusion → vorticity confinement
Coherence-Flow Coupling
L(x,t) = L_intrinsic + κ_flow·|u(x)| + κ_temp·(T_opt - T(x))²
Coral coherence directly modulates local current velocities, creating feedback between biological and physical processes.
IV. Performance Optimizations
To maintain 60 FPS with full CRR dynamics, several computational strategies are employed:
- History Sampling: Field states recorded every 5 frames (12 Hz) rather than every frame (60 Hz)
- Sliding Window: History buffers limited to 200 entries (~16 seconds) to bound memory usage
- Spatial Hashing: O(n) neighbor queries using grid-based spatial partitioning
- LOD System: Visual detail scales with coherence level (low/medium/high)
- Partial Updates: Coral children updated every 2 frames, not every frame
- Object Pooling: Plankton particles reused from fixed pool (zero garbage collection)
V. Theoretical Limitations & Approximations
Limitation 1: Discretization Error
The continuous integral C(x) = ∫₀ᵗ L(x,τ) dτ is approximated by discrete summation. Error scales as O(Δt) = O(1/60 s). For slowly varying L, this is acceptable; for rapidly fluctuating L, higher-order integration schemes would be needed.
Limitation 2: Finite History Window
The theoretical formalism requires integration over all past time. Due to memory constraints, only the most recent ~16 seconds are stored. This truncates long-term memory effects beyond this horizon. A full implementation would use hierarchical temporal summarization or compression.
Limitation 3: Simplified Memory Kernel
The exponential kernel K(t-τ) = exp(-(t-τ)/τ_memory) is a simplification. The theoretical framework allows arbitrary kernel forms. More sophisticated kernels (power-law, oscillatory) could capture multi-scale temporal dependencies.
Limitation 4: Field Resolution
Environmental fields (temperature, nutrients, flow) are discretized on a coarse grid (20px cells). Fine-scale spatial gradients are lost. Higher spatial resolution would require O(n²) computational cost.
Engineering Trade-off: This implementation prioritizes real-time interactivity and visual clarity over perfect mathematical fidelity. The approximations made preserve the essential CRR structure—coherence accumulation, threshold-triggered rupture, and validated memory regeneration—while achieving 60 FPS performance on standard hardware.
VI. Memory Signatures in the Simulation
Different agents exhibit characteristic temporal patterns (memory signatures):
- Coral (Fragile): Long coherence buildup → catastrophic bleaching → slow recovery
- Fish (Resilient): Moderate coherence with frequent small ruptures (predator encounters) → rapid reorganization
- Predator (Oscillatory): Hunting coherence builds and decays rhythmically with feeding cycles
- Ecosystem (Dialectical): Multiple coherence fields interfere, creating emergent collective patterns
VII. Testable Predictions
The CRR implementation generates empirically testable predictions:
- Recovery rate after bleaching should scale as exp(C_pre/Ω), not linearly with time
- Branching events should cluster around high historical coherence periods, not uniform distribution
- Fish school cohesion should persist for time ~ τ_memory after predator departure
- Ecosystem coherence should exhibit 1/f noise spectrum (scale-free temporal correlations)
These predictions distinguish CRR from alternative models (Markovian dynamics, simple threshold systems, random walk models).