CRR Mathematical Framework
Coherence Integration
C(x) =
∫ L(x,τ) dτ
Coherence builds as the fish learns. Each experience contributes to accumulated knowledge.
Higher coherence = more sophisticated behavior.
Memory Density
L(x,τ) =
φ₊ -
φ₋
Memory density captures learning from experience. Predator encounters and food discoveries
create strong positive memories. Energy depletion creates negative stress.
Rupture Detection
δ(t-t₀) when
C(x) >
C_threshold
When coherence exceeds threshold, fish undergoes "rupture" - a learning breakthrough where
behavior reorganizes. Coherence resets but knowledge is preserved through regeneration.
Regeneration Operator
R[χ](x,t) =
∫ φ(x,τ)·e^(C(x)/Ω)·Θ(t-τ) dτ
Fish rebuild behavior using exponentially-weighted past experiences. Higher coherence means
stronger memory influence. The Heaviside function Θ ensures only past affects present (time's arrow).
Coherence C(x):
0
Memory L(x,τ):
0
Regeneration R:
0
Ruptures:
0
Total Learning Events
0
Predator Encounters
0
Food Discoveries
0
Learning Breakthroughs
0