Wave torsion

A vortex rotates every point about its centre by an angle that depends only on the radius. The exponential and Rankine profiles make that angle fall off monotonically; the wave torsion makes it oscillate:

$$\theta(r) = A\,\sin(k\,r - \varphi)\,e^{-r/R}, \qquad \text{rotate } P \text{ about the centre by } -\theta(r).$$

Concentric rings turn alternately one way and the other — a crest at $k r - \varphi = \pi/2$, a trough half a wavelength further out — and the whole pattern fades with the e-folding length $R$. Because $r' = r$, the map is a rotation on every circle: exactly area-preserving, exact at any amplitude, no sub-stepping ever — the vortex lemma verbatim. Advance $\varphi$ over time and the rings travel outward at speed $\omega / k$: a shockfront, a diffraction pattern, the discharge ring of the Anod medium.

Static torsion: +A just off the top-left cluster, −A off the bottom-right, ramped in 24 passes (the composition invariant). k is the wavenumber (CC 104), φ the phase (CC 105), R the decay length. Set ↯ = 1 to strike a note at the centre instead and watch its outward sweep fade over about two seconds.open full size ↗

Invariants

• Class: exact. A rotation by $\theta(r)$ preserves $r$, so $\det J = 1$ everywhere and at any $A$; the $(+A, -A)$ pair inverts analytically. Measured on the desktop harness: two markers on a crest and a trough turn by $0.364$ and $-0.320$ rad against the analytic $0.376$ and $-0.332$, and return to within $3\cdot10^{-4}$ rad after the pair.
• Oscillatory shear resamples rougher than smooth shear. The pre-image of an exact torsion pair wanders 5.5 texels over a thousand passes where the smooth vortex profiles wander 1.5–2.7 — the medium’s, not the operator’s (the markers come home). Recorded in the conservation gate’s bar.
• The medium gains a little mass under it. At the default wavelength a 6000-pass stream grows ink mass by 2.3 %, all of it interior at the ring boundaries, an order of magnitude below edge-clamp duplication and the reason the gate’s fabrication criterion is a rate.

Ownership

• A third vortex profile, Torsion, reachable through every vortex route: the Marble-mode right-drag or two-finger twist, the mod wheel and the Airwave’s left hand with the profile set to Torsion, and the gesture call sumi_add_vortex.
• $k$ and $\varphi$ are flavour controls — the torsion’s wavenumber (rests mid-range, $2\pi\cdot4$ to $2\pi\cdot40$ per canvas height) and phase ($0$ to $2\pi$), unmapped in the core like the ripple’s wavelength; the desktop maps CC 104 and 105 to them. Under the Anod medium’s binding table the per-note bend can play the wavenumber (Per-note bend → Torsion wavelength), and poly pressure tunes it from its rest upward when nothing else owns it.
• The note-on sweep (Torsion sweep on note-on, off by default — under Anod the strike is the spark, and pressure spends torsion around the note): every strike also fires an outward wave of torsion around its drop — $\varphi$ advancing at $2\pi\cdot1.5$ rad/s, the amplitude decaying with $\tau = 0.6$ s, reach three strike radii — emitted as per-frame rotation increments, never a total, so two strikes in the same place add up and a frame the budget merges carries its increment over. The episode runs on its own clock: releasing the note does not cut it. It was the engine’s first time-driven episode; the burst’s age envelope and the spark’s decay reuse the pattern.

Lineage

Jaffer’s vortex lemma — a rotation by any function of the radius is exactly area-preserving — is the whole proof; the torsion only chooses an oscillating function. The look is the electric medium’s: concentric diffraction rings and travelling shockfronts, drawn with the same operator the sumi medium stirs its water with.