TITLE: Sand on Steel — Chladni Plate

FULL PROMPT:
Create a self-contained, single-file interactive page titled "Sand on Steel: Chladni Plate". On an 880×500 brushed-steel plate simulate 2600 sand grains migrating under a standing-wave mode v(x,y)=cos(nπx)cos(mπy)−cos(mπx)cos(nπy) on the unit square: each grain performs gradient descent on v² (force = −v∇v with analytic derivatives), so grains settle exactly on nodal lines. An EXCITE slider injects kinetic energy proportional to its setting so grains visibly jump off their lines and re-settle; n and m sliders retune the mode (resetting energy). Pointer-sprinkle adds grains (capped 5200), MORE SAND adds a batch, SWEEP clears and re-dumps a thinner field, NODAL GLOW toggles a 56×32 heatmap whose brightness is |v| inverse (nodal zones glow amber). Live stats show grain count, mode pair and the REST meter — fraction of grains currently on a node, drawn as a bar under the plate. Everything inline, no external resources.

INTENDED DESIGN DESCRIPTION:
A dark lab bench with one steel plate and one bow: the pattern is never drawn, it is where the sand refuses to be shaken. The craft is making an invisible vibration field legible through accumulation.

MAJOR VISUAL TECHNIQUES:
- Analytic Chladni mode field with closed-form gradients
- Per-grain gradient descent on v² (nodal capture) plus excitation jitter
- Nodal heatmap compositing (brightness = 1−|v|)
- Rest-fraction instrumentation as a physical meter
- Plate chrome: brushed gradient, corner bolts, swept-shake transition

INTENDED INTERACTION MODEL:
Sprinkle sand, drag n/m and EXCITE to break and rebuild patterns, watch the REST bar fall and climb, glow or hide the underlying field to compare what you see with what is there.