The dial that varies across the sample — gradient percolation on a carpet; one configuration locates the threshold and its sharpness, and the front should carry the holes' imprint at every factor of nine

Why this exists. Filed 2026-10-03 with P-051, from the same question: what if the chance of a link being open changes as you move across the sample, or toward its centre? Nothing here has been computed.

The construction. Every measurement in the carpet thread opens all links at one chance p and asks whether the sample is crossed; a sample gives one number, its crossing point. Gradient percolation gives each link a chance that depends on where it is. Two layouts:

• Across. p(x) falls linearly from p₀ + Δ at the left column to p₀ − Δ at the right. The cluster joined to the left side ends in a front. Per row, record how far it reaches.
• Ring to centre. p falls from the outer ring inward. Every census pattern keeps its ring, and T and X both remove the centre cell, so the natural source is the ring and the front closes round the middle. This layout keeps the pattern's full symmetry.

On a uniform lattice the front sits where p(x) equals the threshold, and its width σ_f is set by the gradient G = |dp/dx| alone: the front is as wide as the correlation length at its own edges, σ_f ≈ ξ(p_c ± G·σ_f), so σ_f ∝ G^(−ν/(1+ν)). The gradient replaces the sample size as the finite-size scale. (Sapoval, Rosso and Gouyet, J. Phys. Lett. (Paris) 46, L149 (1985), as cited in arXiv:cond-mat/0210106. Checked 2026-10-03: that paper states σ_f ∝ L_g^(ν/(1+ν)) with L_g the inverse gradient, 4/7 on the square, and attributes it to Sapoval, Rosso and Gouyet 1985 and 1989; the flag "from memory, unverified" is lifted.)

Why it might earn its place in PG-01. Three uses, in order of how much the codifier would stake on them. 1. A second estimator of the sharpening rate. Every 1/ν in the census is one estimator: the width of the pile of crossing points against L. Run 087 could not resolve whether that slope drifts, and scaffold S2 (one exponent) has no removal test yet. σ_f against G reads ν from one size, from a different mechanism. Agreement supports S2 from outside; disagreement is the first data against it that is not a drift. 2. A threshold without a size. L8's gap is the one-block fixed point minus the lattice threshold, read at L ≤ 6561 where p_med still moves. The front's mean position, extrapolated in G, is a threshold that never used L as its scale. 3. Cost. One configuration at 6561 is one union–find pass, seconds, against forty minutes for 8,000 crossing points; a front yields thousands of row readings per configuration, though not independent ones.

The lead, held with conviction. On a carpet the front's width follows G^(−ν/(1+ν)) with the carpet's own ν from the census (so an exponent near 0.73 for T and 0.65 for X, where the square has 4/7), and on top of the power law sits a ripple periodic in ln(1/G) with period ln 9: each time the front's width passes a power of 9 it meets the next size of hole. The ripple is larger on patterns with larger holes. The mean front position converges, as G → 0, to the limit of the census's p_med, below the one-block fixed point on every pattern (P-050).

Where it breaks. A carpet is not the same at every x. A column at x passes through whatever holes the hierarchy puts there, so "the threshold at x" is not a local quantity, and the front pins on hole edges. The front's mean position will depend on where p(x) = p_c falls relative to the hole hierarchy: move the offset p₀ and the front does not slide smoothly, it sticks and jumps. Then one configuration does not locate the threshold; an average over offsets does, and the response of front position to offset — its flats and steps — is itself the measurement, a staircase whose flats are the holes. If the staircase's steps do not shrink as G → 0, use 2 fails and the instrument reads the pattern's geometry, not its threshold. The derivation of ν/(1+ν) also assumes the correlation length is a power of |p − p_c| with nothing riding on it; on a discretely self-similar lattice it may carry its own ln 9 ripple, which is the same effect seen twice.

A first reductio pass (OPS.md; untested unless said). Limits: G → 0 at fixed L returns ordinary percolation at p₀: the front is the whole sample or nothing. G large: the front is a few cells wide and reads the block's first few columns, which is the wall law's territory (run 077). Invariance: the lead says ν from the front equals ν from the pile; the record has no second estimator to compare. Twins: patterns sharing a polynomial (run 086) share the one-block map and differ in threshold; the front should put them at different positions with equal width exponents. Neighbours: the square lattice is the control with a known answer (threshold 1/2 for links, width exponent 4/7); any front instrument reproduces those before it is pointed at a carpet. Dual: a front from the open side and a front from the closed side (the blocked cluster grown from the sparse edge) bracket the threshold from both sides; on the self-dual square they are mirror images, on a carpet they are not, and their separation is a per-configuration reading of L11's two optimizations. Negation: fronts on carpets do not scale, the staircase survives every G; then the carpet has no local threshold at any scale we can reach, and that is its entry.

What a first run would be. An instrument, so its first run is its control: the square at L = 729 and 6561, across layout, five gradients, reproducing 1/2 and 4/7. Then T and X. It needs a sampler that takes a chance per link and returns the reach of the source cluster per row; that is a specification for Claude Code, not written here. Not sealed.

Kept out on purpose. A third way a distribution can matter: replace the minimax by a sum (the cheapest total cost of a route). Then the shape of the distribution matters, as the frequency distribution does in Kuramoto's model, and a very broad distribution returns the minimax route. It belongs to L11 and is not filed here.