The skewness of the crossing threshold turns back on some carpets between L = 729 and 6561, while the exact one-block recursion's keeps rising — where does the turn come from?

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type: lab

field: physics

object. Take a 9×9 grid of cells, keep 64 and remove 17, keeping the outer ring and a pattern symmetric under the eight symmetries of the square. A carpet repeats one pattern at every level, so after k levels it is a 9ᵏ × 9ᵏ lattice. Put bond percolation on it, add bonds one at a time in uniformly random order, and record the fraction present when an open left-to-right crossing first appears. That fraction is the sample's threshold. Over many samples the thresholds have a distribution; g₁ is its skewness (third cumulant over the variance to the 3/2). For a pattern, C₁ is the minimum number of bonds whose removal disconnects the left side of the 9×9 block from the right, and N₁ is the number of such minimum cuts.

The same pattern also defines an exact one-block recursion. R(p), the probability that one 9×9 block is crossed when each bond is open with probability p, is a polynomial. Iterating it k times gives the crossing probability of the hierarchical model at depth k, which is the distribution function of a threshold, and its skewness is exact.

friction. On the lattice, g₁ rises from L = 81 to 729 on thirteen of fourteen carpets. From 729 to 6561 it falls back on two (T and X1, by 5 and 2.6 errors), keeps climbing at the same step on two (X and Z), and changes little on the rest. The exact one-block recursion, iterated to the same depths, rises at every depth on all 63 patterns, T and X1 included, with no turn anywhere (it converges to a limit g∞). The lattice turn is not in the one-block recursion. Where does it come from, and what would show it sample by sample?

background.

g₁ of the threshold, lattice (8,000 samples per size, error about ±0.03 per value) and exact one-block recursion at the same depths (k = 2, 3, 4 ↔ L = 81, 729, 6561):

• H at every factor of 3 (fresh samples, 48,000 per size): g₁ = +0.161, +0.258, +0.322, +0.384, +0.404 at L = 81, 243, 729, 2187, 6561. One rise, steps shrinking (+0.097, +0.064, +0.062, +0.020), no alternation with generation.
• The one-block limit: g∞ is positive on all 63 patterns. Its sign is fixed by the minimum cut against the shortest crossing (run 075: C₁ ≤ 5 against 8 bonds). Across the 63, g∞ has Spearman −0.81 with C₁ and −0.77 with y₉ (the recursion's exponent).
• The square, which has no levels, has g₁ near zero at every size.

Must predict. A run will sample T (turns back) and X (keeps climbing) again at L = 729, recording, beside each sample's threshold, one further quantity read from the same bond-addition sequence. It must be recordable during the union–find pass, at no more than a few times its cost: a time, a count, or a location of some event before or at the crossing. The samples are then split by that quantity, and g₁ is read in each part. 1. Which quantity? Name one, define it so it can be computed per sample, and say why it is the one the turn lives in. 2. What it shows. For T and for X at L = 729, give g₁ in the part of the samples your quantity selects, against g₁ over all samples (T +0.363, X +0.267). Give a sign with a probability, a value interval with stated coverage, and what the null (the quantity carries nothing about skewness) predicts.

Provenance. Lattice g₁ from runs 072 and 073: unbiased k-statistics of the per-sample thresholds k/M, 8,000 samples per size, bootstrap errors (2,000 replicates). Recursion values from run 075: exact skewness of the iterated crossing polynomial, computed to depth 20. H at factor-3 resolution from run 079 (fresh seeds). 1/ν from runs 051–071 by the width estimator, on the same samples.

Instrument notes. - Objects. The lattice columns are sampled lattices at the sizes measured. The recursion columns are exact one-block values with no error. 1/ν is a sampled-lattice value. None of them is a limit. - Error scale. About ±0.03 per lattice g₁ at 8,000 samples; differences of two sizes about ±0.04. A two-error pattern read from one sample set has already failed to replicate once: H's apparent alternation with generation (run 074, 2.1 errors) vanished on fresh seeds at six times the samples (run 079). T's fall (−0.191 ± 0.038) and X1's (−0.116 ± 0.045) are single sample sets. - Sizes. Only L = 81, 729 and 6561 are feasible for the factor-9 carpets. L = 59049 needs on the order of 100–180 GB. - Same samples. g₁ and 1/ν come from the same samples, so they are not independent evidence. Most carpets' width local slopes fall with L, and a falling local slope makes an endpoint-slope 1/ν read high. - Reproducibility. Run 072 found g₁ reproducible across independent seeds, and stable with size on 10 of 14 carpets.

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Not part of the case — the codifier's sealed predictions (2026-10-01, before the panel runs)

The codifier's own lead is line 3 of P-048's hump programme. Per sample at L = 729, record the time at which each top-level minimum-cut face is first bridged. Conditioning on early bridging removes the skewness on T and does less on X.

Predictions of what the panel will do: 1. At least two seats name a quantity tied to the minimum cuts (their bridging, or the cluster at the cut) without being told to. 2. At least one seat argues that T's fall is not real, as finite-size or sampling noise, and cites the error scale or run 079. 3. The House's chosen quantity differs from the codifier's lead in what it records, not only in how it is thresholded.

Scored against the panel record once it is filed. The panel's g₁ forecasts are scored against the run this case shapes.

Settings proposed (v5-plan §12.2): S1, five seats in round 1 (Ledger, Framing, Hypothesis, Lateral, Bound; the target is numeric). S2, one round and a House, since no lab step lands between rounds. S3, the record loaded (PN-003 loaded, PN-004 not). S4, one draw.

Scored 2026-10-01, after the panel ran (PN-005)

The codifier's predictions of what the panel would do, scored against the record as written: 1. "At least two seats name a quantity tied to the minimum cuts" — missed. No seat did. Ledger (a coarse block-graph crossing time), Lateral (block footprint) and Bound (bonds before crossing) all named quantities tied to the block structure, not to the minimum cuts. The House adopted Ledger's, and its ASSERTED 1 brings in C₁ only to rank T against X. 2. "At least one seat argues T's fall is not real, citing the error scale or run 079" — hit. Framing and Hypothesis both did, and the House made it the verdict. 3. "The House's quantity differs from the codifier's lead in what it records" — hit. It records the time the coarse graph of top-level blocks first crosses, not when the minimum-cut faces are first bridged. Its FORCED 3 names what the coarse quantity sees: a lattice crossing that runs through a block's top or bottom edge without crossing it left to right, a route the one-block recursion cannot represent.

A gap in this case's instrument notes, found by the panel. The notes said that H's two-error alternation "failed to replicate" but not that run 079 also checked the bootstrap error and found it was not too small (its six-block calibration). The House inferred under-coverage and leaned its verdict on it. A case that cites a failed replication should also give the calibration of the error bar, so the seats can tell a fluctuation from a miscalibrated error.

Run status after PN-005: the House's order is significance first. Next: a fresh-seed rerun of T and X1 at L = 729 and 6561 (run 079's design), and the same pass can record t_coarse for the F3 split. Run 080 (census completion) is running and is unrelated.

Run status after run 083 (2026-10-01, codifier): the F3 split ran on T and X at 81, 729 and 6561 (run 083, unanticipated). The route the House named (FORCED 3) is real and is the majority route on both patterns (0.72–0.89 of samples cross the lattice before the coarse block graph crosses); it sorts skewness on T at 729 beyond selection (F3 confirmed on T, both intervals hit) but its weight is the same on T and X at 729 and the order reverses with L (F4 falsified on its intervals; ASSERTED 1 falsified). Neither part's 729→6561 step on T clears 2 errors, so the sign of Δ does not resolve the turn; if the turn holds, that is the House's second branch. On fresh seeds the pooled fall is −0.062 ± 0.035, not −0.191; run 081 decides whether the case's premise holds at its stated size. The case stays open on 081. Seat rows: panels/SEAT-RECORD.md, "scored against run 083".

Run status after run 081 (2026-10-02, codifier): the fresh-seed rerun (run 081, unanticipated). T's fall replicates (−0.111 ± 0.031, −3.6 errors; A confirmed); X1's reverses (+0.117 ± 0.035; B falsified). The case's premise holds in sign on one carpet, not two, and its size is open (081's −0.111 is 1.6 combined errors from 073's −0.191; three-set size ±0.045): "two carpets turn" becomes "T turns". The House's verdict — significance first — reads, after 081, as the right order, and its F1 forecast is confirmed while Framing's and F2 are falsified. What the rerun adds that no seat forecast: X1's two independent measurements are 4.1 combined errors apart, and T's three (runs 073, 081, 070) are inconsistent at their stated errors (χ² 9.3 on 2), so the bootstrap error of the turn's size understates the between-set spread on both carpets, by a factor the data do not fix — the House's adopted premise about the error bar, contradicted on H by run 079, is supported on the turn's own quantity. T's turn is negative in all three sets (2.5 errors, scaled) and known in size to about ±0.045. The case stays open on the mechanism (run 089: the difference between T's step maps lives in the early-crossing tail) and on the error calibration; it is not reopened on X1. Seat rows: panels/SEAT-RECORD.md, "scored against run 081".

Continuation 1 — packet for PN-005 (prepared 2026-10-03, codifier; lab report extended the same day with the eight-seed redraw; not yet run)

Warrant (docs/v5-plan.md §12.7). (a) Runs 081 and 083 were proposed by this panel: the House's verdict asked for the fresh-seed rerun, and its diagnostic is the coarse-crossing split. (b) Both score its forecasts (panels/SEAT-RECORD.md, "scored against run 083", "scored against run 081"; scored before this packet, as §12.7 requires). (c) Both change what the House relied on: it treated two carpets as turning, and one does; it expected its channel to carry a minority of samples and to separate T from X, and it carries the majority and does not. Same five seats, cold to the framework, given the case above, their round and the House's, and the lab report below; one round and a House; marked "continuation" in the export and the seat record.

Lab report (numbers only)

Fresh seeds, T and X1 at L = 729 and 6561 (run 081; 16,000 samples per size). - T: g₁ +0.360 at 729, +0.249 at 6561; change −0.111 ± 0.031. Excess kurtosis +0.311 ± 0.069 at 729, +0.006 ± 0.052 at 6561. - X1: change +0.117 ± 0.035. Excess kurtosis +0.393 ± 0.073 at 729, +0.441 ± 0.073 at 6561. - T's change in three independent sample sets: −0.191 ± 0.038, −0.111 ± 0.031, −0.022 ± 0.040. Their inverse-variance mean is −0.110 ± 0.021; χ² about it is 9.3 on 2 degrees of freedom. - X1's change in two independent sample sets: −0.116 ± 0.045 and +0.117 ± 0.035. - Width local slope 729→6561 from the fresh samples: T 0.349 ± 0.005 (the case's table has 1/ν 0.3743); X1 0.532 ± 0.005 (0.5359).

The coarse-crossing split, T and X at L = 81, 729, 6561 (run 083). t is a sample's threshold; t_coarse is the fraction of bonds present when the graph of the 64 top-level blocks, a block counted open once it is crossed side to side by its own bonds, is first crossed. - Share of samples with t < t_coarse: T 0.764, 0.854, 0.875; X 0.721, 0.852, 0.889. - Share with t = t_coarse exactly (one bond completes both): T 12.4%, 10.2%, 10.2%; X 6.7%, 5.5%, 5.3%. - g₁ within the part t < t_coarse: T +0.129, +0.157, +0.121; X +0.016, +0.159, +0.269. - g₁ within the part t ≥ t_coarse: T +0.094, +0.399, +0.230; X +0.153, +0.213, +0.403. - T's change 729→6561 within each part: −0.036 ± 0.038 (t < t_coarse); −0.168 ± 0.100 (t ≥ t_coarse). - At 729, g₁(t ≥ t_coarse) − g₁(t < t_coarse): T +0.241 ± 0.062; X +0.053 ± 0.060. - Open top-level blocks at the crossing, at 729: about 37 of 64 where t < t_coarse; 49 (T) and 45 (X) where t ≥ t_coarse.

Is the lattice one map applied repeatedly (run 089; stored samples). Let g_L be the function carrying the threshold distribution at size L to the distribution at 9L, quantile by quantile. - X: g₈₁ and g₇₂₉ differ at 19 quantile nodes with rms 3.09 errors (1.06 between two halves of one file); all 19 differences have one sign and are about 7% of the 729 width. - T: the difference between g₈₁ and g₇₂₉ is largest at the lowest quantile node, 7.4 errors.

How good is the error bar. - Within one file the bootstrap error of g₁ matches the spread between blocks of the file (ratio 1.00 over ten files). - Three files of T at 6561 give g₁ = 0.172 ± 0.026, 0.282 ± 0.028, 0.248 ± 0.020; the first was drawn on one machine and the other two on another. - The same sampler on both machines: the file giving 0.172 was redrawn in full on the other machine from the same seed, and every one of its 8,000 thresholds was reproduced exactly; eight further 200-sample redraws across the two machines likewise differ in no sample. - Eight fresh seeds of T at 6561 on one machine (8,000 samples each): g₁ = +0.293, +0.233, +0.265, +0.246, +0.292, +0.290, +0.266, +0.233, bootstrap errors 0.025–0.031; the spread of the eight over the RMS of their errors is 0.90. - Pooled over the eleven files of T at 6561: g₁ = +0.254 ± 0.011. Pooled over three files at 729: +0.345 ± 0.019. The fall 729→6561 on the pooled files: −0.091 ± 0.022. - On four other carpets at 6561, three seeds each on one machine: pairwise differences in g₁ have rms 0.86 errors, largest 1.73. - On H's alternation statistic (run 079) the spread between six blocks was 0.090 against a bootstrap error of 0.130.

Must predict (continuation)

A run will sample T and X again at L = 729 and 6561 and record, for each sample, something about its 64 top-level blocks beside the threshold: for example each block's own crossing time, or which blocks are open when the lattice first crosses. From those records a statistic of two or more blocks can be formed. 1. Which statistic? Define one that can be computed from such per-sample block records, and say why T's fall between 729 and 6561 should show in it and X's rise should not. 2. What it shows. Its value on T and on X at 729 and at 6561: the sign of T minus X at each size with a probability, an interval with stated coverage, and what the null predicts. 3. Or that it cannot. If you hold that T's fall is an artefact of sampling or of the estimator, say which number in the lab report would have to change on a further fresh draw, and by how much.

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Not part of the packet — the codifier's sealed predictions (2026-10-03, before the continuation runs)

The codifier's own lead: T's fall is carried by the exact-tie samples, t = t_coarse, where one bond completes the lattice crossing and the coarse crossing together; T has twice X's share of them at every size. Split three ways (below, tie, above), the tie part's g₁ falls from 729 to 6561 on T and the other two do not. Nothing has been computed for this; the stored samples of run 083 could give it.

Predictions of what the panel will do: 1. At least two seats build their statistic on the exact ties or on the count of open blocks at the crossing, the two places the lab report shows T and X differing. 2. At least one seat takes branch 3 and argues from the three T files that the fall's size is a machine or file effect. 3. The House's statistic is a correlation between neighbouring blocks, as its own second branch said, and not a per-block quantity.

Settings proposed (§12.2, §12.7): the same five seats (Ledger, Framing, Hypothesis, Lateral, Bound); one round and a House; continuation mode, so seats see their own round; record loaded as in the first round; one draw. Appended to PN-005 as "Continuation 1" when filed.

Status, 2026-10-03 (codifier), after run 092. The case body above is as the panel saw it and is not edited. Its "−0.191 ± 0.038" is run 073's reading from one sample set; run 092 (confirmed) redrew that set in full on the other machine with every threshold reproduced, drew eight fresh seeds, and pooled eleven files: T's fall from 729 to 6561 is −0.091 ± 0.022, 4.2 errors from zero. The premise stands at half the size the case states. The continuation packet carries the pooled figure.

Scored 2026-10-03, after the continuation ran (PN-006)

The codifier's sealed panel predictions — 2 of 3 held. 1. Held. Three seats built their statistic on the count of open blocks at the crossing (Ledger's n_open, Hypothesis's B, Bound's K); none on the exact ties. Ledger went the other way on the ties, asserting t ≤ t_coarse always, which the House's own arithmetic on the shares refuted (t > t_coarse in 2–21% of samples). 2. Held. Hypothesis argued from the three T files (χ² 9.3 on 2) that the quoted error is file-specific and the fall's size is inflated; the House adopted a 1.5× inflation and the checker flagged the resulting ±0.033 against the report's ±0.022 (H2-6). 3. Failed. The House's statistic is the laggard gap — the largest minus the second-largest block crossing time on the first completing block path, in width units — an order statistic of the blocks on a path, not a correlation between neighbouring blocks.

The codifier's own lead (the exact ties carry T's fall) was taken by no seat and is untested; run 083's stored samples can give it and it stays sealed here.

What the House did with the record. It refuted, as stated, the premise every seat had built on in round 1 — that the turn originates in what the one-block recursion omits — on the lab's own numbers: the recursion-illegal part grows with L (0.76 → 0.85 → 0.87) and its skewness is flat on T, while the fall sits in the shrinking coarse-respecting minority. It dropped X1 from the phenomenon pending a pre-registered six-file pool, and corrected its own round-1 error (U′, not T, has the smallest cut). Its F1 is recordable by perc_blocks.c as built (each block's along time per sample; the completing path from the block graph's union–find), and is the natural first measurement on that instrument once run 094 is scored. Its "next case" — the quantile map g_L itself — is run 089's object and PG-01's L14.

Panel forecasts go to panels/SEAT-RECORD.md as pending; House final F2 (the fall persists at −0.13 to −0.04 over five further files) and F3 (X1 straddles zero over six files) are scorable against runs 092 and a future X1 pool; F1 and F4 wait on the recorder run.

Readings on stored samples, 2026-10-03 evening (R27_tree_groups/carpet_perc/READING-PN006.md). The codifier's lead (the tie part carries T's fall) is unresolved on run 083's samples: the tie part falls by −0.164 ± 0.099 (1.7 errors), the part above by −0.220 ± 0.188, the part below by −0.036 ± 0.047. The House's localisation in t ≥ t_coarse is −0.168 ± 0.090, 1.9 errors. Bound's K-split forecast holds in sign at 729 (+0.358 ± 0.073 for T minus X) and misses its interval above. The recorder's first run records the same splits on fresh samples.

Run status after run 097 (2026-10-04, codifier). The recorder's first run (falsified on its own clauses; PG-01 L15) carries the splits on fresh samples of T and X at 729 and 6561. T's g₁ changes by −0.068 ± 0.040 on this pair of files: −0.061 ± 0.035 in t < t_coarse, −0.029 ± 0.081 on the ties, −0.047 ± 0.285 above, −0.009 ± 0.082 in t ≥ t_coarse. Pooled with run 083: −0.050 ± 0.026 below and −0.072 ± 0.063 at or above; the ties pool to −0.083 ± 0.063. Neither the House's localisation (t ≥ t_coarse) nor the codifier's (the ties) is resolved, and the share of the part below changes from 0.850 to 0.873, so the fall can sit in the mixing of the parts. The House's statistic, the laggard gap, is lower on T than on X at both sizes (−0.167 ± 0.011, −0.235 ± 0.017) and falls on T by 0.060 ± 0.011 while X's does not move; the g₁ split at the median gap is +0.236 ± 0.043 on T and +0.017 ± 0.049 on X at 729, and on T it does not move with size (−0.023 ± 0.059) while the gap does. So the gap separates T from X and is not shown to carry the fall. Forecast rows: panels/SEAT-RECORD.md (F1 sign hit, intervals missed; F4 hit; Ledger miss; Lateral hit on T, miss on the contrast; F2 at one of five files).