The crossing threshold is the weakest link of the strongest wall — does the number of near-strongest walls set the lean of the threshold pile, and its turn with size?

---

type: lab

field: physics

object. Bond percolation on a carpet: a square grid of side L = 9^g whose cells are kept or removed by a 9×9 generator iterated g times; every bond between two kept cells carries an independent uniform number u in [0, 1]; a bond is open at dial p when u < p. A sample's threshold t is the fraction of bonds open when the left column is first joined to the right column. Two exact facts about t hold on every planar graph. (1) t = min over left–right paths of the largest u on the path (the bottleneck of the best path). (2) t = max over top–bottom walls of the smallest u in the wall, where a wall is a set of bonds whose removal separates left from right (the weakest link of the strongest wall). (1) and (2) are the same number by bottleneck duality; the paths live on the lattice, the walls on its dual, where a wall is a top-to-bottom route that walks through holes for free and crosses a bond at a cost. The minimum walls at generation g — the fewest bonds a wall can cross — number N(g), and C(g) is their size; both are exact counts (run 054). The pile is the distribution of t over samples; its middle p*, width w (the 0.2–0.8 quantile gap) and lean g₁ (the standardized third cumulant, as a k-statistic) are read at L = 81, 729, 6561 on 8,000 samples per size.

friction. The lean is positive on every carpet at 729 and 6561 and zero on the square lattice at every size, which is self-dual: its paths and walls are the same objects, so (1) and (2) give one distribution and no lean. On the carpets the paths and the walls are different objects and the pile leans. Between 729 and 6561 the lean rises on most carpets, is flat on some, and falls on T and X1 in some sample sets and not others. No account of the lean's sign or of its change exists. The exact one-block recursion — the carpet reduced to one bit, "spans or not", and iterated — leans more at every depth on all 63 census generators and never turns. Run 054 found that among carpets of one minimum-wall size, the one with more minimum walls has the lower exponent 1/ν (confirmed, H > U > V); run 071 found the same ordering by N₁ across the census (Spearman −0.76, sealed); run 107 found it on six fresh patterns, six of six. So the number of minimum walls already orders the sharpening. Whether it also sets the lean — the threshold as the maximum over many near-strongest walls of their weakest link, an extreme-value statistic whose asymmetry depends on how many walls compete and how dependent they are — is the question. The walls are not independent: on H, N(g) = 2^{C(g)} exactly, 2^64 at L = 729, sharing bonds everywhere.

background.

The lean by size, fourteen carpets and the square (run 073's k-statistics on run 072's files, 8,000 samples per size; errors are bootstrap; the last column is the 729 → 6561 change with its combined error).

The turn, replicated (runs 081, 092). T's 729 → 6561 change in three independent sample sets: −0.191 ± 0.038 (8,000 per size), −0.111 ± 0.031 (16,000), −0.022 ± 0.040 (8,000); inverse-variance mean −0.110 ± 0.021, χ² = 9.3 on 2 degrees of freedom. X1's in two sets: −0.116 ± 0.045 and +0.117 ± 0.035. Run 092: the bootstrap error of g₁ is right within a file (block-to-bootstrap ratio 1.00 pooled over ten files), and files of one graph at 6561 differ beyond their errors by a factor r₈ ≤ 1.42 (sealed bound, confirmed) — the between-seed spread of g₁ at 6561 is up to 1.4 times the within-file error.

H at larger sizes (runs 102, 105; the threshold variable there is k/M). At L = 2187 on 1,500,000 samples, g₁ = +0.351043 ± 0.002596, 2.48 combined errors below the value the one-exponent form fitted to the sizes 81–729 predicts for it; at L = 19,683 on 48,000 samples, 1.00 error below the form's prediction. In every comparison the record holds, H's lean at a new size sits below what the smaller sizes extrapolate.

The walls, counted exactly (run 054; C(g) the minimum-wall size, N(g) the number of distinct minimum walls).

Run 054 (confirmed): at one C, more minimum walls, lower 1/ν (V 0.477 < U 0.524 < H 0.547). Run 071 (confirmed): Spearman(1/ν, N₁) = −0.76 over the census. Run 107 (unanticipated on its margin, six of six on its ordering): the six cut-2 patterns with N₁ = 4 at 0.292–0.302, with N₁ = 16 at 0.256–0.259.

The one-block recursion. Its g₁ rises at every depth on all 63 generators (runs 075, 081); it has one wall and one path.

Coincidences this case leans on. (a) The square's lean is zero at every size: explained — self-duality makes paths and walls one object. (b) Every carpet leans positive: unexplained; the case asks whether (2) explains the sign. (c) More minimum walls, lower exponent, now on 19 of 19 cut-2 patterns and across the census: partly explained — the codifier read the cut-2 generators on 2026-10-19/20 and found N₁ = 4 exactly when the wall of holes runs through the midline column and 16 when it runs through a corner or off-axis, i.e. N₁ counts the wall's freedom to be drawn; why fewer ways to be cut means faster sharpening is not explained. (d) The one-block recursion never turns: explained — it has one wall; a turn needs more than one.

Must predict. Run 109, sealed and not yet computed, reads on stored samples: for each sample, besides its threshold t, the second crossing t₂ — the fraction of bonds open when the left and right columns are first joined after the single bond that completed the first crossing is deleted and the same bond order continues. t₂ ≥ t always; the gap (t₂ − t)/w in widths is small when many near-strongest walls (equivalently many near-best paths) compete and large when one wall dominates. Run 109 reads it on H, U, V, T, X, X1 and the square at 729 and on T at 6561, re-deriving each stored file from its seed.

1. The mechanism. State what identity (2) — the threshold as the maximum over walls of the weakest link — predicts for the sign of g₁ on a carpet with many near-strongest walls, and why the same identity gives zero on the self-dual square. Say whether the prediction depends on N(g) alone or on how the walls overlap, and what it predicts for a lattice where every wall shares the same few bonds (the one-block recursion's limit).
2. The run. Run 109's quantities, at L = 729: (i) the order of the mean gap in widths across H, U, V (one minimum-wall size, N(1) = 16, 48, 64), with a direction; (ii) the sign of Spearman(t, gap) within a file on T; (iii) whether the lean of t₂'s pile is above or below the lean of t's pile on T at 6561, and by how much in combined errors. Give each as a value or an interval with stated coverage.
3. One computation. Name one exact computation, at most a few hours on a laptop, whose result your account and its strongest rival predict differently. The diamond hierarchical lattice (the exact rung of 2026-10-17) is available: its crossing distribution is exact at every level, its walls are enumerable, and its limiting lean is −0.1042 (two paths of two bonds) and −0.3717 (three paths of two bonds) — negative, where the carpets' is positive.

Provenance. Lean values from cumulant073.json (run 073, k-statistics, 1,000-resample bootstrap) on run 072's files; replications from RESULTS081 and run 092's calibration note; H at 2187 and 19,683 from RESULTS102 and RESULTS105 (threshold k/M, 30-block errors); wall counts from RESULTS054 (brute force and contracted dual agree where both fit); N₁ orderings from RESULTS071 and RESULTS107; the one-block statement from runs 075 and 081; the diamond limits from RESULTS106. Nothing here is a limit except where marked; every lattice number is one sample set unless a replication is quoted.

Instrument notes. - Objects. Three kinds of number: sampled lattice values (the tables), exact counts (walls), exact limits (the diamond rung). The one-block recursion is a fourth object, a map on one number; its lean is not a lattice lean. - The lean. g₁ is a k-statistic of the per-sample threshold; runs 072–081 use perc's Beta-drawn p_c as the variable, runs 090–105 use k/M; the two agree on the lean to the third decimal at 729 (RESULTS081, two columns, difference 1.9 × 10⁻¹⁰ on T). Bootstrap errors are right within a file; between seeds at 6561 the spread is up to 1.42 × the within-file error (run 092), so a 6561 difference under 1.5 combined errors is not a difference. - The turn. T's fall is −0.110 ± 0.021 pooled across three sets that disagree with each other (χ² 9.3 on 2); X1's two sets have opposite signs. A reader should treat "T turns" as established on the pooled value and "X1 turns" as open. - Which way an unconverged estimator reads. The lean's change with size is itself a correction to scaling; three sizes a factor of 9 apart cannot distinguish a turn from a slow approach with a bump (run 074 on H at five sizes: unanticipated, neither clean). - The walls. N(g) counts distinct minimum walls; near-minimum walls (one or two bonds larger) are far more numerous and are the ones the maximum in (2) competes among at a sample's threshold. No count of near-minimum walls exists in the record. - The pending reading. t₂ is defined by deleting one bond — the one whose addition completed the first crossing — and continuing the same order; a sample whose first crossing is completed by a bond that is the only link between two large pieces has a large gap. The gap is in widths of the first crossing's pile.

Not part of the case — the codifier's sealed predictions (2026-10-20, before any panel runs and before run 109 computes; the first three are run 109's sealed prediction)

1. The mean gap (t₂ − t)/w at 729 orders H > U > V — fewer minimum walls, larger gap — with the H–V difference above 3 combined errors.
2. Spearman(t, gap) within a file on T at 729 is negative (a late first crossing has many near-complete routes and a small gap), below −0.2.
3. The lean of t₂'s pile on T at 6561 is above the lean of t's pile by more than 2 combined errors: the turn lives in the first crossing, not in the second.
4. The panel's strongest rival to the wall mechanism will be the localisation of PN-005/PN-006 (the fall sits in the mixing of the parts split by the coarse crossing), and the House will not choose between them on the numbers above; it will ask for the gap reading, which is what G1 says the panel is for.

Scored 2026-10-20, after the panel ran (PN-009)

The House against the case. The case leaned on "more minimum walls → more competitors → the lean"; the House read the case's own table and found the lean at 729 is H +0.349 > V +0.277 > U +0.146 while N₁ is 16 < 48 < 64 — not monotone — and computed the diamond (3,3) rung (27 minimum walls against 3 paths) leaning −0.162. Its rule: the sign follows the minimum-wall size C against the shortest-path length l (C < l positive, C ≥ l negative); on the diamond grid it reads (2,2) −0.1047, (3,2) −0.3716, (2,3) +0.111, (3,3) −0.162, (4,2) −0.507; on the carpets C(g) ≪ l always, consistent with every carpet positive and discriminating nothing there. On run 109's reading it made a correction the codifier accepts: t₂ is the maximum over walls through the completing bond of their second-weakest link, so the gap reads redundancy local to that bond, not the global count of competing walls; and the paired difference lean(t₂) − lean(t) should carry a paired bootstrap error, not combined errors.

The codifier's faults. (1) G6: the case leaned on an N₁-ordering of the lean it never checked against the lean column of its own table — the ordering is false at 729. (2) The sealed prediction 4 ("the House will not choose between the mechanisms and will ask for the gap reading") missed: the House chose, against the case, and ordered eight tests with the gap reframed. (3) Run 109's clause (C) names combined errors where the House is right that a paired bootstrap applies; the seal stands as written and the lab's paired error is added as a reported item.

The codifier's sealed predictions, scored. 1 (gap orders H > U > V, H − V > 3 errors): miss — no order resolved. 2 (Spearman below −0.2): sign right, magnitude short (−0.149). 3 (lean(t₂) above lean(t) by 2 errors): miss — zero within error. 4 (the House will not choose): miss. Run 109 scored unanticipated on 2026-10-20.

What leaves the panel. Run 110 (sealed 2026-10-20): the House's TEST 1, the diamond (b,s) grid at (2,4), (3,4), (4,3), scored on the House's own sign rule. Next case named by the House: the sign of g₁ against C(1) versus l across the 63 generators — on the carpets l = 8 at generation 1 for every pattern and C(1) ≤ 5, so the rule predicts positive everywhere and the census cannot falsify it; the diamond grid can.

After run 110 (2026-10-20, codifier). The House's sign rule, sealed as run 110's prediction, is falsified by one exact rung: (3,4) has b < s and leans −0.0606. Seven of eight rungs obey it. So neither the case's wall count nor the House's wall-size-against-path-length sets the sign by itself; the lab's table shows λ(b, s) = λ(s, b) to twelve digits on three pairs with different p* and g₁, which PG-03 takes up as an exact question. The case stays open on its original question with both proposed mechanisms scored and neither standing.