The Operators

midi-sink never solves a fluid on a grid. Grid advection diffuses, and diffusion turns crisp suminagashi rings into mud within seconds. Instead the ink is an area-preserving, continuous deformation of the sheet, evaluated by inverse lookup: for every output pixel $P$ the engine asks where that water came from, $P_{src} = f^{-1}(P)$, and reads the ink there. Two half-float textures hold the accumulated pre-image coordinates $(u, v)$ plus the ink phase and a per-drop hue selector; each gesture is one more pass that reads one texture and writes the other. This is Aubrey Jaffer’s mathematical marbling (Lu, Jaffer, Jin, Zhao & Mao, 2012), extended with two of his own later closed-form fields and a few of ours — see the citations.

Every operator page follows the same shape:

• The formula, exactly as the shader evaluates it, in aspect-corrected normalized coordinates (non-square canvases are corrected before the mathematics, or rings become ellipses; distances are in canvas heights).
• Invariants — why it preserves area, its class — exact at any magnitude, or sub-stepped to stay away from a fold — and what composes with what. Every operator declares its class and passes the same four-part conservation gate (the desktop bench’s --soak: area, mass, the pre-image’s wander, fabrication as a rate); the composite is gated per backend.
• Ownership — which MIDI dimension or gesture drives it, and the one-consumer rule: a MIDI dimension has exactly one consumer at a time (bend_mode, press_mode, slide_mode choose it) so two operators never fight over one stream.
• A live demo — the release wasm, the same engine bytes every platform ships, embedded through the scene API. The sliders are the formula’s symbols. There is no second implementation anywhere on this site.

Two conventions run through every demo. The scenes are paced — the ⏱ slider is the number of frames between steps, and the operators that compose exactly are applied as a run of small passes, so you watch the transformation happen rather than see only its result (set it to 0 for the result alone). And every scene works on two ring clusters, top-left and bottom-right, so one operator can show two orientations or two signs at once: a horizontal and a vertical tine, two counter-rotating vortices, perpendicular fold axes, a drop beside a pressure feed. A third: every scene carries the medium toggle (M) — flip it to 1 and the same script runs from its first strike as the Anod discharge: every strike in a scene is the medium’s, so under Anod the ring clusters are struck as sparks on their charge, as a tap is in the app, and the deformation is read as strain instead of bands. The toggle is permanent for the run, never a switch at the end.

The drop, the first operator: six expansions on each cluster, one per step.open full size ↗

The nine of 1.0

Operator	Kind	Class	Driven by
Drop — and the pressure feed	expansion	exact	strike, pressure feed, breath
Tine	shear	exact (Jacobian 1)	per-note bend (glide), master bend
Vortex — exponential, Rankine, torsion	rotation	exact at any angle	mod wheel / Airwave CC (exponential), two-finger twist (Rankine)
Wake — inviscid doublet, viscous Stokeslet	potential-flow doublet · 2-D Stokes impulse	sub-stepped ≤ a/4	every stylus stroke; a wind legato (the sounding drop dragged to the next note)
Pinch — saddle, crossed tines	Hamiltonian saddle	exact (det = 1)	CC 74 with Slide → Pinch, pen azimuth, two-finger pinch
Ripple	sine shear	exact (Jacobian 1)	per-note bend with Per-note bend → Ripple, CC 102/103
Swirl — Lamb–Oseen	viscous rotation	exact at any angle	poly pressure 0xA0, channel pressure with Channel pressure → Swirl
Scroll	translation with ingress	exact	the piano-roll layouts’ tempo

The five of the electric medium

The second medium, Anod, reads the same field as a discharge and brought five operators with it — every one of them a shear or a rotation composed from the first nine’s lemmas, and every one of them playable under suminagashi too:

Operator	Kind	Class	Driven by
Wave torsion	oscillating rotation by θ(r)	exact at any amplitude	the vortex routes with the Torsion profile; the note-on sweep; Anod’s press
Chladni — an eddy in every key	exact disc rotations on the layout’s cells · the inverse Chladni’s summed flow	exact (discs) · first-order (inverse)	the stir and balance CCs 106/107; Anod’s per-note bend
Burst — the viscous multipole, m ≥ 2	time-integrated impulsive multipole	sub-stepped (first-order, under the wake’s criterion)	the gesture; the middle piece of the Anod strike
Spark — the composed strike	drop + burst + jagged shears	exact shears; sub-stepped by inheritance	the Anod strike on its charge; the slide (CC 74) its frequency
Chirikov — the standard map	sinusoidal kick + linear drift	exact; capped at K = 1.25 per step	the throw: CC 109, Anod’s mod wheel

The two media and their look — the washi and the glass, the three palettes of each and the palette model behind them — are the medium guide’s.

Two rules that hold everywhere

Continuous feeds are deltas. Pressure, breath, swirl and pinch arrive as absolute controller values, but the operators consume rates: a pass per frame whose magnitude is the smoothed value × Δt. Feeding an absolute value integrates into runaway strain; a delta relaxes when the hand relaxes.

Paper is screen-locked. The washi fibre and grain are sampled in screen space, never through the deformed field. Ink moves; the paper it sits on does not. If fibres ever drift with a tine, that invariant has been broken.