1. COHERENCE INTEGRATION: C(x,t) = ∫₀ᵗ L(x,τ) dτ
Mathematical: Temporal integration of memory density field L(x,τ)
Memory Density: L[idx] = growthActivity[idx]×0.1 + branchingActivity[idx]×0.2 + hyphalDensity[idx]×0.05
Integration: coherence[idx] += L[idx] × dt (each frame)
Decay: coherence[idx] ×= 0.9998 (slow coherence dissipation)
Result: Each grid cell accumulates coherence C(x,t) representing integrated history of mycelial activity
2. RUPTURE DETECTION: δ(t-tᵢ) at threshold crossing
Mathematical: Dirac delta impulse at critical stress σ(t) > σ_crit
Stress Calculation: σ_combined = weighted_avg(σ_temp, σ_pH, σ_moisture, σ_O₂, σ_CO₂)
Rupture Threshold: if(σ > 0.75) { probabilistic_death(); ruptureCounter++ }
Immediate Lethal: if(σ > 0.9) { immediate_death(); ruptureCounter++ }
Result: Biologically accurate rupture events at discrete times tᵢ when stress exceeds tolerance
3. EXPONENTIAL Regeneration: R[χ](x,t) = ∫ φ(x,τ)·exp(C(x,τ)/Ω)·Θ(t-τ) dτ
φ(x,τ) Historical field: signal = nutrients[idx]×2 + moisture[idx]×0.5 - density[idx]×0.3
exp(C(x)/Ω) Coherence amplification: RegenerationWeight = exp(coherence[idx] / OMEGA)
Implementation: effectiveSpeed = baseSpeed × RegenerationWeight × (1-stress)
Memory Kernel K(t-τ): 10-frame exponentially weighted sensor history
Result: Hyphal growth rates exponentially amplified by accumulated coherence - high C regions grow much faster
4. MEMORY KERNEL: K(t-τ) = exp(-|t-τ|/τ_memory)
Sensor History: sensorMemory[10] array stores last 10 nutrient/moisture readings
Exponential Weighting: weight[i] = exp(-i × 0.15) (older memories weighted less)
Convolution: memSignal = Σᵢ sensorMemory[i] × weight[i]
Navigation: Hyphal tips use memory-weighted signal for directional growth
Result: Non-Markovian dynamics - decisions depend on accumulated sensory history, not just current state
5. EULER-LAGRANGE VARIATIONAL STRUCTURE
d/dt(∂L/∂ẋ) - ∂L/∂x = ∫ K(t-τ)φ(x,τ)exp(C(x,τ)/Ω) dτ + Σ ρᵢ(x)δ(t-tᵢ)
LEFT SIDE: Classical Variational Dynamics
• d/dt(∂L/∂ẋ): Momentum evolution → velocity updates via chemotactic gradients
• ∂L/∂x: Force from coherence field → ∇nutrients, ∇moisture guide growth
• Coherence Lagrangian L(x,ẋ,t): Environmental signal strength integrated over time
RIGHT SIDE TERM 1: Memory Integral (Regeneration)
• φ(x,τ): Historical field → nutrient×2 + moisture×0.5 - density×0.3
• exp(C(x,τ)/Ω): Exponential coherence weighting → OMEGA = 0.3 (temperature scale)
• K(t-τ): Memory kernel → exp(-Δt/τ_memory) with τ = 10 frames
• ∫ ... dτ: Temporal convolution → weighted history influences present dynamics
• Observable Effect: High-coherence regions exhibit 2-10× faster growth than low-coherence regions
RIGHT SIDE TERM 2: Impulsive Terms (Rupture)
• δ(t-tᵢ): Dirac delta at rupture → discrete stress-threshold events
• ρᵢ(x): Rupture amplitude → energy loss proportional to stress severity
• Σ: Multiple rupture events → ruptureCounter tracks cumulative damage
• Biological Correspondence: Hyphal death, network fragmentation, resource loss
KEY SIGNATURES YOU CAN OBSERVE:
1. Exponential Divergence: Dense mycelial regions grow exponentially faster than sparse regions (exp(C/Ω) term)
2. Non-Markovian Navigation: Hyphae "remember" past 10 nutrient readings, creating smooth adaptive trajectories
3. Threshold Ruptures: Network suddenly fragments when stress crosses σ = 0.75 (discrete δ-function events)
4. History-Weighted Recovery: After stress relief, high-coherence regions recover first and fastest
6. PERFORMANCE OPTIMIZATION
Float32Array fields: coherence[], hyphalDensity[], nutrients[], moisture[]
Coherence integration: O(N) per frame where N = 700×700 grid cells
Exponential caching: RegenerationWeight computed per hyphal tip, not per grid cell
Memory kernel: Fixed 10-frame window (not full history) for O(1) access
Result: Maintains 60fps with full CRR mathematical rigour