Module 6 — High-dimensional dynamics and Arnold diffusion

Lecture notes written against midi-sink as shipped (libsumi 1.12.0, Voxo 0.26.0, 2026-10-25) and checked claim by claim against the engine before this fold (2026-10-26; libsumi is unchanged since, and Voxo’s later versions add the sampler’s second generation and touch no synth claim here). Code citations and gate numbers refer to docs/PROJECT_SPEC.md (spec v6), docs/DECISIONS.md Part XIII (cited DECISIONS_13 #n), voxo/src/suzu.h, and site/src/content/docs/operators/. The lab beside the module is a standalone teaching re-implementation, never the engine; the engine’s own lab is the Suzu lab.

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Lecture Notes

6.1 Topological Obstruction ($n = 2$) vs. Escape ($n \ge 3$): Why KAM Tori Cannot Trap Orbits in High Dimensions

In Modules 4 and 5, we studied the Kolmogorov–Arnold–Moser (KAM) Theorem and saw that for an autonomous 2-DoF Hamiltonian system (or a 1-DoF time-periodic twist map like the Chirikov Standard Map), surviving Diophantine KAM tori act as impenetrable topological dams for $K < K_c \approx 0.971635$. Even though resonant islands and chaotic homoclinic tangles form a dense web between the KAM curves, no chaotic trajectory can ever cross a surviving KAM curve, so the action variables remain permanently trapped near their initial values: $\|I(t) - I(0)\| = \mathcal{O}(\sqrt{\epsilon})$ for all $t \in (-\infty, +\infty)$.

In 1964, Vladimir Arnold discovered a startling dimensional divide: this topological trapping works ONLY for $n \le 2$ degrees of freedom! For $n \ge 3$ degrees of freedom, KAM tori no longer divide the energy surface, and chaotic trajectories can wander across the entire phase space for arbitrarily small perturbations $\epsilon > 0$.

The Codimension Formula on the Energy Hypersurface

Let $(M^{2n}, \omega, H)$ be an autonomous Hamiltonian system with $n$ degrees of freedom:

• Conservation of total energy $H(q, p) = E$ restricts any trajectory to a regular isoenergetic hypersurface $\Sigma_E := H^{-1}(E) \subset M^{2n}$ of dimension: $\dim \Sigma_E = 2n - 1$
• By the Liouville–Arnold and KAM theorems (Module 4, §4.1–§4.2), every surviving invariant KAM torus $\mathcal{T}_\omega \subset \Sigma_E$ is a Lagrangian submanifold of dimension: $\dim \mathcal{T}_\omega = n$
• Therefore, the codimension of a KAM torus $\mathcal{T}_\omega$ inside the energy hypersurface $\Sigma_E$ is: $\boxed{\; \operatorname{codim}_{\Sigma_E}(\mathcal{T}_\omega) = \dim \Sigma_E - \dim \mathcal{T}_\omega = (2n - 1) - n = n - 1 \;}$

By the Jordan–Brouwer Separation Theorem (and Alexander Duality in algebraic topology), a closed submanifold separates an ambient manifold into disconnected “inside” and “outside” regions if and only if its codimension is equal to $1$! Let us compare the dimensions side by side:

Degrees of Freedom $n$	Phase Space $\dim M = 2n$	Energy Surface $\dim \Sigma_E = 2n - 1$	KAM Torus $\dim \mathcal{T}_\omega = n$	Codimension in $\Sigma_E$: $n - 1$	Topological Consequence & Stability
$n = 1$ (Autonomous Pendulum / Oscillator)	$2$	$1$	$1$	$0$	$\mathcal{T}_\omega = \Sigma_E$: the energy curve is the invariant 1-torus. Completely integrable.
$n = 1\tfrac{1}{2}$ (Periodically Kicked Rotor / Standard Map)	$2$ (per strobe)	$2$ (cylinder $\mathbb{T}^1 \times \mathbb{R}$)	$1$ (circle $\mathbb{T}^1$)	$1$	1D circles separate the 2D cylinder. Topological trapping for $K < K_c$.
$n = 2$ (Hénon–Heiles, Stadium Billiard, Double Pendulum)	$4$	$3$	$2$ (2-torus $\mathbb{T}^2$)	$1$	2D tori $\mathbb{T}^2$ are closed surfaces inside $\Sigma_E^3$ (like nested doughnuts in $\mathbb{R}^3$). Topological trapping: $\|I(t) - I(0)\| = \mathcal{O}(\sqrt{\epsilon})$ forever!
$n = 3$ (3-Torus Modal Lattice, 5-Vortex Braid, Asteroid Belt)	$6$	$5$	$3$ (3-torus $\mathbb{T}^3$)	$2$	Codimension $5 - 3 = 2$ (like 1D curves/circles in $\mathbb{R}^3$!). No topological trapping: $\Sigma_E^5 \setminus \bigcup_\omega \mathcal{T}_\omega$ is path-connected!
General $n \ge 3$ (Suzu 16-Mode Lattice, Toda Perturbations)	$2n$	$2n - 1$	$n$	$n - 1 \ge 2$	Complement of all KAM tori is a single connected network (Arnold Web). Global Arnold Diffusion occurs for any $\epsilon > 0$!

Think of the analogy in ordinary 3D space $\mathbb{R}^3$:

• A 2D spherical shell (codimension $3 - 2 = 1$) completely seals off its interior from its exterior; a point inside can never escape without piercing the shell. That is $n = 2$ ($\mathbb{T}^2 \subset \Sigma_E^3$).
• A 1D closed loop of wire (codimension $3 - 1 = 2$) cannot trap a point in $\mathbb{R}^3$! Even if you fill a room with a dense forest of 1D wires of positive total measure, a trajectory can always thread its way around the wires without touching a single one. That is $n = 3$ ($\mathbb{T}^3 \subset \Sigma_E^5$)!

How High-Dimensional Phase Spaces Appear in midi-sink

1. $N$-Vortex Kirchhoff Chord Braiding (SUMI_DEFORM_BRAID, PROJECT_SPEC.md §10.8.2): Recall from Module 2 (§2.1) that in libsumi’s free chord-braiding mode (braid_mode = SUMI_BRAID_FREE, the D key; sumi_braid_step takes $n \le 16$ vortices, fifteen voices in practice), $N$ held MIDI notes spawn $N$ interacting point vortices governed by Kirchhoff’s Hamiltonian $H_{\text{Kirchhoff}}(x_1, \dots, x_N)$ on a $2N$-dimensional phase space ($n = N$ degrees of freedom). Because $H_{\text{Kirchhoff}}$ is translation- and rotation-invariant in the plane, Noether’s theorem provides 3 independent integrals ($H$, angular impulse $M = \sum_k \Gamma_k \|x_k\|^2$, and two commuting combinations of linear impulse $P_x, P_y$), reducing the system by 2 degrees of freedom to $n_{\text{eff}} = N - 2$:
   • 1, 2, or 3 Held Notes ($N \le 3 \implies n_{\text{eff}} \le 1$): Complete Liouville integrability! Every 3-vortex triad moves on invariant tori and never exhibits chaos.
   • 4 Held Notes ($N = 4 \implies n_{\text{eff}} = 2$): Chaotic layers appear, but surviving 2-tori have codimension $1$ inside the reduced 3D energy-impulse manifold, trapping vortex excursions.
   • 5 to 15 Held Notes ($N \ge 5 \implies n_{\text{eff}} = N - 2 \ge 3$): Invariant tori have codimension $n_{\text{eff}} - 1 \ge 2$, opening the Arnold Web across vortex configuration space!
2. Coupled 16-Mode Resonator Lattice (lattice_kick & lattice_rotate_simd, voxo/src/suzu.h): In Suzu, each voice’s modal resonator bank couples up to $n = 16$ second-order cells ($\dim M = 32$) via the shared-potential Laplacian kick $y_k \mathrel{+}= (\kappa \varepsilon_0^2 / \varepsilon_k)(L x)_k$ combined with the nonlinear restoring shear $s(y_k / A_k)$. For $n \ge 3$ active modes and $\kappa > 0$, invariant tori have codimension $n - 1 \ge 2$, allowing slow resonant action transfer across the partial bank while the joint symplectic integrator holds the bank’s energy — the gate: the bell’s eight modes with every decay zero, ten minutes, $-0.0026$ dB on the energy ledger, the partials within 0.08 cent at any admitted coupling. (Machine precision, $2 \times 10^{-16}$ in the $Q$-norm, is the Dirac network’s claim in Module 2, not the lattice’s.)

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6.2 The Arnold Web ($\sum_{i=1}^n k_i \omega_i(I) = 0$), Whiskered Tori & Transition Chains

If trajectories for $n \ge 3$ can bypass the surviving KAM tori, where in action space $(I_1, \dots, I_n)$ do they travel? They travel along the resonant channels of the Arnold Web!

Geometry of the Arnold Web in Action Space $(I_1, I_2, I_3)$

Consider a 3-DoF near-integrable Hamiltonian $H(I, \theta) = H_0(I_1, I_2, I_3) + \epsilon H_1(I, \theta)$ with unperturbed frequency map:

$\omega(I) = \nabla H_0(I) = \big(\omega_1(I),\, \omega_2(I),\, \omega_3(I)\big)$

For every non-zero integer vector $k = (k_1, k_2, k_3) \in \mathbb{Z}^3 \setminus \{0\}$ (with $\gcd(|k_1|, |k_2|, |k_3|) = 1$), the condition of rational commensurability defines a 2-dimensional resonance surface $\mathcal{R}_k \subset \mathbb{R}^3$:

$\mathcal{R}_k := \Big\{ I = (I_1, I_2, I_3) \in \mathbb{R}^3 : k \cdot \omega(I) = k_1 \omega_1(I) + k_2 \omega_2(I) + k_3 \omega_3(I) = 0 \Big\}$

To make this concrete, take the canonical strictly convex unperturbed Hamiltonian:

$H_0(I_1, I_2, I_3) = \frac{1}{2}\big(I_1^2 + I_2^2 + I_3^2\big) \quad \Longrightarrow \quad \omega(I) = (I_1, I_2, I_3)$

Here:

• The unperturbed isoenergetic surface $H_0(I) = E$ in 3D action space is the sphere $I_1^2 + I_2^2 + I_3^2 = 2E$ of radius $\sqrt{2E}$.
• Each resonance surface $\mathcal{R}_k$ is the plane $k_1 I_1 + k_2 I_2 + k_3 I_3 = 0$ normal to $k$, which slices the energy sphere $H_0(I) = E$ along a great circle $\mathcal{C}_k = \mathcal{R}_k \cap S^2_{\sqrt{2E}}$!
• Under perturbation $\epsilon H_1(I, \theta) = \epsilon \sum_{k} V_k \cos(k \cdot \theta)$, each great circle $\mathcal{C}_k$ thickens into a resonant strip (channel) of half-width: $\Delta I_k = 2\sqrt{\frac{\epsilon |V_k|}{\|k\|_2}} \sim \sqrt{\epsilon}\,e^{-\rho \|k\|_1 / 2}$
• Because all great circles on a sphere intersect one another, the union of all resonant strips $\mathcal{W}_\epsilon := \bigcup_{k \neq 0} \mathcal{C}_k$ forms a single connected, dense spiderweb covering the energy sphere — the Arnold Web!

Look closely at the two distinct directions of motion inside a resonance channel $\mathcal{R}_k$:

1. Fast Transverse Libration ($\parallel k$, across the channel width $\mathcal{O}(\sqrt{\epsilon})$): If only a single resonant harmonic $\cos(k \cdot \theta)$ were present, then $\dot{I} = \epsilon k V_k \sin(k \cdot \theta)$ would point purely along the integer vector $k$! On the energy sphere $H_0(I) = E$, the vector $k$ is perpendicular to the resonance plane $\mathcal{R}_k$ ($\nabla(k \cdot I) = k \perp \mathcal{R}_k$) and tangent to $H_0(I) = E$ precisely on $\mathcal{R}_k$ (since $k \cdot \nabla H_0(I) = k \cdot \omega(I) = 0$ on $\mathcal{R}_k$!). Thus, the single-harmonic pendulum motion oscillates back and forth across the narrow width $\mathcal{O}(\sqrt{\epsilon})$ of the channel and can never move along the channel!
2. Slow Longitudinal Arnold Drift ($\perp k$ and $\perp \omega(I)$, along the channel length $\mathcal{O}(1)$): Because $H_0(I)$ is strictly convex ($k^T D^2 H_0(I) k = \|k\|^2 > 0$), moving in the direction of $k$ immediately pushes $k \cdot \omega(I)$ away from zero and detunes the resonance. How, then, does a trajectory travel $\mathcal{O}(1)$ distances along the web? It drifts along the tangent vector of the resonance curve $\mathcal{C}_k = \mathcal{R}_k \cap \Sigma_E$: $\boxed{\; \mathbf{t}_k(I) := k \times \omega(I) \;}$ Notice that $\mathbf{t}_k(I) = k \times \omega(I)$ is simultaneously orthogonal to $\omega(I) = \nabla H_0(I)$ (so it stays on the constant-energy sphere $H_0(I) = E$!) and orthogonal to $k$ (so it stays locked on the resonance plane $k_1 \omega_1 + k_2 \omega_2 + k_3 \omega_3 = 0$!) until it reaches a resonance junction (hub) $\mathcal{R}_{k^{(1)}} \cap \mathcal{R}_{k^{(2)}}$, where $\omega(I)$ is parallel to $k^{(1)} \times k^{(2)}$ and the trajectory can switch onto a new resonance line!

Arnold’s 1964 Theorem: Whiskered Tori & Transition Chains

How does the slow longitudinal drift along $\mathcal{C}_k$ actually happen mechanically? Through whiskered tori and Arnold transition chains!

Consider Arnold’s celebrated 1964 model Hamiltonian (written in autonomous 3-DoF form with $\dot{\theta}_3 = 1$):

$H(I_1, I_2, I_3, \theta_1, \theta_2, \theta_3) = \frac{1}{2}I_1^2 + \frac{1}{2}I_2^2 + I_3 + \epsilon\big(\cos\theta_1 - 1\big)\Big(1 + \mu\big(\sin\theta_2 + \cos\theta_3\big)\Big)$

Let us trace Arnold’s mechanism in three steps:

1. Unsplit Whiskered Tori ($\epsilon > 0, \mu = 0$): When $\mu = 0$, the system decouples into a pendulum $H_{\text{pend}}(I_1, \theta_1) = \frac{1}{2}I_1^2 + \epsilon(\cos\theta_1 - 1)$ along the $(1, 0, 0)$ resonance $\mathcal{R}_{(1,0,0)}$ ($I_1 = 0$) and a 2-torus rotor in $(\theta_2, \theta_3)$ where $I_2 = \omega_2$ is strictly constant! For every fixed value of the longitudinal action $I_2 = c$, the hyperbolic saddle point $(I_1 = 0, \theta_1 = 0)$ of the pendulum sweeps out an unstable invariant 2-torus $\mathcal{T}_c = \{I_1 = 0, \theta_1 = 0, I_2 = c, (\theta_2, \theta_3) \in \mathbb{T}^2\}$. Because $\mathcal{T}_c$ is hyperbolic in the transverse $(I_1, \theta_1)$ plane and neutral along $\mathbb{T}^2$, it possesses a 3-dimensional stable manifold $W^s(\mathcal{T}_c)$ (trajectories approaching $\mathcal{T}_c$ as $t \to +\infty$) and a 3-dimensional unstable manifold $W^u(\mathcal{T}_c)$ (trajectories leaving $\mathcal{T}_c$ as $t \to -\infty$). Arnold named $\mathcal{T}_c$ a whiskered torus, and $W^s(\mathcal{T}_c), W^u(\mathcal{T}_c)$ its incoming and outgoing whiskers. When $\mu = 0$, $\dot{I}_2 = 0$ everywhere, so the outgoing whisker $W^u(\mathcal{T}_c)$ coincides identically with the incoming whisker $W^s(\mathcal{T}_c)$ of the same torus $c$, and no drift in $I_2$ can occur.
2. Whisker Splitting & Heteroclinic Transition Chains ($0 < \mu \ll \epsilon \ll 1$): When $\mu > 0$, the coupling term $\epsilon\mu(\cos\theta_1 - 1)(\sin\theta_2 + \cos\theta_3)$ leaves every whiskered torus $\mathcal{T}_c$ completely untouched (because $\cos\theta_1 - 1$ and its gradient vanish to second order at $\theta_1 = 0$!). However, along the whiskers $W^u(\mathcal{T}_c)$ and $W^s(\mathcal{T}_{c'})$ away from $\theta_1 = 0$, $\cos\theta_1 - 1 \neq 0$, so the whiskers split apart! Using the multi-dimensional Melnikov potential, Arnold proved that the outgoing whisker $W^u(\mathcal{T}_{c_m})$ of torus $c_m$ intersects transversely the incoming whisker $W^s(\mathcal{T}_{c_{m+1}})$ of a neighbouring torus $c_{m+1}$ whenever $|c_{m+1} - c_m|$ is smaller than the Melnikov splitting amplitude: $W^u(\mathcal{T}_{c_m}) \pitchfork W^s(\mathcal{T}_{c_{m+1}}), \qquad |c_{m+1} - c_m| \le \Delta I_{\text{split}} \sim \mu\,\exp\!\left(-\frac{\pi\,\omega_2}{2\sqrt{\epsilon}}\right)$
3. Shadowing the Chain ($\mathcal{O}(1)$ Drift in $I_2$): A sequence of whiskered tori $\mathcal{T}_{c_0}, \mathcal{T}_{c_1}, \dots, \mathcal{T}_{c_M}$ connected by transverse heteroclinic whisker intersections $W^u(\mathcal{T}_{c_m}) \pitchfork W^s(\mathcal{T}_{c_{m+1}})$ is called an Arnold Transition Chain. By Smale’s $\lambda$-Lemma (Inclination Lemma), the unstable manifold $W^u(\mathcal{T}_{c_0})$ accumulates on $W^u(\mathcal{T}_{c_1})$, which accumulates on $W^u(\mathcal{T}_{c_2})$, and so on all the way to $\mathcal{T}_{c_M}$! Thus there exists an open neighborhood of initial conditions near $\mathcal{T}_{c_0}$ that shadows the entire transition chain, drifting from $I_2(0) \approx A$ to $I_2(T) \approx B$ ($|B - A| = \mathcal{O}(1)$) for arbitrarily small $0 < \mu \ll \epsilon \ll 1$!

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6.3 Nekhoroshev’s Exponential Stability Theorem (1977) & Fast Lyapunov Indicators (FLI)

Look at the Melnikov whisker-splitting formula above: the step size $\Delta I_{\text{split}}$ between consecutive tori in Arnold’s transition chain contains the factor $\exp(-\pi\omega_2 / 2\sqrt{\epsilon})$, which is exponentially small in $1/\sqrt{\epsilon}$! Does this mean all Arnold diffusion in analytic quasi-convex systems is exponentially slow?

In 1977, Arnold’s student Nikolai Nekhoroshev proved that the answer is yes: while KAM theory gives perpetual stability ($\forall t \in \mathbb{R}$) for a Cantor set of initial conditions (Diophantine tori), Nekhoroshev’s Theorem gives exponentially long stability ($|t| \le T_0 \exp((\epsilon_0/\epsilon)^a)$) for ALL initial conditions in phase space!

Theorem 6.1 (Nekhoroshev’s Exponential Stability Theorem, 1977; Lochak–Pöschel 1992–1993): Let $H(I, \theta) = H_0(I) + \epsilon H_1(I, \theta)$ be real-analytic on $D \times \mathbb{T}^n$, and suppose $H_0(I)$ is steep (in particular, quasi-convex: on every isoenergetic tangent plane $v \cdot \nabla H_0(I) = 0$, the Hessian quadratic form $v^T D^2 H_0(I) v \ge m \|v\|^2 > 0$ is positive-definite). Then there exist positive constants $\epsilon_0, I_0, T_0 > 0$ such that for all $0 < \epsilon < \epsilon_0$ and every initial condition $(I(0), \theta(0)) \in D \times \mathbb{T}^n$: $\boxed{\; \|I(t) - I(0)\| \le I_0\,\epsilon^b \qquad \text{for all } |t| \le T_{\text{Nekh}}(\epsilon) := T_0\,\exp\!\left( \left(\frac{\epsilon_0}{\epsilon}\right)^a \right) \;}$ with optimal stability exponents (Lochak 1992, Pöschel 1993): $a = \frac{1}{2n}, \qquad b = \frac{1}{2n}$

Why Nekhoroshev Stability Requires Both Analyticity and Convexity (Geometry of the Proof)

Why does Nekhoroshev’s bound hold even right inside the resonant channels $\mathcal{R}_k$ of the Arnold web?

1. Analytic Ultraviolet Cutoff & Exponential Remainder: Because $H_1(I, \theta) = \sum_{k \in \mathbb{Z}^n} \hat{H}_{1,k}(I) e^{i k \cdot \theta}$ is real-analytic on a strip of width $\rho$, its Fourier harmonics decay exponentially: $|\hat{H}_{1,k}| \le C e^{-\rho \|k\|_1}$. If we truncate the Fourier series at an ultraviolet cutoff $\|k\|_1 \le K_{\text{UV}} \sim \epsilon^{-a}$ ($a = 1/2n$), the truncated high-frequency tail is already exponentially small: $\mathcal{O}\big(\exp(-(\epsilon_0/\epsilon)^a)\big)$.
2. Covering Action Space by Resonant Blocks (Geography of Resonances): For the finitely many low-order harmonics $\|k\|_1 \le K_{\text{UV}}$, we partition action space into non-resonant regions, single-resonance channels $\mathcal{R}_k$, double-resonance junctions $\mathcal{R}_{k^{(1)}} \cap \mathcal{R}_{k^{(2)}}$, up to $(n-1)$-fold resonances. Inside each block of multiplicity $r$, normal-form averaging eliminates all harmonics outside the $r$-dimensional resonant sublattice $\Lambda$, leaving an integrable resonant normal form plus an exponentially small remainder of size $\sim \exp(-(\epsilon_0/\epsilon)^a)$!
3. Why Quasi-Convexity Blocks Fast Drift Along $\Lambda$: Inside the resonant normal form of multiplicity $r$, the fast action drift $\dot{I}$ lies strictly in the $r$-dimensional subspace $\operatorname{span}(\Lambda)$ while conserving the unperturbed energy $H_0(I)$ up to $\mathcal{O}(\epsilon)$. Because $H_0(I)$ is strictly convex, the intersection of the affine plane $I(0) + \operatorname{span}(\Lambda)$ with the convex energy shell $\big|H_0(I) - H_0(I(0))\big| \le C\epsilon$ is a small closed ellipsoid of diameter $\mathcal{O}(\epsilon^b)$! Therefore, a trajectory can escape from its $\mathcal{O}(\epsilon^b)$ resonant trap only by climbing out under the exponentially small normal-form remainder $\sim \exp(-(\epsilon_0/\epsilon)^a)$, which requires at least $T_{\text{Nekh}}(\epsilon) \sim \exp((\epsilon_0/\epsilon)^a)$ time!

Counterexample when Convexity Fails: If $H_0(I) = \frac{1}{2}(I_1^2 - I_2^2 + I_3^2)$ has an indefinite Hessian (not quasi-convex!), then along the null cone $I_1 = \pm I_2$, the resonant vector $k = (1, 1, 0)$ lies inside the energy surface ($k^T D^2 H_0 k = 0$), allowing fast $\mathcal{O}(\epsilon)$ drift without exponential suppression!

Illuminating the Arnold Web on the GPU: The Fast Lyapunov Indicator (FLI)

Because Arnold diffusion takes $\sim \exp((\epsilon_0/\epsilon)^a)$ steps to move $(I_1, I_2, I_3)$ macroscopically, how can a 60 Hz WebGL2 shader visualize the 3D Arnold web in real time? By computing the Fast Lyapunov Indicator (FLI) introduced by Claude Froeschlé, Elena Lega, and Rudolphe Gonczi (1997, Science 2000)!

At each point $I_0 = (I_1, I_2, I_3)$ in 3D action space (with fixed initial angles $\theta_0$), we evolve the symplectic map $z_{m+1} = M(z_m)$ together with a unit tangent vector $v_0 \in T_{z_0} M$ ($\|v_0\| = 1$) under the exact linearized Jacobian map $v_{m+1} = DM(z_m)\,v_m$ for a short horizon of $N$ steps ($N \approx 20\text{–}50$), and record the supremum of the logarithmic stretch:

$\boxed{\; \operatorname{FLI}(I_0; N) := \sup_{1 \le m \le N} \ln\,\|v_m\| \;}$

Look at how $\operatorname{FLI}(I_0; N)$ acts as an X-ray of the Arnold Web:

• Diophantine KAM Tori: Tangent vectors shear only linearly in time ($\|v_m\| \sim m\,\|D^2 H_0\|$), so $\operatorname{FLI} \approx \ln N$.
• Centers of Resonant Islands: Librating orbits have periodic tangent matrices, so $\operatorname{FLI}$ dips below the KAM background!
• Whiskered Tori & Separatrices of the Arnold Web: Along the hyperbolic borders of every resonant channel $k_1 \omega_1 + k_2 \omega_2 + k_3 \omega_3 = 0$ and especially at resonance junctions $\mathcal{R}_{k^{(1)}} \cap \mathcal{R}_{k^{(2)}}$, $\|v_m\| \sim e^{\lambda m}$ grows exponentially, making the resonant filaments glow in 3D action space!

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Lab Instructions (module_06_lab.html)

Open the module’s lab (a standalone page — a teaching re-implementation, never the engine) in a WebGL2 + WebAudio browser. This capstone lab combines a 3D Raymarching FLI Shader over Action Space $(I_1, I_2, I_3)$ with a High-Speed ($10^5$ steps/frame) Arnold-Web Drift Triadic Synthesizer.

Part 1: Shader Lab — 3D Volumetric Raymarching of the Arnold Web via Fast Lyapunov Indicators (FLI)

The top WebGL2 canvas raymarches a 3D volume of action space $(I_1, I_2, I_3) \in [0.15, 1.85]^3$ (drag with your mouse to orbit the 3D camera in azimuth and elevation). Along each camera ray, the fragment shader evaluates the local Fast Lyapunov Indicator (FLI) and multi-order resonance proximity for the coupled 3-DoF Froeschlé / Arnold map:

$$\begin{aligned} I_i^{(m+1)} &= I_i^{(m)} + \epsilon\,\frac{\partial V(\theta^{(m)})}{\partial \theta_i}, \qquad \theta_i^{(m+1)} = \theta_i^{(m)} + 2\pi I_i^{(m+1)} \qquad (i = 1, 2, 3) \\ V(\theta_1, \theta_2, \theta_3) &= \cos\theta_1 + \cos\theta_2 + \cos\theta_3 + \mu\big(\cos(\theta_1 - \theta_2) + \cos(\theta_2 - \theta_3) + \cos(\theta_1 + \theta_2 - 2\theta_3)\big) \end{aligned}$$

1. Explore the 3D Arnold Web & Energy Shell $H_0(I) = E$:
   • Start in 3D Volumetric Raymarched Arnold Web (FLI Filaments) mode and drag the canvas to rotate the 3D action cube $(I_1, I_2, I_3)$.
   • Observe the glowing gold and cyan resonant planes $\sum_{i=1}^3 k_i I_i = 0$ intersecting the spherical isoenergetic shell $I_1^2 + I_2^2 + I_3^2 = 2E$ along great-circle channels, and notice the bright Resonance Hubs (Junctions) where three or more resonant sheets cross!
   • Switch to Spherical Energy Surface Slice ($H_0(I) = E$) + FLI Web or 2D Cross-Sectional FLI Tomography ($I_3 = \text{slice}$) to inspect individual resonant channels ($1:1:0$, $1:2:-1$, $2:-1:-1$, $3:-2:0$, etc.) and their double-walled whiskered separatrix borders!
   • Sweep the Perturbation $\epsilon$ slider from 0.005 (Nekhoroshev regime: razor-thin, isolated filaments) to 0.080 (Chirikov overlap regime: thickened, merging chaotic channels).

Part 2: Sound Lab — Hearing Arnold Diffusion Along the Web ($10^5$ Steps per Audio Frame)

Because true Arnold diffusion along a transition chain is exponentially slow ($\sim \exp(-(\epsilon_0/\epsilon)^{1/6})$), running 1 step per audio sample would take hours to move an inch along the web. Instead, Part 2 runs a High-Speed Symplectic Burst Engine executing up to $100,000$ ($10^5$) map steps per frame along the 3D Arnold web while mapping the instantaneous 3D action vector $(I_1(t), I_2(t), I_3(t))$ to a Three-Voice Musical Triad:

$f_1(t) = f_{\text{base}}\,I_1(t), \qquad f_2(t) = f_{\text{base}}\,I_2(t), \qquad f_3(t) = f_{\text{base}}\,I_3(t)$

1. Hear the Triad Migrate Along a Single Resonance Line & Pivot at a Junction:
   • Click Start 3-Voice Arnold Triad Synth.
   • Watch the glowing trajectory marker on the 2D Projected Arnold Web Map (and in the 3D Raymarcher) while listening to the three-voice chord $(f_1, f_2, f_3)$:
   • While the trajectory travels along a single resonance channel $\mathcal{R}_k$ (for instance, $\mathbf{k} = (2, -3, 1)$, so $2 f_1(t) - 3 f_2(t) + f_3(t) = 0$), all three pitches $f_1(t), f_2(t), f_3(t)$ glide slowly and continuously across the spectrum, yet they maintain the exact linear harmonic interval constraint $k_1 f_1 + k_2 f_2 + k_3 f_3 = 0$!
   • The moment the drifting action vector reaches a Resonance Junction (Hub) $\mathcal{R}_{k^{(1)}} \cap \mathcal{R}_{k^{(2)}}$ (such as $(I_1 : I_2 : I_3) = (3 : 4 : 6)$ or $(2 : 3 : 4)$), heteroclinic whisker scattering pivots the trajectory onto a new intersecting resonance channel $\mathcal{R}_{k^{(2)}}$, changing the interval rule of the gliding triad!
   • Use the Burst Acceleration ($10^2 \dots 10^5$ steps/frame) slider and the Nekhoroshev $\epsilon$ slider to verify Nekhoroshev’s exponential scaling law $T_{\text{Nekh}} \sim \exp((\epsilon_0/\epsilon)^{1/6})$ live in the telemetry bar!

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Problem Set

Problem 6.1: Codimension of KAM Tori & Longitudinal Drift Vector on the Arnold Web

1. Codimension & Separation in $n$ Degrees of Freedom: For an autonomous $n$-DoF Hamiltonian system on $(M^{2n}, \omega, H)$, prove that if $H(q, p) = E$ is a regular energy value, then $\dim \Sigma_E = 2n - 1$, any Lagrangian KAM torus $\mathcal{T}_\omega \subset \Sigma_E$ has codimension $n - 1$ inside $\Sigma_E$, and $n = 2$ is the unique integer $n \ge 2$ for which $\operatorname{codim}_{\Sigma_E}(\mathcal{T}_\omega) = 1$.

   • For libsumi’s $N$-vortex Kirchhoff Hamiltonian $H_{\text{Kirchhoff}}(x_1, \dots, x_N)$ (PROJECT_SPEC.md §10.8.2), where the conserved linear and angular impulses reduce the effective degrees of freedom to $n_{\text{eff}} = N - 2$, determine the minimum number of vortices $N$ required for invariant tori to have codimension $\ge 2$ inside the reduced energy-impulse manifold.

2. Isoenergetic Tangent Direction of a Resonance Channel ($n = 3$): Let $H_0(I_1, I_2, I_3)$ be a smooth integrable Hamiltonian on $\mathbb{R}^3$ with frequency vector $\omega(I) = \nabla H_0(I)$ and Hessian matrix $Q(I) = D^2 H_0(I)$. Suppose $I^*$ lies on a single resonance surface $\mathcal{R}_k = \{I : k \cdot \omega(I) = 0\}$ for $k \in \mathbb{Z}^3 \setminus \{0\}$.

   • Show that the normal vector to the energy surface $\Sigma_E = \{H_0(I) = E\}$ at $I^*$ is $\omega(I^*)$, and the normal vector to the resonance surface $\mathcal{R}_k$ at $I^*$ is $Q(I^*)\,k$.
   • Conclude that the 1-dimensional Arnold drift channel $\mathcal{C}_k = \mathcal{R}_k \cap \Sigma_E$ has tangent vector: $\mathbf{t}_k(I^*) = \omega(I^*) \times \big(Q(I^*)\,k\big)$
   • Show that if $H_0(I)$ is strictly convex ($Q(I^*) \succ 0$), then the fast single-harmonic resonant oscillation vector $k$ is never tangent to $\mathcal{R}_k$ ($k \cdot (Q(I^*) k) > 0$), proving that fast $\mathcal{O}(\sqrt{\epsilon})$ motion can only oscillate across the width of $\mathcal{R}_k$, while any $\mathcal{O}(1)$ drift along $\mathbf{t}_k(I^*)$ requires multi-harmonic coupling!

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Problem 6.2: Whiskered Tori & Melnikov Splitting in Arnold’s 1964 Model

Consider Arnold’s 1964 Hamiltonian on $(I_1, I_2, \theta_1, \theta_2, t) \in \mathbb{R}^2 \times \mathbb{T}^2 \times S^1$:

$H(I_1, I_2, \theta_1, \theta_2, t) = \frac{1}{2}\big(I_1^2 + I_2^2\big) + \epsilon\big(\cos\theta_1 - 1\big)\Big(1 + \mu\big(\sin\theta_2 + \cos t\big)\Big)$

1. Invariance of the Whiskered Tori for All $\mu$: Compute Hamilton’s equations for $\dot{I}_1, \dot{\theta}_1, \dot{I}_2, \dot{\theta}_2$, and show that for every $c \in \mathbb{R}$ and every $\epsilon, \mu \in \mathbb{R}$, the 2-torus $\mathcal{T}_c := \big\{ (I_1, I_2, \theta_1, \theta_2, t) : I_1 = 0,\; \theta_1 = 0 \pmod{2\pi},\; I_2 = c \big\}$ is an exact invariant torus of the full flow with $\dot{\theta}_2 = c, \dot{t} = 1$! Why does the perturbation $\epsilon\mu(\cos\theta_1 - 1)(\sin\theta_2 + \cos t)$ vanish along with all its first derivatives on $\mathcal{T}_c$?

2. Unperturbed Whisker Separatrix & First-Order Action Jump $\Delta I_2$: When $\mu = 0$ and $\epsilon > 0$, the unstable and stable whiskers of $\mathcal{T}_c$ coincide along the pendulum separatrix: $\theta_1^0(t - t_0) = 4\arctan\!\big(e^{\sqrt{\epsilon}(t - t_0)}\big), \qquad I_1^0(t - t_0) = \frac{2\sqrt{\epsilon}}{\cosh\!\big(\sqrt{\epsilon}(t - t_0)\big)}, \qquad \theta_2^0(t) = \theta_{2,0} + c\,t$ Using $\dot{I}_2 = -\frac{\partial H}{\partial \theta_2} = -\epsilon\mu\big(\cos\theta_1 - 1\big)\cos\theta_2$ and $\cos\theta_1^0(\tau) - 1 = -\frac{2}{\cosh^2(\sqrt{\epsilon}\tau)}$, integrate $\dot{I}_2$ along the unperturbed separatrix from $t = -\infty$ to $t = +\infty$ to show that to first order in $\mu$, a trajectory traversing the whisker experiences a net longitudinal action jump: $\Delta I_2(\theta_{2,0}) = \int_{-\infty}^{\infty} \frac{2\epsilon\mu\,\cos(\theta_{2,0} + c t)}{\cosh^2(\sqrt{\epsilon} t)}\,dt = \frac{2\pi \mu c}{\sinh\!\left(\dfrac{\pi c}{2\sqrt{\epsilon}}\right)}\cos(\theta_{2,0})$ Verify that for $0 < \sqrt{\epsilon} \ll c$, the maximum action jump per whisker transit scales exponentially as: $\max_{\theta_{2,0}} |\Delta I_2| \approx 4\pi \mu c\,\exp\!\left(-\frac{\pi c}{2\sqrt{\epsilon}}\right)$

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Problem 6.3: Nekhoroshev’s Stability Exponents & Variational Equations for the Fast Lyapunov Indicator (FLI)

1. Optimal Ultraviolet Cutoff in Nekhoroshev Theory: Suppose an analytic perturbation $H_1(I, \theta) = \sum_{k \in \mathbb{Z}^n} \hat{H}_{1,k}(I) e^{i k \cdot \theta}$ has Fourier coefficients decaying as $|\hat{H}_{1,k}| \le C e^{-\rho \|k\|_1}$. By Dirichlet’s Box Principle (or Lochak’s periodic-orbit approximation), within a ball of radius $R \sim \epsilon^b$ around any point $I(0) \in \mathbb{R}^n$, rational frequency vectors of period $T$ occur with spacing $\sim T^{-1/(n-1)}$. Balancing the resonant normal-form domain radius $\epsilon^b \sim (\epsilon K_{\text{UV}})^{-1}$ against the Fourier truncation remainder $\epsilon e^{-\rho K_{\text{UV}}}$ leads to $K_{\text{UV}} \sim \epsilon^{-a}$ with $a = b = \frac{1}{2n}$.

   • For $n = 3$ degrees of freedom ($a = 1/6$), if reducing $\epsilon$ by a factor of $64$ ($2^6$) increases $(\epsilon_0/\epsilon)^{1/6}$ by a factor of $2$, by what factor does Nekhoroshev’s stability time $T_{\text{Nekh}}(\epsilon) = T_0 \exp((\epsilon_0/\epsilon)^{1/6})$ increase when $(\epsilon_0/\epsilon)^{1/6}$ goes from $5$ to $10$?

2. Exact Variational Map for the 3-DoF Fast Lyapunov Indicator (FLI): For the 3-DoF symplectic map $I_i' = I_i + \epsilon \frac{\partial V(\theta)}{\partial \theta_i}$, $\theta_i' = \theta_i + I_i'$ ($i = 1, 2, 3$), write the $6 \times 6$ tangent Jacobian matrix $DM(\theta, I)$ acting on a tangent vector $v = (\delta\theta, \delta I)^T \in \mathbb{R}^6$: $\begin{pmatrix} \delta\theta' \\ \delta I' \end{pmatrix} = \begin{pmatrix} I_3 + \epsilon\,D^2 V(\theta) & I_3 \\ \epsilon\,D^2 V(\theta) & I_3 \end{pmatrix} \begin{pmatrix} \delta\theta \\ \delta I \end{pmatrix}$ Prove that $(DM)^T \Omega_6 (DM) = \Omega_6$ whenever $D^2 V(\theta)$ is symmetric (so the tangent map is itself symplectic with $\det DM = 1$!), and explain why $\operatorname{FLI}(N) = \sup_{1 \le m \le N} \ln\|v_m\|$ grows as $\ln N$ on regular KAM tori ($D^2 V \to 0$ in action-angle coordinates) but grows linearly $\propto \lambda N$ along hyperbolic whiskered tori.