The lessons

The engine is a course in symplectic geometry that happens to draw and to sound. Every operator on the water is an area-preserving map with its class declared and gated; every voice in Suzu is a Hamiltonian system stepped by a symplectic integrator whose invariant is read live. These six modules teach the mathematics on that material: each claim about the engine was checked against the code and the suites as shipped (libsumi 1.12.0, Voxo 0.26.0 at the writing, 2026-10-25; reviewed before the fold on 2026-10-26), and every number quoted is a gate’s printout, never an estimate. They are graduate-level notes: exterior calculus and a first course in Hamiltonian mechanics are assumed.

Module	What it teaches	Where it lives in the engine
1 — Exterior calculus, the exactness rule and area-preserving phase space	forms and symplectomorphisms in 2D and 4D; why the field stores displacements; the exactness rule and its operator families; reading the Jacobian; the magic circle	the deformation chain, Sumi / Anod / Kessho, suzu::Cell
2 — Hamiltonian vector fields and symplectic integrators	Hamiltonian flows and their splittings; the port-Hamiltonian network; Euler, Störmer–Verlet and RK4; the shadow Hamiltonian at audio rate; what still breaks	the Chladni, braid and wake passes, suzu::Dirac, suzu::TodaRing, the membrane
3 — Darboux’s theorem, Moser’s trick and generating functions	Riemannian strain against symplectic flatness; Darboux coordinates on the drop; Moser’s homotopy; generating functions and the engine’s uses of them	Kessho’s two layers, the crystal drop, the Arnol’d kick, the sheared cell, the retune
4 — Integrable systems, action-angle coordinates and KAM theory	Liouville integrability; the perturbed torus; Poincaré sections and rotation numbers; islands and Greene’s threshold; the tongue on the water	the billiard, the Toda ring, the modal bank, the Arnol’d lock
5 — Resonance overlap, cantori and phase-space diffusion	Chirikov’s criterion; the breakup of the golden torus; cantori and turnstiles; the regimes of diffusion	the Chirikov operator, suzu::Rotor
6 — High-dimensional dynamics and Arnold diffusion	the dimensional divide; whiskered tori and the Arnold web; Nekhoroshev stability; the FLI	the free braid, the coupled modal lattice

The labs

Each module ends with a lab, a single HTML file beside it (module_0N_lab.html) that runs in any WebGL2 + WebAudio browser: a ping-pong displacement field in RGBA32F with the module’s operators, and a 48 kHz audio thread with the module’s integrators. They are teaching re-implementations, never the engine — the site carries no second implementation of any operator or voice, and the labs are the one exception, declared as such in each file’s header. Their arithmetic was checked in node against the engine’s for module 1 and they all render headless without error; their code was not gated against the engine. The engine’s own lab, the synth compiled to WebAssembly, is the Suzu lab, and the water itself is the marble app the operator pages embed.

Conventions

• Every §n is docs/PROJECT_SPEC.md — spec v6 since the fold of 2026-10-26: §10.7–§10.10 Kessho, the harmony operators and the strobe, §13.8–§13.14 the voices; KESSHO §n is the Kessho proposal, in git history (git show 21ff974:_work/specs/medium_specs_v3.md), its substance §10.7.
• DECISIONS_10 #n … DECISIONS_15 #n are Parts X–XV of docs/DECISIONS.md, the record of what shipped (the v3.0 arc’s roadmap is its Part 10 of docs/ROADMAP.md).
• Gate numbers are the synth suite’s (tests/voxo_suzu_tests.cpp); the canvas’s conservation gates are the bench’s --soak <operator>; the Kessho flatness gate is --kessho-flat (as measured, DECISIONS_10 #4).
• Where a lesson derives or interprets beyond what the record says (the wake’s fold, the gong’s Mathieu reading, the billiard’s collision map), it says so.