A dial for every block — each block has its own crossing point and its neighbours shift it; the shift is a small table with one entry per pair of sides, and how fast the table settles is the correction the one-exponent scaling lacks

Why this exists. Filed 2026-10-03 from the owner's proposal: the dial (the chance p that a link is open) is one global setting; give each block its own, and let neighbouring blocks' dials affect each other. The reference is Kuramoto's model as told in Strogatz's Sync: oscillators with their own natural frequencies lock together once the coupling between them is large against the spread of the frequencies. P-051 and P-052 came from the same conversation and vary the dial by level and by position; this one makes it a property of the block. Part of it is already in the record under other names. One look at stored samples is reported below; nothing new has been drawn.

The construction. A carpet of side 9ᵍ is 64 blocks of side 9ᵍ⁻¹ laid out on the pattern. Open the links in a random order. Each block, on its own links alone, has a moment at which its left side is first joined to its right: its own crossing point τ, the block's dial. The 64 are independent draws from one law, the law of the carpet one level down, with a middle and a width w. The lattice has its own crossing point t.

• No coupling. If a route could pass a block only once the block is crossed, t would be the moment the crossed blocks first connect left to right on the pattern. That is run 083's coarse time t_coarse. Because the blocks are identical and independent they open in a uniformly random order, so the share of blocks open at that moment, q_c, is a constant of the pattern, the same at every size.
• What the lattice does. It crosses earlier: t < t_coarse in 85% of samples on T and X at 729 (run 083) and in 69% to 99% over the census (run 085), by a mean lag of 0.46 to 1.92 widths depending on the pattern (run 085's ρ). A route passes through blocks that are not crossed. That is the coupling: a block's neighbours let it be used before its own dial says so, and it is measured in units of the spread, as Kuramoto's is.
• One number. Model the lattice as the uncoupled rule with every block's dial advanced by κ widths. A level is then two numbers, the width w and the coupling κ, and a step up the hierarchy maps (w, κ) to (w′, κ′). The width shrinks by the sharpening rate. κ moves toward a fixed value at a rate of its own, and that second rate is a correction exponent.
• A small table (the generalisation). One number is too few, because how a block can be used depends on the sides a route enters and leaves by. Record for each block, on its own links, the moment each pair of sides is first joined: along (left–right), across (top–bottom) and the four corners (two adjacent sides, each taken without its end thirds, since the corner cell belongs to both). The rule becomes: a route passes a block between two sides once those two sides are joined inside it. Every census pattern has the square's full symmetry (63 of 63, checked on census068.json's masks), so the six times have two laws: along and across share one, the four corners share the other. The table is the corner's offset from the along time, in widths, and the dependence between the times of one block. At level 1 each entry is an exact polynomial: along is R, and the corner is CASE-29's corner polynomial (which takes the middle cell of each side).

The pattern enters through which pairs of sides have neighbours. A kept cell with exactly two kept neighbours has them on opposite sides (it can only be passed straight) or on adjacent sides (only round a corner). T has 20 of the first kind and 12 of the second; X has 28 and 4; both have 24 cells with three neighbours and 8 with four (counted from the masks). The same table weighs differently on the two.

Why it might earn its place in PG-01. 1. A second chart on L14's object. L14 carries a law over the ways twelve terminals can be joined, at every setting of the dial. Joining is monotone in the dial, so each event has a time, and six times per block say most of what the law says. The linearization L14 asks for has the same growth rates in either chart. 2. A mechanism for scaffold S2's missing term. Run 087, and the reading of run 079's record on H, ask whether the local slope drifts and have no account of why it would. Here the drift is the table settling, and its rate can be measured on a quantity other than the width. 3. The record CASE-28's continuation needs. Its must-predict is a statistic of two or more top-level blocks, and run 083's score says that if T's turn holds it is the House's second branch: "the next object would be a two-sample correlation across neighbouring blocks". Both need each block's times kept per sample. 4. L8's gap. Run 085 found the gap ordered by the mean lag at one size (Spearman +0.98 over 49). The table says which entry carries the lag.

A look at stored samples (codifier, 2026-10-03; arithmetic on samples083/ and lag085.json; errors from 20 consecutive blocks of samples, no random numbers drawn; not read by a second reader).

w is the 20%-to-80% span of t, as in run 085. The step ratio is (value at 6561 − value at 729) / (value at 729 − value at 81). The errors of ρ are 0.005 to 0.021, of q 0.001 to 0.002, of q_c 0.0004 to 0.0013.

• q_c does not move with size, as the construction says it cannot. The stored samples pass that check.
• The lattice crosses with 57% of its blocks crossed, where the uncoupled rule needs 78% (T) or 74% (X).
• The coupling is still moving at 6561 on both patterns, by every reading.
• The readings do not share one rate. On T the lag has nearly settled (0.18) and the crossed share has not (0.47). On X those two agree (0.47, 0.45) and s₋ does not (0.29). One coupling number settling at one rate would give every reading the same ratio. If the approach were geometric, 0.47 per level is a correction exponent of 0.34 and 0.18 is 0.78 (−ln ratio / ln 9).
• T and X have nearly the same q at each size. That is those two patterns: over the 49 census members at 729 (run 085) q runs from 0.373 (#60) to 0.681 (#38).
• Over those 49, q and ρ order the patterns alike (Spearman +0.82; +0.90 within cut 4, n = 28). Against the number of cells of each kind in the pattern, q gives −0.70 for three-neighbour cells, +0.49 for straight cells, +0.19 for corner cells and +0.10 for four-neighbour cells. #60 and #63, the two largest lags, have no straight cells. #38, the smallest lag, has 20 corner cells, so the counts alone do not order the census.

Three sizes give one ratio per reading and no check that the approach is geometric. The counts are five looks (the fifth, the corner share of two-neighbour cells, gives −0.28) at 49 patterns that come in families. A look, not a result.

The lead, held with conviction. 1. The coupling is a table, and the corner entry does the work. With blocks passed by the table's rule, the lattice's lead over the block-level crossing at 729 falls from 0.8 widths to under 0.3 on T and X. Keeping along and across and dropping the corners leaves more than half the lag in place. 2. The entries settle at different rates, and those rates are the correction exponents. In widths, each entry tends to a constant of the pattern. T shows two rates already. A skewness that rises and then falls needs two corrections of opposite sign and different rate, which one coupling number cannot give: T's turn (CASE-28) is the fast entry finishing while the slow one continues. 3. Twins differ in the corner. Patterns that share R share the along entry at level 1 exactly. Where their thresholds differ (CASE-29), the corner entry or the layout of cell kinds differs. 4. X turns too, later. Its lag is still moving at the slow rate at 6561. If the slow entry carries a turn, it lies at 59,049 or beyond, which no machine here reaches; this part of the lead carries no weight.

Where it breaks. - A side is not a point. The table treats a side as one terminal. A route enters at a particular cell, and two sides joined inside a block need not be joined at cells that line up with the neighbour's. So the table's rule can cross before the lattice, and the lag may change sign instead of shrinking. Run 084's lesson was that which rows carry the contact matters as much as how many. If the entries depend strongly on the thirds chosen, the table grows into the ladder's twelve terminals and nothing is saved over L14. - Times are not shifts. "Advanced by κ widths" is the one-number model. The entries of one block are dependent random times and the lattice takes a best route over all of them. The exact ties of run 083 (one link completes the block and the lattice: 10% on T, 5% on X) are that dependence showing. A table of medians may miss what the joint law holds. - The parallel with Kuramoto is loose where it matters. There the coupling is a knob and the transition happens as it is turned. Here the lattice fixes the coupling and the only knob is the size. There is no locking transition to find. What carries over is the unit: coupling measured against spread. - Three sizes. At 81 a block is a single 9 × 9 pattern, far from any scaling. "Two rates on T" could be one rate plus the smallness of level 2. A fourth size is the Mac Studio's.

A first reductio pass (OPS.md; untested unless said). Limits: at level 1 the blocks are single cells and there is no table. On the square lattice with no holes the same construction applies and the crossing chance of a square at threshold is 1/2, so q should sit near 1/2 there; that is the control with a known answer (from reasoning, not from the record). Invariance: q_c is the same at every size (holds in the stored samples); the entries in widths should be, in the limit (untested). Twins: lead 3. Dual: a block is crossed along by open links exactly when no chain of closed links cuts it from top to bottom. On the self-dual square that ties the along and across times of one block together; on a carpet the dual is a different graph (L11), so their dependence is a free entry of the table. Negation: the table's rule leaves the lag where it was. Then the coupling sits in where on the side a route enters (the ladder) or in several blocks at once (the House's second branch), and the same per-block record measures the second.

What a first run would be. An instrument, so its first run is its controls. The coarse sampler already finds each block's along time and keeps only the moment the block graph crosses. A recorder variant keeps, per sample, the six times of each of the 64 blocks and runs on until every block has all six. Controls: the along times reproduce run 083's coarse times on the same seeds exactly; the opening order gives q_c; at 81 the along times follow R to sampling error. Then T and X at 81 and 729, where run 083's sampling took about a minute a graph, so it fits in the cloud or on BernoullisBook while the M5 is busy. 6561 took 1.5 to 6.4 hours a graph in run 083 and is the M5's. The recorder is a specification for Claude Code, not written here. Not sealed. The look above is committed with this probe, so a later seal has it on record as known.

Added the same day — the table's shape is the square's (the owner's remark: the places that need coupling are tied to the symmetry of the square the carpet is built from).

• Three rows, and the square fixes the number. R keeps one event, left joined to right, and that event picks a direction. The square's eight symmetries carry it to top joined to bottom, and a route that turns enters by one side and leaves by another. Close the crossing event under the symmetries and under turning and what comes out is pairs of sides. Under the eight symmetries the pairs fall into three classes: opposite, adjacent, and the same side twice. The third was left out above. It is the route that enters and leaves by one side (the explainer's "in and out"); its event is two parts of one side joined inside the block.
• Why run 093 found nothing. It looked for a symmetry of the one-block map and found that on 34 of 49 polynomials the map commutes with nothing but itself. The square's symmetry is not a symmetry of R. It relates R's event to the events R dropped, and the coupling is what puts it back (OPS.md, "The operator algebra you choose determines what you cannot see").
• To second order the table is six numbers, each with a symmetry label. A block's six times (two opposite pairs, four corners) can vary in two fully symmetric ways, all opposite pairs together and all corners together, and in four that are not: along against across; corners alternating round the block; and a direction with two components, which corner is early. A symmetric law has one variance for each kind and one cross term between the two symmetric ones. Six numbers, five in units of the width.
• A neighbour. A triangle has one class of pairs of distinct sides; a square has two. One class cannot give two rates and two can. The usual account of why gaskets are exact and carpets are not is the size of the contact between blocks; this counts something else, and the two are not separated here.
• What the census cannot say. All 63 patterns were built with the full symmetry. A pattern that keeps the two mirror flips and loses the diagonal one gives along and across different laws. Which measured regularities split with it is a control the census has no member for.

Where it breaks. Symmetry gives the table's shape and its labels, not its values, and it does not make a side a point. Along a side the nine rows fall into five mirror classes (0 with 8, 1 with 7, 2 with 6, 3 with 5, and 4), and a table resolved by row has an entry for every class of pairs of those. The ladder's two sets, {1, 4, 7} and {2, 4, 6}, are two choices of three of the five. The exact level-1 weight of each row is computed inside rg058.py's transfer matrix and summed away (PG-01, status of 2026-10-03).

Kept out on purpose. The literal reading — draw a different chance p for every block and let neighbouring blocks pull theirs together — is a different model, a random field of dials. It is nearer P-052's gradient and is not what the stored samples ask for. Not filed here.

Lead 1 against run 097 (2026-10-04, codifier). Run 097, falsified. With blocks passed by the table's rule the lattice's lead at 729 is 0.62 widths on T and 1.68 on X, not under 0.3. The second half (dropping the corners leaves more than half the lag) cannot be scored as worded: the like-for-like rule without corners lags by 1.32 and 2.45, more than run 083's rule did (0.76 and 0.87); the corners move it by 0.70 and 0.77. The table was read with the opposite pairs on middle thirds, which are later than the full sides by 0.42 and 1.30 widths in median; the same table with full sides for the opposite pairs has not been computed. Leads 2 to 4 are untouched: the medians of the events in widths all move later between 729 and 6561, by 0.05 to 0.23, and the corner's shift falls by 1.1 and 1.4 errors.