Fluid simulation · mimo-v2.6-pro

Let me build a stable fluids (Jos Stam) style simulation with Canvas 2D, using ImageData rendering at a bounded grid resolution (e.g., 128x85 scaled up via canvas drawImage from offscreen). That's reliable and fast.

Let me write it.

Grid: SIM_W = 128, SIM_H = 85 (aspect 1200x800 = 1.5). Use offscreen canvas at grid resolution, putImageData, then draw scaled to main canvas with imageSmoothingEnabled = true for smooth look.

Fields: u, v (velocity), r,g,b (dye density). Use Float32Array.

Steps per frame:
1. add sources from pointer (splat of velocity + dye)
2. diffuse velocity (or skip; advect with dissipation works)
3. project (pressure solve, Gauss-Seidel ~ 20 iterations)
4. advect velocity
5. project again
6. advect dye
7. dye dissipation, velocity dissipation

Use SW advection with bilinear? Semi-Lagrangian with bilinear interpolation is fine.

Perf: 128*85 = 10880 cells. Gauss-Seidel 20 iterations = 217k ops, fine. Per frame we do maybe 2 projections → 40 iterations. Fine.

Actually let me use N=128 for width, height derived. To keep the code simple, use a grid of NX=128, NY=85. Indexing: i + j*NX.

Boundary handling: simple wrap or clamp — clamp velocity to zero at walls (set boundary cells to reflect). Simplest: keep dye within, velocity damp at borders. I'll implement set_bnd for velocity with reflection on walls.

Rendering: map dye RGB to pixels, clamp to 255. Add slight brightness. Also maybe render velocity magnitude subtle — no, keep dye only but with a subtle background gradient.

Pointer: track pointer position, on move compute dx,dy, splat dye and momentum along drag. Radius ~ 0.06 of width in grid cells... Let's use radius in grid units, e.g. 5 cells radius with smooth falloff.

Color selector: palette of swatches + maybe hue slider. Provide swatches: cyan, magenta, orange, green, blue, yellow, white, rainbow (cycles hue over time).

Controls: buttons Pause/Resume (toggle), Clear.

Seeding initial: a few splats with velocity swirls.

Pause: stop stepping but still render. Also keep splat injection while paused? Probably allow injection but not stepping — actually better: while paused, don't inject either? Spec doesn't say. I'll allow injection but simulation frozen... that would look odd (dye appears at a spot). Simpler: while paused, still process pointer splats but don't advect — actually let's just freeze everything except rendering; pointer drag while paused just paints dye... Hmm. I'll do: paused → no simulation step, but pointer still adds dye/momentum which appears immediately and then when resumed it flows. That seems reasonable and harmless.

Clear: zero all fields.

Canvas sizing: full-screen, resize handler re-creating grid? Keep grid fixed (independent of screen size) so work per frame bounded; just scale canvas CSS to window with fixed backing resolution? Better: set canvas width/height to window size * dpr capped, but render offscreen grid scaled. Rendering cost is the drawImage which is fine. Cap dpr to 1 or 1.5 to bound work. Actually simplest: canvas backing = window size (CSS pixels, dpr capped at 1.5), and we draw scaled grid image. Fine.

Let me write the code carefully.

Constants:
- NX = 132, NY = 88? Let's use NX=128, NY=86 ≈ aspect 1.49. Fine.

Array indexing helper idx(i,j) = i + j*NX.

Velocity advection: backtrace x = i - dt*u[idx], y = j - dt*v[idx]. dt fixed 1 (grid units) — with grid coordinates, dt=1 means velocity in cells/step. Splat velocity magnitude: drag dx in pixels → convert to grid units: dxGrid = dx * (NX / cssWidth) * some force multiplier, say 60? Let's compute force = drag delta in grid cells * 8. Then u magnitude maybe ~ 5-20 cells/step which is fine for semi-Lagrangian.

Actually semi-Lagrangian unconditionally stable but high velocities cause smearing; ok.

dissipation: dye *= 0.994 per step; velocity *= 0.995.

Hmm "persistent swirling flow" — with 0.995 velocity decay per step at 60fps, half-life ~ 138 steps ≈ 2.3s. Use 0.9975 → half-life ~4.6s... Let's use velocity decay 0.998, dye decay 0.995 (dye persists but slowly fades). Actually dye "gradually dissipates" — fine.

Add slight vorticity confinement? Not necessary but adds swirl. Might skip for simplicity; the projection creates swirls anyway. Maybe add small vorticity confinement for nicer curls — it's cheap (~ a few passes). I'll add it, small epsilon.

Actually keep code manageable: diffuse? Skip diffusion (viscosity dissipation suffices) — but then velocity may get noisy. Projection + dissipation is ok.

Let me implement:

function addSource(field, ...) — handled inline in splat.

splat(x, y, dx, dy, color, radius): loop over bounding box cells, weight = exp(-(d2/r2)) or (1 - d2/r2)^2. Add to u,v and r,g,b.

Pointer handling: pointerdown → set last position, active. pointermove → compute delta, splat at new position with velocity from delta and dye color. Also interpolate along segment for fast drags (do 2-3 splats along path). pointerup → active false; momentum remains.

Rendering: create ImageData once at NX*NY, fill from dye, putImageData to offscreen canvas, then ctx.drawImage(offscreen, 0,0, w,h) with smoothing on. Also for a nicer look, apply a slight tone mapping: c = 1 - exp(-dye) style, and add background color.

Also draw a subtle vignette/background? The CSS background can be dark; canvas opaque black-ish base.

Maybe also render velocity as faint streaks? Skip.

Now the palette: swatch buttons setting current color as RGB triple in 0..1 (bright). Rainbow mode: hue cycling.

Color selector required — swatches + hue slider input range. Use both: a set of swatch buttons and an <input type="color">? Native color input works offline. I'll do swatches + hue slider (custom) for reliability. Actually `input type=color` is fine and simple. I'll do swatches + native color input.

UI: top-left panel with title, buttons, swatches. Keep small.

Let me also add a small "sample" debug? Not needed.

Now write index.html.

Careful details in solver:

set_bnd(b, x): for walls: x[0, j] = b===1 ? -x[1,j] : x[1,j]; etc. Corners average.

lin_solve for pressure: p_new = (div + neighbors)/4 iterated.

project(u,v,p,div):
for each interior cell: div = -0.5*(u[i+1,j]-u[i-1,j] + v[i,j+1]-v[i,j-1]) / N?? Standard Stam: 
div[idx] = -0.5*(u[i+1]-u[i-1] + v[j+1]-v[j-1])/N; p=0;
lin_solve(0, p, div, 1, 4);
u[idx] -= 0.5*(p[i+1]-p[i-1])*N ... 

Since our grid is non-square (NX≠NY), using Stam's formula with single N is approximate but works if spacing equal in both directions. Use h-based formulation: div = -0.5*h*(u[i+1]-u[i-1] + v[j+1]-v[j-1]) with h=1: div = -0.5*(du + dv); then solve ∇²p = div with standard 5-point: p = (sum neighbors + div)/4... Let me just derive: with h=1, ∇²p = div → (p[i-1]+p[i+1]+p[j-1]+p[j+1] - 4p)/1 = div → p = (sum - div)/4. Hmm sign: 4p = sum - div → p = (sum - div)/4. Gauss-Seidel: p = (sum of neighbors - div)/4? Let's check standard Stam: lin_solve(b,x,x0,a,1/a_sum) does x[idx] = (x0[idx] + a*sum)/a_sum with x0 = div, a=1, a_sum=4: p = (div + sum)/4. And Stam's div = -0.5*(du+dv)/N and correction u -= 0.5*N*(p[i+1]-p[i-1]). These are consistent within his scaling. To avoid mistakes, I'll use my own formulation:

h = 1 (grid spacing).
div[idx] = -0.5*(u[i+1,j]-u[i-1,j] + v[i,j+1]-v[i,j-1]);  // = -∇·u * h... ∇·u = (du + dv)/2 with central differences and h=1. So div = -∇·u.
Solve ∇²p = div → p = (sum neighbors - div)/4 via Gauss-Seidel (since ∇²p = sum - 4p = div → 4p = sum - div).
Then u[i,j] -= (p[i+1,j]-p[i-1,j])/2 (since grad p central diff with h=1 → ∂p/∂x = (p[i+1]-p[i-1])/2).
Then the divergence of u_new = ∇·u - ∇²p = ∇·u - div = ∇·u + ∇·u?? Wait div = -∇·u, ∇²p = div = -∇·u. u_new = u - ∇p → ∇·u_new = ∇·u - ∇²p = ∇·u + ∇·u = 2∇·u. Wrong sign! Let's redo: we want ∇·u_new = 0 → ∇·u - ∇²p = 0 → ∇²p = ∇·u. So div should be +∇·u = 0.5*(du+dv), and p = (sum - div)/4, u -= grad p. Good. Or equivalently keep div = -∇·u and p = (sum + div)/4 = Stam's form. I'll use: div = 0.5*(u[i+1]-u[i-1]+v[j+1]-v[j-1]); p solve: p = (sum - div)/4; u[i,j] -= 0.5*(p[i+1]-p[i-1]); v similarly.

Gauss-Seidel iterations: 16-24. Fine.

Boundary: after computing u,v, set_bnd for u (b=1) and v (b=2). For pressure p, Neumann (b=0). For div, doesn't matter much.

Velocity advection with set_bnd after.

Vorticity confinement: compute curl w[i,j] = 0.5*((v[i+1,j]-v[i-1,j]) - (u[i,j+1]-u[i,j-1])). Then N = ∇|w| normalized, force = eps * (N × w) → f_x = eps * Ny * w, f_y = -eps * Nx * w. eps ~ 0.1-0.3. Adds cost but ok. I'll include with eps = 0.15 maybe. Let's include but keep simple, maybe skip if it complicates. I'll include — it's ~30 lines and gives nicer swirls.

Hmm, given time budget, maybe skip vorticity confinement and rely on the projection + advective swirly. Actually curl noise from drags naturally creates vortices. I'll include it anyway — it's simple and helps "persistent swirling".

Let me write the file now.

Rendering tone mapping: 
r = dyeR, etc. pixel = 255 * (1 - exp(-dye * 1.2))? For values near 1 that's ~0.7. Or simpler: min(255, dye*255). Use smooth: c = 255 * (1 - exp(-1.6*d)). Add background dark navy: base = (8,10,18). Final = base + c*(color normalized). Since dye carries color, just map.

ImageData with Uint8ClampedArray, index p = (i + j*NX)*4.

Now the drag → force conversion: pointer coords in CSS pixels relative to canvas rect. Convert to grid: gx = (x/rect.width)*NX, gy = (y/rect.height)*NY.

delta in grid units: dgx = (dx/rect.width)*NX.

force multiplier: velocity injected = dgx * FORCE where FORCE ~ 12? If user drags 200px across 1200px width in one frame, dgx = 200/1200*128 = 21 cells * 12 = 252 cells/step — way too fast. Typical pointermove events give ~5-20px per event → dgx ~ 0.5-2 cells → *12 = 6-24 cells/step. That's decent but high. Use FORCE = 6 and cap injected velocity magnitude to ~30 cells/step.

Actually better to inject based on event delta but normalize: velocity contribution = clamp(dg * 8, -25, 25).

Dye amount: per splat add ~0.6 to color channels scaled.

Radius: R = 6 cells (in grid units), covering 12 cells diameter → visually ~ 6% of width. Good.

Interpolation along drag path: compute number of steps = ceil(dist / 2 cells), splat along.

Splat function:

function splat(gx, gy, vx, vy, col, amount) {
  const r = R, r2 = r*r;
  const i0 = max(1, floor(gx-r)), i1 = min(NX-2, ceil(gx+r)), similarly j.
  for j, for i: dx = i - gx, dy = j - gy, d2 = dx*dx+dy*dy; if d2 < r2: w = 1 - d2/r2; w = w*w*(or exp). u[idx] += vx*w; v += vy*w; dye += col*amount*w.
}

Now seeding: on load, splat ~6 random spots with random velocity directions and colors.

Now the loop: requestAnimationFrame; accumulate time but fixed step: one sim step per frame (dt=1 in grid units). If paused, skip step.

Work per frame bound: NX*NY*(iterations*2 + passes) ≈ 11k*60 = 660k ops → fine.

Resize: set canvas.width = innerWidth * dpr, etc. Keep grid fixed.

Let me write it.

Also add keyboard shortcuts: space = pause, C = clear. Nice.

Pointer events: use pointerdown/move/up with setPointerCapture, and touch-action: none on canvas.

Write the code now.

For the color input: swatches set `curCol` as [r,g,b] in 0..1 with dye strength. Native color picker: parse hex.

Rainbow mode: button "Rainbow" sets rainbow=true, color cycles hue per splat based on time.

I'll write the file.

Edge cases: pointermove before pointerdown → ignore.

Let me write.

For ImageData creation: `const img = new ImageData(NX, NY)` — supported in modern browsers. Or use offCtx.createImageData(NX,NY). Use createImageData for safety.

render():
for each cell: d = dye arrays; 
rr = 1 - Math.exp(-1.7*dR) etc. Actually Math.exp per pixel * 11k = 11k exp calls, fine. Or precompute LUT of 256 entries for values 0..4. Use LUT: tone = LUT[min(255, floor(d*64))]. Let's just do Math.min(255, d*255) plus a soft knee via sqrt? Keep simple: c = 255 * min(1, d) is linear and looks fine, but highlights clip. Use: c = 255*(1 - Math.exp(-2.2*d)) — I'll do the exp, cost is negligible (33k exps per frame... 11k*3 = 33k exp, ~0.3ms). Fine.

Background: pixels where dye=0 show dark base (10,12,20).

OK writing.

One more: drawImage with imageSmoothingEnabled true and 'high' quality for smooth upscale.

Also, to make it look nicer, maybe add a subtle static gradient background drawn under. Just fill black then drawImage grid — since grid has background color built in, just drawImage covering whole canvas.

Now code:

```js
const NX = 128, NY = 86;
const size = NX*NY;
const u = new Float32Array(size), v = new Float32Array(size);
const u0 = ..., v0 = ...;
const r = ..., g = ..., b = ...;
const r0,g0,b0;
const p = new Float32Array(size), div = new Float32Array(size), curl = new Float32Array(size);
```

idx(i,j) = i + j*NX.

setBnd(bnd, x):
for j in 1..NY-2: x[idx(0,j)] = bnd===1 ? -x[idx(1,j)] : x[idx(1,j)]; x[idx(NX-1,j)] = bnd===1 ? -x[idx(NX-2,j)] : x[idx(NX-2,j)];
for i: x[idx(i,0)] = bnd===2 ? -x[idx(i,1)] : x[idx(i,1)]; similarly bottom.
corners: average of neighbors.

project():
div.fill(0); p.fill(0)  (p initial guess 0 fine)
for j 1..NY-2, i 1..NX-2: div[k] = -0.5*(u[k+1]-u[k-1] + v[k+NX]-v[k-NX]);  

Hmm sign: I determined div = +∇·u and p = (sum - div)/4. Let me use Stam-consistent version to avoid errors: 
div[k] = -0.5*(u[k+1]-u[k-1] + v[k+NX]-v[k-NX])  // = -∇·u
p solve: p[k] = (p[k-1]+p[k+1]+p[k-NX]+p[k+NX] + div[k]) * 0.25;  // Stam: x = (x0 + a*sum)/asum → p = (div + sum)/4
then u[k] -= 0.5*(p[k+1]-p[k-1]); v[k] -= 0.5*(p[k+NX]-p[k-NX]);
Check: u_new = u - ∇p; ∇·u_new = ∇·u - ∇²p. ∇²p = sum - 4p = -div = ∇·u. So ∇·u_new = ∇·u - ∇·u = 0. ✓ Great, Stam's form is self-consistent.

linSolve for pressure: iterations 20, with setBnd(0,p) after each (or after all; do after each for stability at boundaries — cost fine).

advect(dst, src, u, v, dt):
for j 1..NY-2, i 1..NX-2:
 x = i - dt*u[k], y = j - dt*v[k];
 clamp x to [0.5, NX-1.5], y likewise;
 i0 = floor(x), i1 = i0+1, s1 = x-i0, s0 = 1-s1; similarly j.
 dst[k] = s0*(t0*src[i0+j0*NX] + t1*src[i0+j1*NX]) + s1*(t0*src[i1+j0*NX]+t1*src[i1+j1*NX])
setBnd after.

Order per step:
- applySwirl? no
- addSources (from pointer queue)
- vorticity confinement (curl, force) → uses u,v
- project
- advect velocity (u0 ← u advected... careful: advect u using u,v: copy u→u0? advect(dst=u0, src=u), then swap arrays. Since I use fixed arrays, do: u0.set(u); v0.set(v); advect(u, u0, u0, v0)? No — velocity must be advected by the pre-advection velocity field. So: u0.set(u); v0.set(v); advect(u, u0, u0, v0); advect(v, v0, u0, v0). Good.
- project again
- advect dye: r0.set(r); advect(r, r0, u, v) — dye advected by current velocity. Do all three channels.
- decay: multiply arrays by factor. Combine into advect? Separate loop over whole array with multiply — cheap.

That's 2 projects + 3 dye advections + 2 velocity advections + curl pass. ~ maybe 7 passes of 11k + 40 iterations of 11k = ~500k cell ops. Fine.

Wait — should I also diffuse dye? No.

Now let me write the actual file.

UI styling: dark glass panel, rounded, buttons.

Also show a hint text that fades after first drag.

Let me produce the file.

Also need "Bound simulation resolution and work per frame" — mention NX/NY fixed, fixed timestep, capped iterations.

Time to write.

const PARAMS: dyeDecay = 0.994, velDecay = 0.998, epsilon vorticity 0.12? tune.

Actually with dt=1 grid unit per frame and velocity magnitudes up to 25 cells/step, advection is fine.

FORCE: velocity injection = deltaGrid * 4? Let's think: to make a satisfying swirl, after drag of ~100px over ~10 frames, per-frame delta ≈ 10px → dg = 10/1200*128 = 1.07 cells → v = 1.07*FORCE. With FORCE=8 → 8.5 cells/step. Over 10 frames the fluid travels... reasonable. Cap at 30.

Dye amount per splat: 0.35 * weight, with color (r,g,b) each up to 1.3 scale so dye accumulates to saturation. Let's say amount = 0.5 per splat; with overlapping splats along the path dye reaches ~1 quickly. Good.

Now writing.

Also handle pointer leaving canvas (pointerup/cancel).

Clear: zero u,v,dye, arrays. And maybe reseed? No — spec says Clear removes dye and resets velocity so next drag starts fresh. So no reseed.

But after Clear the screen is black — fine.

Write now.

For hex parse: `#rrggbb` → components /255.

I'll implement palette swatches with data-color and data-rainbow.

Final render: I'll add a subtle bloom? Skip.

Code writing:

```js
function step(){
  // sources already applied in real time via queue
  vorticity();
  project();
  u0.set(u); v0.set(v);
  advect(u, u0, u0, v0);
  advect(v, v0, u0, v0);
  project();
  r0.set(r); g0.set(g); b0.set(b);
  advect(r, r0, u, v);
  advect(g, g0, u, v);
  advect(b, b0, u, v);
  decay();
}
```

decay: for k: u[k]*=VD; v[k]*=VD; r[k]*=DD; ... plus maybe subtract small constant for dye to clear haze: r[k] = max(0, r[k]*DD - 0.0008)? Multiply is enough.

vorticity(): 
w[k] = 0.5*((v[k+1]-v[k-1]) - (u[k+NX]-u[k-NX]));
then for interior: Nx = |w[i+1]| - |w[i-1]|, Ny = |w[i+NX]| - |w[i-NX]|; len = sqrt(Nx²+Ny²)+1e-5; Nx/=len; Ny/=len; u += EPS * Ny * w[k]; v -= EPS * Nx * w[k];
EPS ~ 0.3.

Hmm sign convention: force = eps * h * (N × ω) where ω = w ẑ → (Nx, Ny, 0) × (0,0,w) = (Ny*w, -Nx*w, 0). ✓ matches.

OK. Now write the file for real.

Let me also make sure the sample points for verification: maybe I'll just check visually with screenshots and motion.

Write file.

Let me write index.html.

There are problems in the code I wrote:

1. `setBnd` last corner line: `x[N*NX === 0 ? 0 : N-1] = ...` — weird but works (N-1). Ugly, let me clean.
2. The fade slider logic is broken/unfinished: `DYE_DECAY_OVERRIDE` referenced before declaration in `frame` (hoisted `let` → TDZ error on first frame!). Actually `frame` runs via requestAnimationFrame after script executes, so `let DYE_DECAY_OVERRIDE = DYE_DECAY;` is already initialized by then. But the decay in `step()` uses DYE_DECAY constant, not the override. So the fade slider does nothing. Let me just use a mutable variable `dyeDecay` used inside step(), and remove the junk.

Let me rewrite those parts properly.

Also `pointerleave` calls endPointer which deletes pointer — with pointer capture, leave fires? With setPointerCapture, pointerleave may fire when moving outside canvas, releasing the stroke. Better to not listen to pointerleave, or only on mouse without capture. I'll remove pointerleave.

Also hideHint is defined after use in inject — function declaration hoisted, fine.

Let me fix these.

Wait: `dyeDecay` is declared with `let` at the top of the IIFE in the "Solver parameters" section, and `fadeRange` handler sets it — fine since same scope.

Also `hideHint` defined later with function declaration → hoisted. Fine.

Now run in browser.

Now run in the browser and check initial output + console.

The sim looks good — dye blobs with swirl. Issues to fix quickly:
1. FPS shows 14 fps — that's low. Likely because screenshot environment throttles rAF, or solver is heavy. Let me consider: 2 projects × 18 iters × 11k = 400k ops... plus advections. Should be fast in real browsers. The 14 fps may be from the harness's screenshot throttling (frame 1 timing). Hmm, it says "space = pause · c = clear · 14 fps". Possibly the sandbox runs slowly. To be safe, reduce PROJECT_ITERS to 12 and skip one of the two projects? Actually two projections matter for quality. Let me reduce PROJECT_ITERS to 12 and see. Also the render loop with Math.exp per pixel ×3 — could use a LUT to cut cost.

Given limited time, do quick optimizations:
- PROJECT_ITERS 18 → 12
- tone map via LUT (precomputed 512-entry).
- Reduce work: skip vorticity confinement's extra pass? Keep, it's cheap.

2. Swatch buttons appear empty/dark — the swatch colors aren't showing! Because `.swatch` has no background color set from data-color. The first one shows white-ish border active but no color. I need to set background-color inline or via JS. That's an essential UI bug (color selector). Fix: set style background from data-color in JS at startup.

Let me fix both quickly, then re-screenshot and test drag/clear.

Also the swatch for rainbow shows gradient (works since CSS sets it) — confirms background not set for others.

Let me apply edits now.

Wait — the swatches have `background` set via inline style, but the `.swatch` CSS also sets no background, good. However the `#rainbow` uses conic-gradient — inline style only applied to data-color ones. Good.

Now reload and test: drag to inject, then sample flow after release, second drag, clear, drag again. Let me use browser drag + motion and inspect via window.__fluid sampleVelocity.

Swatches now show colours. Now test the drag interaction and post-release flow. Let me perform a drag across the canvas, then sample velocity after release via inspect (window.__fluid). The browser tool supports drag action.

Drag from (700,250) to (950,450) maybe. Then sample velocity at a few points.

The drag injected a huge amount of dye/momentum — way too strong. The drag tool apparently produced a very fast drag (large delta per event), saturating the field. Dye got blown off the right side and covers the screen with haze. The amount injected is too much: `DYE_AMOUNT / steps * 1.6` with 8 steps ≈ 0.124 each... but drag events with huge deltas → velocity clamp at 34 cells/step → very fast advection blowing dye around.

Also the whole screen filled with faint dye — that means dye amount accumulated broadly. Actually looks like dye got smeared across by huge velocity.

Problems:
1. Injected velocity too high (MAX_FORCE 34 cells/step = 34/128 of screen per step). Reduce MAX_FORCE to ~14 and FORCE to 4.
2. Dye amount per stroke appears too much / too big radius. Actually the drag path splat covers a long stroke — fine.

Also the harness's drag is instantaneous (one big delta), producing one giant splat along path. In real use, pointer moves are smaller. But I should still clamp more gracefully.

Let me reduce FORCE to 4.0 and MAX_FORCE to 16, and reduce DYE_AMOUNT to 0.5. Also the burst of dye at the right edge is from clamping to the boundary — acceptable.

Also fps 23 now. Hmm, that's the harness's measurement; likely rAF throttling in headless. Fine.

Let me tune and retest with a shorter drag, and verify post-release persistence via sampling velocity with window.__fluid.

Actually one important check: does dye continue moving after release? The tool's drag ends quickly and then screenshot is taken at frame 1? "Frame 1, +0ms" — screenshots seem to be captured quickly. Let me use `inspect` to read window.__fluid.sampleVelocity at several points and check nonzero velocities after release.

Let me first tune params.

Also cap per-event delta so a single huge synthetic drag event doesn't blow everything: clamp dxg, dyg to e.g. 8 grid cells each in `inject`. Let me add that in inject: 

```
if (dxg > 8) dxg = 8; else if (dxg < -8) dxg = -8;
```
And steps limited to 8 already.

Let me add clamp.

Now reload, clear, drag, and verify flow persists after release using window.__fluid sampling. Let me do: reload → click Clear (to start clean) → drag → sample velocities → wait (motion frames) → sample again → second drag in another direction → Clear → drag again.

I'll use browser drag then inspect via `browser inspect` with a JS eval? The `inspect` action presumably returns DOM/console. Can I run arbitrary JS? Not documented. The `type`/`click` actions interact. Hmm — inspect returns console messages; I could log to console... but I can't execute arbitrary JS directly.

Alternative: verification via screenshots and motion frames. Use `motion` to see evolution after release. That demonstrates continuing swirl.

Actually I could add temporary console logging... simpler: rely on visual checks. But sampling the flow would be good. Perhaps `browser` `inspect` executes page JS? Let me just try passing text? No.

I'll do: reload, clear via clicking Clear button, drag, screenshot, then motion frames to see continued swirl after release. Then second drag in another direction, screenshot. Then click Clear, screenshot (should be blank), then drag again, screenshot.

Let's proceed quickly.

Now drag with a moderate stroke and then check continued motion. Let me pick a magenta swatch first (click at ~87,204), then drag.

Interesting: swatch selection works (magenta active + picker updated). But the seeded scene got smeared into a huge diffuse haze — the seed velocities (7.5 cells/step) plus vorticity and low dye decay have diffused everything into a fog covering the screen. It's been running for a while between screenshots (the harness continues running). The dye is fading to a haze rather than staying as coherent swirls.

Issues:
1. Numerical diffusion from semi-Lagrangian advection smears dye over time — inherent. To keep swirls crisper, I could do dye advection with BFECC/MacCormack (2 extra advections) — costs2 more passes. That gives much crisper dye. Given remaining time (~2 min?), maybe skip.

2. The overall haze across the whole screen suggests dye diffused broadly. Adding a slight dye threshold subtraction (e.g. subtract 0.0015 per step) would clean up the haze and keep swirls visible.

Actually, the more important thing: dye dissipation is multiplicative so very low values persist forever as haze. Add a subtractive floor: dr[k] = max(0, dr[k]*decay - 0.0008). That removes the haze.

Also reduce seed velocity slightly to avoid immediate full-screen smear? The scene looks fine initially and slowly diffuses — that's physically OK.

Let me apply the subtractive floor quickly and then do the drag verification.

Also let me quickly verify flow persistence with a motion frame after a drag.

Time check: I should wrap up fast. Apply edit, reload, drag, screenshot, then clear → drag, screenshot. Report.

Note: with subtractive floor, the "Dye fade" slider at max (dyeDecay 0.9796) will clear fast — fine.

But careful: dye injected then decays at 0.9945 - 0.0011 → values below ~0.2 decay fast. Hmm: a = d*0.9945 - 0.0011. Equilibrium at d* = 0.0011/(1-0.9945) = 0.2. So dye below 0.2 decays to zero relatively fast (~1/(0.0055) ≈ 180 steps for small values, faster for tiny). Values above 0.2 persist with multiplicative fade. Hmm, that might make thin dye vanish quickly. Reduce floor to 0.0006 → equilibrium 0.11. Actually the haze in the render was visible at probably0.05-0.15 values. Equilibrium 0.2 might erase faint swirls too fast.

Let me use 0.0006 floor (equilibrium 0.11). Hmm, still kills stuff below 0.11 quickly. Let me instead use 0.0004 → eq 0.073. Time constant for small values: d_{n+1} = 0.9945 d - 0.0004; for d=0.1: d_n = 0.0995-0.0004 = 0.09905... decay per step0.00095 → over 100 steps loses 0.095 → dies in ~100 steps (1.7s). That kills faint dye quickly. Hmm.

Better approach: subtract only a tiny amount (0.0003) — faint dye dies in ~2-3 s which is "gradually dissipates" — acceptable and keeps the scene clean. Since tone map multiplies low dye to visible (LUT at 0.05 → 255*(1-exp(-0.0775))=19/255 visible-ish), the haze was mild.

I'll use 0.0005 compromise. Fine — visual check will tell.

Change0.0011 → 0.0005 in all three.

Actually, let me keep it simple and just edit the constant once — I'll rewrite the loop values.

Now reload and do the verification sequence:
1. Clear (start clean)
2. Drag (dye follows)
3. Screenshot shortly after → check dye follows drag
4. Motion to see continued swirl after release
5. Second drag in another direction
6. Clear → screenshot (blank)
7. Drag again → new flow

Let me be efficient: reload, click Clear, drag, screenshot.

Clean seeded scene, no haze. Now do the verification sequence:
1. Click Clear (clears), then drag to start fresh flow — actually the requirement sequence: drag → sample after release → second drag → Clear → drag again.

Let me do: drag A (right-up diagonal), screenshot, motion (post-release swirl), drag B (opposite direction), screenshot, Clear, screenshot (blank), drag C, screenshot.

Time is short (~2-3 min). Let me combine: drag then screenshot then motion, then drag opposite then screenshot, then Clear + screenshot, then drag + screenshot. That's ~6 calls. Do it fast.

The dye follows the drag direction (up-right) — good — but it's blown too far right and covers a big region. The harness drag is one huge instantaneous delta clamped to 8 cells × FORCE 4 = 32? No: dxg clamped to 8 → vx = 32 → clamped by MAX_FORCE 15. Still 15 cells/step is fast (12% of screen per step). Over many frames it carries dye to the wall. That's OK-ish but a bit extreme. Reduce MAX_FORCE to 9 and FORCE to 3 for more graceful behavior; real pointer drags are small deltas so they'd be weaker too... Real drag delta per event ~2-6 px → dg = 0.2-0.6 cells → vx = 0.6-1.8 cells/step. That's quite slow actually. Hmm, FORCE4 gives reasonable real-use behavior; the synthetic instantaneous drag is the extreme case.

Compromise: keep FORCE 4, lower MAX_FORCE to 10. Also lower DYE_AMOUNT slightly to 0.42 so strokes don't oversaturate.

Then verify post-release swirl via motion. Time is tight — make this one edit, reload, drag, motion screenshot, and finish. I still need to verify Clear + fresh drag. Let me combine: after edit, reload, click Clear? No — verification order:

1. drag (already seen dye follows drag)
2. motion after release → swirl continues
3. second drag other direction
4. Clear → blank
5. drag → fresh flow

I'll do: edit → reload → drag → motion → click Clear → screenshot → drag → screenshot. That's 5 calls, fast.

Actually to save calls: motion returns frames showing evolution. Let me just do it.

Now verify post-release motion and clear/drag. Reload first.

Actually to save time, I can skip reload — no, the script is loaded in the page; changes need reload. Reload → then drag → motion → clear → drag.

Let me do reload + drag in parallel? They're dependent (order matters). Sequential.

Actually I want to check motion AFTER release. `motion` action returns motion frames. Let me do reload, drag, then motion.

Now: Clear first (so the test is unambiguous), drag, then motion to check post-release swirl.

Actually the verification asked: drag and sample after release, then second drag, then Clear, then drag again. Let me do drag → motion (post-release swirl) → click Clear → screenshot (blank) → drag → screenshot (fresh flow). The second drag in another direction I can fold in: drag opposite direction before clear. That's 5 calls. Go.

Dye follows the down-right drag direction and shows curl structures. Now check post-release continuation with motion frames (the pointer is already released after drag).

The dye fades far too quickly — within2.5s nearly everything is gone. The problem: the drag's dye values were thin (a single stroke at amount 0.42 spread over radius), so values hover below the subtractive-floor equilibrium and get eaten. Also DYE_DECAY at "normal" = 0.9945 with subtract0.0005 → equilibrium 0.09, and dye below that dies within ~2-3 seconds. Plus vorticity/velocity decayed.

Also note: the dye gets advected off toward the right edge quickly (velocity 10 cells/step) — it left the visible center. Actually it faded everywhere.

Fixes needed for "persistent swirling flow":
1. Reduce subtractive floor to 0.00015 (haze removal but slow fade for thin dye).
2. Increase dye decay to 0.9975 base ("normal") so dye persists ~ longer. Set fade slider mapping: dyeDecay = 0.9995 - t*0.012 → at t=0.22 → 0.9969. Hmm at "normal" we want ~0.997.
3. Reduce VEL_DECAY less aggressive? Currently 0.9975 → half life 275 steps ≈ 4.5s at 60fps (but the harness runs at ~20fps so steps are slower). The flow does keep moving (motion frames show dye drifting). Fine.
4. Make injected dye stronger so the stroke is clearly visible: DYE_AMOUNT 0.42 → 0.6.

Also the tone map: LUT at low values gives visible color; but the haze issue was earlier when dye values were ~0.05 across the screen. With floor 0.00015 equilibrium = 0.00015/0.0025 = 0.06 at decay 0.9975... that's still haze level but the tone map makes 0.06 → 255*(1-exp(-0.093)) = 23/255 visible slightly. Acceptable — the earlier full-screen haze came from the initial giant splats spreading.

Let me apply: floor 0.00015, dyeDecay base 0.9975 with slider mapping t:0.9996 - t*0.012 (t=0.22 → 0.9970), DYE_AMOUNT 0.6.

Then verify Clear + drag again quickly. Time is very tight. Make edits via sed, reload, drag, motion short, then Clear, screenshot, drag, screenshot. Maybe skip motion to save time; but persistence check matters. Motion took ~3s. I'll do one drag + one motion, then clear + drag in fewer calls.

Actually I can verify persistence more cheaply: after drag, take screenshot twice? Each screenshot returns Frame 1 +0ms which suggests it captures right away after reload... unclear timing. Motion gave frames at 900ms intervals — good for persistence. But budget: ~2 min left? The instruction says finalize soon. Let me do: sed edit, reload, drag, motion (persistence), click Clear, screenshot, drag, screenshot. That's 5 tool calls. Then report. Go fast.

Actually I can combine Clear-click and its screenshot in one call? No. I'll cut: after motion, click Clear (which returns an image) → then drag (returns image) → done. That gives evidence of blank after clear and fresh flow after drag. 4 calls.

Wait: the fade slider sets dyeDecay = 0.9996 - t*0.012; at t=0.22 → 0.9970. Good, matches base.

Now reload, drag, motion (persistence), then Clear + drag. Go.

Now drag in an opposite direction (e.g., up-left) with a different colour to also test the second-drag requirement, then motion to confirm persistence, then Clear and drag again.

Sequence (4 calls):
1. drag (up-left, magenta selected? currently cyan default) — just drag.
2. motion — persistence/swirl.
3. click Clear → returns screenshot showing blank.
4. drag → returns screenshot showing fresh flow.

Go.

Dye follows the up-left drag and shows curling tendrils. Now motion to verify post-release persistence (dye continues moving/swirling).