Module 6: High-Dimensional Dynamics & Arnold Diffusion (Graduate Capstone)

3D Volumetric FLI Raymarcher of the Arnold Web & 10⁵-Step/Frame Triadic Drift Lab — midi-sink (libsumi + Suzu)
codim(𝕋^n ⊂ Σ_E) = n − 1 ≥ 2  |  ∑ k_i ω_i(I) = 0  |  T_Nekh ∼ exp((ε₀/ε)^(1/2n))

Part 1: Shader Lab — 3D Raymarching the Arnold Web in Action Space (I₁, I₂, I₃) via Fast Lyapunov Indicators Drag canvas to orbit 3D camera around action space (I₁, I₂, I₃)

For n ≥ 3 degrees of freedom, n-dimensional KAM tori have codimension n − 1 ≥ 2 inside the (2n − 1)-dimensional energy surface Σ_E, so their complement is a single connected network — the Arnold Web ∑ k_i ω_i(I) = 0. This WebGL2 raymarching shader steps through 3D action space (I₁, I₂, I₃), computing the local Fast Lyapunov Indicator (FLI) on the fly to illuminate the whiskered resonant channels and multi-resonance junctions in 3D, together with the live drifting triad marker from Part 2!

View: 3D Volumetric FLI Raymarcher Nekhoroshev Stability T_Nekh ∝ exp((ε₀/ε)^(1/6)): 4.8×10⁵ steps Triad Action I(t): (0.72, 0.84, 1.08) n = 3 DoF → codim(𝕋³ ⊂ Σ_E⁵) = 2 (Escape Open!)
Reading the 3D Arnold Web:
• Gold & Cyan Sheets: Resonant planes k₁I₁ + k₂I₂ + k₃I₃ = 0 where the tangent vector norm ‖v_N‖ (FLI) spikes along whiskered separatrices.
• Bright White-Gold Nodes: Multiplicity-2 Resonance Junctions (Hubs) where two or more resonant sheets intersect on the energy sphere.
• Pulsing Magenta Orb: The live 3D action coordinate I(t) = (I₁, I₂, I₃) of the Part 2 Triad Synthesizer!

Part 2: Sound Lab — Hearing Arnold Diffusion Along the Web (10⁵ Symplectic Steps / Frame) 3D Action (I₁, I₂, I₃) drives a 3-voice triadlocked to k₁f₁ + k₂f₂ + k₃f₃ = 0

Because Arnold diffusion is exponentially slow (∼ exp(−(ε₀/ε)^(1/6))), a GPU/burst accelerator runs up to 100,000 (10⁵) symplectic steps per audio frame while mapping the 3D action vector (I₁, I₂, I₃) to a three-voice triad (f₁, f₂, f₃) = f_base·(I₁, I₂, I₃). Hear the triad slowly migrate along a single resonance line (holding k₁f₁ + k₂f₂ + k₃f₃ = 0 invariant while all three pitches glide!) until it hits a resonance junction and pivots onto a new harmonic path.

Arnold Web on Energy Sphere S²_E & Active Drift Trail Channel: 2f₁ − 3f₂ + f₃ = 0
3-Voice Triad Glissando: f₁(t) [Cyan], f₂(t) [Gold], f₃(t) [Violet] Junction Pivots: 0
Audio: Stopped Triad: (216 Hz, 252 Hz, 324 Hz) Active Law: 2·f₁ − 3·f₂ + 1·f₃ = 0.0 Hz Integrated Steps: 0 (0.00×10⁶)
How to listen to Arnold Diffusion:
• Watch the Active Law pill (e.g. 2f₁ − 3f₂ + f₃ = 0): as the trajectory drifts along the whiskered transition chain on the energy sphere I₁² + I₂² + I₃² = 2E, all three voices glide in opposite directions so that k₁f₁ + k₂f₂ + k₃f₃ stays locked at 0.0 Hz!
• When the magenta dot reaches an intersection hub (white circle on the left map), it scatters onto a new resonance line and pivots its harmonic direction!