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The verification behind Sandpiles, as Percussion

The instrument is honest only because the avalanche it plays is a real property of the pile, not of the order it happens to simulate the toppling in. Everything the page makes audible is proved here from scratch, in the same integer arithmetic the browser runs.

The model

The abelian sandpile on an N×N grid (Bak–Tang–Wiesenfeld 1987). Each square holds an integer count of grains; a square with ≥ 4 grains topples, sending one grain to each of its four orthogonal neighbours. Grains leaving the boundary are lost to the sink. Relax until every square is < 4. The number of times each square topples is its odometer.

What is checked (all 11 ✓, node verify.mjs)

• The abelian property (Dhar 1990). On 200 random configurations (N = 3..8), the final stable grid, the full odometer, and the total topple count are identical whether you relax by parallel waves, by row-major sweeps, or in four independent uniformly-random orders. This is the load-bearing truth: the avalanche, and therefore its rhythm, is order-free.
• Avalanche size = Σ odometer, and the group operation a ⊕ b (stack and relax) is commutative.
• The identity element of the sandpile group, e = stabilize(6·all − stabilize(6·all)) (Creutz 1991), verified for N = 4, 6, 8, 12 to be stable, idempotent (e ⊕ e = e), and to fix the maximal recurrent configuration (e ⊕ c_max = c_max) — and on 16×16 to be the non-trivial fractal.
• Self-organised criticality. Driving a 40×40 grid to its stationary state and recording 30,000 avalanches gives sizes with no characteristic scale (std/mean ≈ 2.1, max ≫ mean) and a near-straight log–log survival function — a power law.
• A landmark avalanche the page exposes (a centre grain on a flat-3 7×7 pile → 84 topplings) reproduced exactly under two different relaxation orders.

The honest caveat

The avalanche-size distribution is a power law, but its precise exponent is genuinely subtle and still debated for the 2-D abelian sandpile (the distribution is multifractal; finite-size effects are strong). The instrument therefore reports the slope it measures on your run as a description, never as a claimed universal constant. What is exact — and what the instrument's claim of honesty rests on — is the order-independence proved above.

The browser instrument

verify-page.mjs loads the built page in a real (headless) browser and asserts the engine the audio reads is the same one proved here: a real avalanche is order-free under five random orders, the 84-topple landmark reproduces, the identity is non-trivial, a centre tap on a full pile triggers an avalanche and the live abelian panel confirms it, the histogram builds, and every control + the URL round-trips, on desktop and mobile, with no errors or overflow. Audio cannot be heard headless, so the sound is verified through the integer rule that drives it.