CRR Learning: Dolphins with Sonar Detection

Core CRR Operators

Coherence Integration C(x,t)

C(x,t) = ∫₀ᵗ L(x,τ) dτ Where: - L(x,τ) = Memory density (accumulated state information) - Non-Markovian: ALL past states influence future behavior

Coherence accumulates continuously as dolphins practice ring passages. Higher coherence = stronger commitment to current behavioral patterns, but also increased rigidity.

Rupture Detection δ(t-t₀)

δ(t-t₀) = Dirac delta at rupture threshold Triggers when: C(x,t) > Cₜₕᵣₑₛₕₒₗ𝒹

When coherence exceeds threshold (100 in this simulation), the system becomes too rigid to absorb new information. Rupture is thermodynamically necessary, not a failure.

Regeneration Operator R[χ](x,t)

R[χ](x,t) = ∫₀ᵗ φ(x,τ)·exp(C(x)/Ω)·Θ(t-τ) dτ Where: - φ(x,τ) = Historical field signal (past successful trajectories) - Ω = System temperature parameter (15.0 for dolphins) - exp(C/Ω) = Exponential memory weighting - Θ(t-τ) = Heaviside function (causality constraint)

Past states gain exponentially increasing influence as coherence builds. Successful ring passages are weighted by accuracy × recency, creating adaptive learning.

CRR Sonar Detection System

Sonar Pulse Emission

Sonar pulse emitted every N frames when: - Dolphin coherence C > 20 (sufficient awareness) - Active exploration or ring-seeking behavior Pulse properties: - Radius: r(t) = v_sound · t - Intensity: I(r) = I₀ · exp(-r/λ) - λ = Coherence-dependent attenuation length

Dolphins with higher coherence emit sonar more frequently and with greater range. The sonar itself is a CRR process - it accumulates information about the environment.

Coherence Field Interaction

When sonar intersects bubble ring: Detection_strength = C_dolphin · exp(-distance/range) · ring_opacity Information gain: ΔI = Detection_strength · (1 + C_dolphin/Ω)

The dolphin's coherence field interacts with detected targets. Higher coherence = better target discrimination and memory-weighted target prioritization.

Memory-Weighted Target Selection

Target priority = distance⁻¹ · similarity(current, past_success) · exp(C/Ω) Where similarity measured by: - Angle relative to swimming direction - Depth similarity to successful passages - Ring size preference from memory

Dolphins don't just detect rings - they weight them by similarity to past successful passages. This is pure CRR: coherence amplifies the influence of historical patterns.

Learning Dynamics

Exploitation-Rigidity Trap

As dolphins successfully pass through rings, coherence increases. This amplifies the weight of successful trajectories via exp(C/Ω), creating increasingly precise approaches. However, excessive coherence makes the system brittle - unable to adapt to new ring positions or sizes. Rupture becomes thermodynamically necessary.

Regeneration After Rupture

Post-rupture velocity regeneration: v_new = (1 - α) · v_current + α · Σᵢ wᵢ · v_memory_i Where: - α = Regeneration strength (0.5) - wᵢ = accuracy_i · recency_i (normalized) - Top 3 most successful memories selected

After rupture, dolphins regenerate movement patterns from their most successful memories. This isn't random exploration - it's targeted sampling of the historical phase space weighted by past performance.

Collective Pod Intelligence

When multiple dolphins are present, their coherence fields overlap. This creates emergent coordination - dolphins tend to approach rings that other pod members have successfully passed. The collective memory exceeds individual capacity.

Observable Predictions

Early trials (C < 30): Erratic approaches, frequent misses, wide distribution of trajectories.

Learning phase (30 < C < 70): Increasing precision, exp(C/Ω) rises from ~7 to ~90, accuracy improves from 40% to 75%.

Pre-rupture (C > 90): Near-perfect centering on familiar ring positions, but rigidity to novel configurations. Success rate peaks then plateaus.

Post-rupture regeneration: Brief performance drop as system explores, then rapid recovery weighted by successful memories.