The ladder made continuous — give the links across block seams their own chance; the contact set between blocks then has a dimension between 0 (a rung) and 1 (the carpet), and ramification becomes a dial

Why this exists. Filed 2026-10-03 from the owner's question while reading the carpets explainer: the dial gives every link a uniform random number; is there any point in another distribution, perhaps one whose parameters change with position? Nothing here has been computed. Run 094 was open when this was written and nothing below reads it.

What a change of distribution cannot do. A draw's crossing point is a minimax: over routes, the smallest value of the largest link number on the route. It uses only the order of the link numbers. Any distribution applied identically to every link gives the same orderings with the same odds, so it relabels the dial and changes nothing measured as a share of links open. The distribution matters only when it differs from link to link. This probe takes the difference by level; P-052 takes it by position across the sample.

The construction. Take the generation-g carpet of a pattern P. A link that crosses a line of 9-adic level j ≥ 1 is a level-j seam link (P-049's terms); every other link is interior. Three families, all subgraphs or thinnings of the carpet, with the carpet and the ladder as members:

• A, a uniform seam weight. Every seam link, at every level, is open with chance λ·p instead of p. λ = 1 is the carpet.
• B, Cantor contacts. Fix a digit set S ⊂ {0, …, 8}, symmetric under c ↦ 8 − c. A level-j seam link at position r is kept iff all j base-9 digits of r mod 9ʲ lie in S. A seam segment of length 9ʲ then carries |S|ʲ links: the contact set between two blocks is a Cantor set of dimension d_s = ln|S| / ln 9. |S| = 9 is the carpet (d_s = 1). P-049's rung is the other end: its contacts are the rows whose top digit lies in S and whose lower digits are all 4, k of them at every scale (d_s = 0). Letting the top m digits range over S and fixing the rest at 4 gives kᵐ contacts, still finitely ramified; letting all of them range gives B. So the ladder is the first step of a longer family, and B is where it leaves finite ramification.
• C, a leak. A rung with its designated contacts kept, and every other seam link open with chance ε·p. ε = 0 is the rung of runs 084 and 094; ε = 1 is the carpet.

The crossing is read between the cells of the left and right columns at rows whose digits lie in S (the whole columns at |S| = 9, the rung's terminals at d_s = 0). B is deterministic: gen.py can build it as it builds the ladder's lattices, and perc.c samples it unchanged. A and C need a sampler that takes a weight per link (a draw's crossing point is then the minimax of u/λ).

What is lost, and what is kept. The ladder's point was that a block talks to its neighbour through finitely many cells, so the state closes and a recursion reproduces the lattice (run 084's controls). For d_s > 0 the contacts grow with the level and nothing closes: these objects are sampled, never recursed. What is kept is that every one of them is the carpet's own graph with links removed, so a draw on the carpet and a draw on a thinned version can share random numbers, and the thinned one can only cross later.

The lead, held with conviction. 1. A uniform weight moves the threshold and nothing else. Seams of every level sit at multiples of 9, so a uniform λ is a period-9 change of couplings on a fixed graph. The sharpening rate is the carpet's. 2. The seam's dimension moves the sharpening rate. 1/ν depends continuously on d_s, reaching the carpet's value at 1 and falling toward 0 as d_s → 0, where the rungs sit with no transition at all (P-049's bound). 3. Any positive seam dimension restores a transition. P-049's bound multiplies factors 1 − (1 − p)^(contacts); with contacts growing as 9^(d_s·j) the product converges to something positive, so the drift of the threshold toward the wall, which every rung shows (run 084: four levels, still rising), stops. 4. The order of T and X is a function of d_s. Run 084 has X below T at d_s = 0 on rows 1, 4, 7; the carpet has X above T at d_s = 1. Somewhere between, the order crosses. That crossing value is what "how much boundary does the carpet's behaviour need" means as a number, where the ladder could only answer in steps of k. 5. The leak has a crossover scale. A level-j seam carries about ε·p·9ʲ open non-contact links, so the rung's behaviour holds while ε·9ʲ ≪ 1 and the carpet's returns above it. Quantities at different ε and g should fall on one curve in ε·9ᵍ.

Where it breaks. (1) is universality on a carpet, which nothing in the record establishes: the census shows the exponent moving from pattern to pattern, and if it also moves with λ then the exponents of the census are functions of more than the pattern, and scaffold S2 has a second problem. (2) and (3) could fail together if a positive d_s below some value still leaves the threshold drifting at every size we reach; four levels cannot tell a slow drift from a limit (run 084, item 2 of its score). (4) assumes one crossing; with the digit set free (which three digits, not only how many) there may be none or several, which is run 094's question asked on a bigger family.

A first reductio pass (OPS.md; untested unless said). Limits: d_s = 1 and λ = 1 are the carpet, on record; d_s = 0 is the ladder, on record at k = 1 and 3. λ → 0 in A disconnects the blocks: threshold → 1. Invariance: (1) says 1/ν is invariant under λ; the nearest thing on record is the census, where it is not invariant under the pattern. Consistent, not supporting. Twins: two digit sets of equal size give equal d_s and different contacts ({1, 4, 7} and {2, 4, 6}); (2) predicts equal 1/ν in the limit and says nothing at reachable sizes. Run 094 is this twin test at d_s = 0. Neighbours: the square (no holes) under B is a Cantor-thinned square lattice; whether its exponent stays 3/4 is a clean question with a known endpoint. Joint consistency: (3) with run 084's rising threshold: no conflict, 084 is d_s = 0. Negation of (2): the seam dimension is irrelevant and every d_s > 0 has the carpet's 1/ν; then ramification is a yes-or-no property and the ladder's whole question collapses to "finite or not".

What a first run would be. Cantor contacts on S = {1, 4, 7} (d_s = 0.5) for T and X at 81, 729, 6561, perc.c unchanged, 8,000 samples, with the carpet's census values and run 084's rung beside it: the threshold's increments by level (does the drift stop), the local slopes, the order of T and X, the skewness. Not sealed; it waits on run 094, whose answer says whether the digit set must be varied as well.

The way of asking is not new. Whether a modulation of couplings changes an exponent is the Harris criterion's question, and for modulations organised by a rule rather than at random it is Luck's: an aperiodic modulation is irrelevant, marginal or relevant by the sign of Φ = 1 + ν(ω − 1), ω its wandering exponent (stated in Iglói and Turban, Europhys. Lett. 27, 91 (1994), arXiv:cond-mat/9406035). Whether a criterion of that kind has been written for hierarchical thinning on a carpet: not known to the codifier, unverified.

Back to the dictionary. P-049 said the ladder makes ramification a parameter where §23–24 and §45 treat it as a yes-or-no property. This makes the parameter a dimension, and puts the carpet and its finitely ramified relatives on one axis.