CRR RAINBOW
scroll/pinch to zoom · adjust CRR parameters at macro · observe live calculations at micro/nano
CRR Mathematics MACRO
Ω Adaptation Rate 0.318
1/π for Z₂ systems · controls fading speed
Cth Rupture Threshold 0.85
Coherence level triggering phase transition
L(λ,τ) Luminance Rate 1.00
Coherence accumulation speed
CV Coefficient of Variation 0.159
CV = Ω/2 · derived from Ω
Coherence
C
→
Rupture
δ
→
Regeneration
R
Snell's Law at Entry
n₁ sin θ₁ = n₂ sin θ₂
nwater(λ) = 1.333
θincidence = 59.4°
θrefraction = 40.2°
Dispersion
n(λ) = A + B/λ²
Δn (red→violet) = 0.014
Angular spread = 1.8°
Each wavelength exits at unique angle → spectrum
Entry
n₁
→
Refract
δθ
→
Separate
λ
Path Integral Through Droplet
C(x,t) = ∫L(x,τ)dτ
Droplet ⌀ = 1.2 mm
Path length = 2.4 mm
Phase = 4.2π rad
Internal Reflection
θc = arcsin(1/n)
Critical angle = 48.6°
Reflection R = 0.98
Light trapped → coherence accumulates
Rainbow Exit (Rupture)
θ = 180° + 2θᵢ - 4θᵣ
Exit angle = 42.0°
exp(C/Ω) = 23.1
Caustic = coherence max → visibility
Traverse
∫L
→
Exit
42°
→
Rainbow
🌈
Wave Phase
φ = 2πnL/λ
λ = 550 nm
Optical path = 2640 nm
Phase φ = 30.2 rad
Interference
Δφ = 2πm (constructive)
Path diff = 412 nm
Fringe order m = 4
I ∝ cos²(Δφ/2)
Memory Weighting
R = ∫φ·exp(C/Ω)dτ
C = π (coherence max)
Ω = 1/π ≈ 0.318
exp(C/Ω) = eπ ≈ 23.1
Same eπ in black holes & neural gain!
Phase
φ
→
Interfere
Σ
→
Fringe
I
λ = --- nm
f = --- Hz
---
1.0×
macro view
C(λ,t) = ∫L(λ,τ)dτ → δ(adaptation) → R(blank field)