One round and a House, closed after House 1 (docs/v5-plan.md §12.2, S2). Run by the runner across two Claude Code sessions: the first was ended by a safeguard flag on the House prompt after checker.r1 had returned; a second session resumed from the inbox and returns/ and dispatched house1, checker.house1, backprop and the matcher. Record NOT loaded (S3 alternation after PN-008). Live roster not compared (--no-live-roster). Body below is the runner's export of 2026-10-20, unedited beyond the id. The blindness caveat above is the dispatcher's own statement and is part of this record. The codifier's post-panel reading is in cases/CASE-31-best-wall-weakest-link.md; the codifier's sealed predictions were stripped from the inbox (0 occurrences in the inbox copy).

One round and a House, closed after House 1 (docs/v5-plan.md §12.2, S2). Run by the runner across two Claude Code sessions: the first was ended by a safeguard flag on the House prompt after checker.r1 had returned; a second session resumed from the inbox and returns/ and dispatched house1, checker.house1, backprop and the matcher. Record NOT loaded (S3 alternation after PN-008). Live roster not compared (--no-live-roster). Body below is the runner's export of 2026-10-20, unedited beyond the id. The blindness caveat above is the dispatcher's own statement and is part of this record. The codifier's post-panel reading is in cases/CASE-31-best-wall-weakest-link.md; the codifier's sealed predictions were stripped from the inbox (0 occurrences in the inbox copy).

PN-nnn — 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? · run 2026-10-20, lab roster (physics) · one round, closed after House 1, no lab step · resolved: local

Case id. CASE-31

Case

Type. lab / physics

Presenting 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.

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.

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.

Premise ledger

• [derived, load-bearing] t = min over left-right paths of the largest u on the path (bottleneck of best path)
• [derived, load-bearing] t = max over walls of the smallest u in the wall; identical to (1) by bottleneck duality on planar graphs
• [assumed, load-bearing] The square lattice is self-dual, so paths and walls are the same objects and the lean is exactly zero
• [measured] Square lean measured: +0.008, +0.009, +0.011 (each +-0.011 to +-0.017), consistent with zero at all sizes
• [assumed, load-bearing] The lean is a property of the maximum over many near-strongest walls (extreme-value asymmetry), not of some other feature of the carpet, e.g. the fractal geometry or the small-sample bottleneck structure
• [assumed, load-bearing] Near-minimum walls (one or two bonds larger than minimum) are the ones competing at a sample's threshold, so N(g) of minimum walls is a proxy for the competition; no count of near-minimum walls exists
• [assumed, load-bearing] N1 (number of minimum walls at generation 1) is a good proxy for the number and overlap of competing walls at L = 729 and 6561
• [assumed, load-bearing] The ordering 'more minimum walls, lower 1/nu' (H>U>V, census Spearman -0.76, 19 of 19 cut-2 patterns) is a causal link to sharpening and carries over to the lean
• [assumed, load-bearing] The lean's change from 729 to 6561 (a turn on T) is a real finite-size trend and not a slow approach with a bump or a between-seed artifact (r8 up to 1.42; T's three sets disagree, chi2 9.3 on 2 dof)
• [measured] Bootstrap errors on g1 are correct within a file (block-to-bootstrap ratio 1.00 over ten files)
• [measured] The p_c variable (Beta-drawn) and k/M give the same lean (difference 1.9e-10 on T at 729), so the tables are comparable across runs
• [assumed, load-bearing] The one-block recursion (one bit, one wall, one path) is a valid model for 'few overlapping walls' and its behaviour (lean rising at every depth, never turning) transfers to the lattice question
• [assumed, load-bearing] The second-crossing gap (t2-t)/w is a monotone measure of the number of near-best competing walls/paths (small when many compete); it is not instead dominated by a single bridge bond between large pieces
• [assumed, load-bearing] Wall overlap (shared bonds, N=2^C on H) does not make the effective number of independent competitors collapse to a small number, so N(g) is informative

Number kinds

• [count] L = 9^g side length; generator 9x9 — structural, fixed by construction
• [count] N(g), number of minimum walls (e.g. 16, 2^64, 1.93e29) — exact integer count from enumeration (run 054); but note 4.0000 vs 4.0163/4.0189 log-ratio is a computed ratio, not a universal exponent
• [count] C(g), minimum-wall size (4, 16, 64; 3, 9, 27) — exact integer, fixed by structure (scales as 4^g or 3^g)
• [unclassifiable] ln N(g+1)/ln N(g) at 3->4 (4.0000, 4.0163, 4.0189) — ratio of exact counts, but no route to the limit stated; not shown to be a structural exponent
• [computed exponent] 1/nu per carpet (0.3743, 0.5508, 0.75 for square, etc.) — square 0.75 is exact (3/4 for 2D percolation, 4/3 nu); carpet values are numerical estimates with errors, none called universal here
• [free] g1 (standardized third cumulant) at each size, e.g. +0.363, +0.172 — set by measurement on 8,000 samples with bootstrap error; no structure predicts it
• [free] Change in g1 729->6561 (e.g. -0.191, -0.110 pooled) — a difference of measured values with errors; between-seed spread up to 1.42x within-file error
• [computed exponent] Diamond limiting leans -0.1042 and -0.3717 — exact limits from the hierarchical recursion (RESULTS106); computed, not dimensional; labelled exact so no error bars on the lean itself
• [count] Number of paths/walls in diamond variants (two paths of two bonds; three paths of two bonds) — integers fixed by the lattice definition
• [free, target] Run 109 mean gap (t2-t)/w in widths for H, U, V — to be measured on samples; the ordering is the target but the magnitude is set by data
• [free, target] Spearman(t, gap) on T — a sign of a rank correlation measured in the run
• [free, target] Lean of t2 pile minus lean of t pile on T at 6561, in combined errors — measured difference; between-seed spread r8 up to 1.42 applies
• [count] N1 values 4 and 16 for the cut-2 patterns; 16, 48, 64 for H, U, V — exact integers at generation 1
• [free] Spearman(1/nu, N1) = -0.76 — measured correlation over the 63-generator census, not structural
• [free] Mean-gap threshold: 1.5 combined errors for 6561 differences; r8 <= 1.42 — r8 bound is an empirical sealed bound set by calibration, not structural; 1.5 is a chosen decision rule
• [computed exponent] Chi-squared 9.3 on 2 dof for T replication — mislabelled by tag convention: a test statistic with fixed dof 2 (count) and measured value; unclassifiable as exponent, treat as a measured statistic
• [unclassifiable] Number of near-minimum walls (one or two bonds larger) — stated to be far more numerous than N(g) but no count or route exists in the record

Target form. behaviour — Whether the number and overlap of near-strongest walls (via identity 2) accounts for the sign and size-dependence of the threshold pile's lean, tested by run 109's second-crossing gap ordering across H, U, V, the sign of Spearman(t, gap) on T, and the lean of t2 versus t on T at 6561, plus one discriminating exact computation on the diamond lattice. Bound: not fired.

Bound note. Bound not added — target is a behaviour question with no threshold in it — outside the number-kind taxonomy, Bound not applicable.

Seat versions. Ledger v1 · Framing v1 · Hypothesis v1 · Lateral v1

Seat record. off (with/without comparison) · seat-record.md · 97 seat rows · SHA-256 a62b86d7f4f5bb8cbc2ef5ef4a901ff4c31efb5aa5220151d191cb83458f5b6a

Record packet. mode: off (with/without comparison) · full: packet-full.txt · 29 facts, cap 40 · SHA-256 548fbc804edc2d9caab34ce6f5554814314eb994d9f570b19d1caa9af1c2af76

Roster

_The seats that sat, with the axis text the House receives for each and the stance each was given (what it refuses)._
- Ledger (Ledger) v1
- axis: What is exactly true here by identity or definition, before any model or heuristic is applied — and what quantity can be computed two independent ways?
- refuses: to interpret. Writes the bookkeeping exactly — identities, counts, what is pinned and what is free — and stops.
- Framing (Framing) v1
- axis: Is the named object the actual object? Is the question as posed the right question — and has the target changed category (a count that varies with a parameter is not a count)?
- refuses: to answer the question as asked when the framing is wrong. Names the object the case should have been about.
- Hypothesis (Hypothesis) v1
- axis: What theorem, tool, identity or approximation does the case (or a seat) invoke — what does each one require, and is it satisfied here? Name the tool, state its preconditions, check each against the case.
- refuses: to let a result stand on an unchecked precondition. Does not accept that a tool applies because it is standard in this area.
- Lateral (Lateral) v1
- axis: Is this a known structure under a different name — and what theorem do the two share?
- refuses: to offer a resemblance without naming the shared theorem, and to present a cross-reference as if it were a result.

Round 1

Ledger (Ledger)
TAKE
Bookkeeping only. (1) and (2) are one number, so any statement about the lean is a statement about the distribution of a max over walls of a min over bonds, or equally a min over paths of a max. Exact: t = max_W min_{b in W} u_b. For one wall of size C, min of C uniforms has a left-skewed-in-p, right-tailed law (Beta(1,C)); a max over K such independent walls is an extreme-value pile whose skew is determined by K and C. Exact on the square: path and wall are the same object under duality, so t's law equals its own reflection t -> 1-t, and the third cumulant is exactly zero (the measured +0.008 to +0.011 are zero within error). On a carpet the dual is a different graph, no reflection symmetry, so zero is not enforced; that is all identity gives. It does NOT give a sign: reflection symmetry fails but the direction of failure is not an identity. N(g) alone is not the right count: the pile depends on the joint law of the wall minima, i.e. the overlap structure (on H, N = 2^C exactly, so ln N / C = ln 2 is pinned, a fully overlapping, product-like family, effective independent count far below N). One-block limit: one wall, one path, K = 1: no maximum over competitors; the lean is whatever Beta-type skew a single min gives and it can only grow with depth; consistent with never turning. Two computable ways for the same quantity: t from paths and t from walls, per sample (identity check for run 109). Also t2 >= t is exact; gap = 0 iff the completing bond is not the unique bottleneck link.

PREDICTION
SIGN: H, U, V mean gap ordered H > U > V (fewer minimum walls, larger gap), p=0.65
VALUE: mean gap (t2-t)/w on H at 729: 0.15 to 0.9 widths, 80% interval
NULL: no dependence of mean gap on N(1): the three means equal within error

ASSUMES
Gap is monotone in the number of near-best competing paths; first-crossing bond is typically a bridge between pieces.

CHANGES MY MIND
A per-sample exact enumeration showing that the gap is dominated by single-bridge bonds regardless of N (gap uncorrelated with the count of near-minimum walls), which would show the competition count is not what the gap measures.

Framing (Framing)
TAKE
The named object is mis-categorised twice. First, "the lean" is not one object. Identity (2) says t is a max over walls of a min over bonds. That is an extreme-value structure, and its skew depends on the effective number of independent competitors, K_eff. K_eff is not N(g). N counts minimum walls at the finest scale, whereas K_eff is set by how many walls lie within a few bond-lengths of the best at the threshold's u-level. On H, N = 2^C is almost entirely shared-bond redundancy, so the count of independent competitors is far smaller than N suggests. Minimum-wall count and K_eff are different quantities, and N1 is a proxy for sharpening (1/nu) only.

Second, a quantity that changes sign or direction with size, with three sizes a factor 9 apart, is not a count or a sign. It is a crossover trajectory. Asking whether N sets "the lean and its turn" treats a drifting correction as a number.

Third, the sign: a max over many weak-link mins gives a Gumbel-type, positively skewed pile only if the min is the dominant randomness. Positive lean on every carpet probably comes from the bottleneck of one dominant geometric bridge (the pending reading), not from competition. So I predict the gap (t2 - t)/w is NOT a clean K_eff monotone. It mixes bridge dominance with competition.

The one-block recursion is not the limit "few walls". It is a one-bit map and a different object.

The diamond lattice has negative lean, so (2) alone does not fix the sign. Geometry does.

PREDICTION
SIGN: mean gap (t2-t)/w ordered H > U > V at 729 (more minimum walls, smaller gap), p=0.6; Spearman(t, gap) on T negative, p=0.6
VALUE: mean gap on H 0.15 to 0.9 widths, 80% interval; the V-to-H difference is under 0.3 widths
NULL: no ordering of mean gap across H, U, V; Spearman(t, gap) = 0

ASSUMES
That t2 is dominated by bridge bonds at least as much as by wall competition, so the gap is not a clean K_eff measure. That the lean on the carpets is geometric (bridge-dominated) and not extreme-value competition.

CHANGES MY MIND
The gap ordering on H, U, V following N1 strongly (monotone, separation over 3 combined errors), together with the diamond lattice's lean tracking its path count in the predicted direction and magnitude.

Hypothesis (Hypothesis)
TOOL
Bottleneck (minimax) duality, identity (2), on a planar graph
REQUIRES
(a) a planar embedding with a dual in which cuts correspond to dual paths; (b) distinct continuous edge weights; (c) the holes handled as zero-cost dual edges; (d) for any statement about the lean, a property of t that is not already symmetric under t -> 1-t.
SATISFIED
(a) yes, a carpet with holes is planar. (b) yes, u is continuous. (c) yes, as stated in the case. (d) NO. (1) and (2) are the same number, so (2) cannot predict a sign that (1) does not. Duality relates the threshold on G to 1 - (threshold on the dual of G). That gives g1(G) = -g1(dual), and nothing more. It carries no sign for G.

TOOL
Fisher-Tippett extreme-value asymmetry (max over walls of the weakest link)
REQUIRES
(a) many competitors; (b) weak dependence, with no clumping (Poisson-like exceedances) at the critical level; (c) a known tail class of the weakest link; (d) convergence to the limit at L = 81 to 6561; (e) the same heuristic applied to (1) must not contradict it.
SATISFIED
(a) yes for the counts, but the effective number is not checkable. (b) no: N = 2^C on H, with walls sharing bonds everywhere, and near-critical closed clusters are clumped. (c) partly. A wall of m bonds has a weakest link with an exponential upper tail, but walls near threshold are not iid. (d) not checkable, and the data say no, since the leans are +0.1 to +0.5 against the limit of +1.14. (e) NO. Applied to (1), the minimum over paths of the path maxima gives a Gumbel-min with skew -1.14. The same number gets opposite signs depending on which side is called the "many". The limit skew is also independent of K, so N cannot set the lean in the limit. N enters only through the speed of convergence and through dependence.

TOOL
Self-duality implies lean = 0 on the square lattice
REQUIRES
The dual of the sampled object is isomorphic to it, boundary conditions included, so that the law of t is symmetric under t -> 1-t.
SATISFIED
Not checkable. The dual of an L x L open-boundary bond grid has a different boundary type (free versus wired). Exact zero holds only for a matched geometry. The three measured values (+0.008, +0.009, +0.011) are all positive and consistent with zero. The premise is "assumed", and it should be a one-line check.

TOOL
Second-crossing gap (t2 - t)/w as a count of near-best competing walls
REQUIRES
The gap is a monotone function of the number of near-best walls that compete.
SATISFIED
No. Deleting the bond e that completed the crossing makes every wall through e cheaper, because W minus e is now a cut. Walls that avoid e have minimum <= t, so they cannot raise t2. Hence t2 = max over walls through e of the second-smallest u. The gap is therefore (i) the spacing of the two smallest u in the winning wall, about (1-t)/m for a wall of m bonds, and (ii) the bypass of e. It does not count competing walls that avoid e. It confounds wall length with the redundancy local to e.

TOOL
One-block recursion as a model of few overlapping walls
REQUIRES
One wall, one path, and the same extreme-value asymmetry as the lattice.
SATISFIED
Partly. One wall of m >= 2 bonds gives t ~ Beta(1,m), skew +0.57 at m = 2, tending to +2. One path gives skew of the same size, negative. So the sign comes from the wall/path asymmetry, not from the number of walls. The recursion is a map on one number, a different object, and it is transferable only for the sign.

TOOL
Differences in "combined errors" (bootstrap plus r8)
REQUIRES
Independent samples.
SATISFIED
No for item 2(iii). t and t2 come from the same samples, so the errors are paired and the difference has a smaller error than the combined error. The between-seed factor r8 does not apply.

TAKE
Identity (2) does not predict the sign. It is an equality with (1), and the extreme-value heuristic gives +1.14 or -1.14 depending on which side is read as the "many". The square is zero by symmetry, not by extreme-value arithmetic. The defensible content is this: a single wall of m >= 2 bonds gives a positive lean (Beta(1,m)). A single path gives a negative lean. Many competitors on one side pull that side's skew toward its Gumbel value, with no dependence on N. Many competitors on both sides cancel. So the carpet's positive lean means the wall side is less competitive than the path side at the critical scale, in effective independent numbers. The lean follows the sign of (effective paths minus effective walls). The census N1 is not a good measure of that.

The case's own table shows N does not order the lean. At 729, H (N1=16) has +0.349, U (48) has +0.146 and V (64) has +0.277. At 6561 the order is H > V > U. This is not monotone in N1 or in 1/nu. The diamond already contradicts "more walls, more positive". Going from (b=2) to (b=3) raises the number of walls from 4 to 8 and moves the lean from -0.104 to -0.372. The case's own wall/N1 premises ("a good proxy", "does not collapse") fail on this evidence. The turn on T is not established as a trend, because there are three sizes only. A one-block recursion cannot turn, since it has one wall.

PREDICTION
(i) SIGN: mean gap in widths ordered H > U > V (fewer minimum walls, larger gap), p=0.6 for the full ordering.
VALUE: mean gap on H in widths 0.05 to 0.8, 80% interval. The ratio H/V is 0.9 to 1.8, 80% interval. The gap is heavy-tailed, so the mean is dominated by rare bridge samples.
NULL: no ordering, with H, U and V within a bootstrap error of each other.
(ii) SIGN: Spearman(t, gap) on T at 729 is positive, p=0.6.
VALUE: -0.05 to +0.30, 80% interval.
NULL: zero. Only the within-wall spacing (1-t)/m would then act, and it changes by a few percent across the pile.
(iii) SIGN: lean of t2 above lean of t on T at 6561, p=0.65.
VALUE: -0.02 to +0.15 in lean units (80%). The paired difference is far smaller than 1 combined error. I expect it to be 0 to 1.5 combined errors if the errors are naively combined as independent.
NULL: equal leans. The gap is small compared to w, so t2 is about t plus a small shift.
Exact computation (item 3). On the diamond family (b parallel branches of s series bonds, b,s in {2,3}), compute exactly or by 10^6-sample Monte Carlo, at levels 3 to 5:
- t_min = the maximum over the minimum walls only, of their weakest link; and
- the true t.
SIGN: lean(t_min) is positive where lean(t) is negative, p=0.7. This is the (3,2) case, which has t = -0.372.
VALUE: lean(t_min) in (3,2) at level 4 is +0.2 to +1.0, 80% interval.
NULL: if minimum walls set the lean, as the rival says, lean(t_min) equals lean(t) in sign and is monotone in N. A second check: the dual lattice must give +0.1042 and +0.3717 exactly, and a mismatch is a code or boundary error, not physics.

ASSUMES
- Wall weakest links have exponential upper tails, as for a minimum of m uniforms.
- Hierarchical diamond conclusions carry over to the sign question on carpets.
- For items 2(i) and (ii), the winning wall at threshold has m of order 100 or more, so the spacing term is small.
- w is of order 0.03 to 0.1.

CHANGES MY MIND
Finding that lean(t_min) matches lean(t) in sign and ordering across the diamond (b,s) grid at levels 3 to 5. Then minimum walls do carry the lean, and my claim that near-minimum walls and dependence control it would be wrong.

Lateral (Lateral)
TAKE
Known structure: t is the bottleneck (minimax) path value, i.e. the MST/invasion-percolation bottleneck. By max-flow/min-cut for bottleneck capacities, t = max over walls of min u. Two theorems carry the sign.

(a) Fisher-Tippett-Gnedenko. A max of many near-independent mins is Gumbel-max, skew +1.14. A min of many near-independent maxes is Gumbel-min, skew -1.14. The same number gets opposite skew depending on which description has more effective competitors. That is why identity (2) alone predicts no sign.

(b) Exact tail duality. P(t<x) ~ N_paths·x^ℓ, with ℓ the shortest-path length. P(t>x) ~ N_walls·(1-x)^C, with C the minimum-wall size. The square has ℓ = C = L, so the pile is symmetric and the lean is 0 (Kesten self-duality; t ~ 1-t). On carpets C(g) = 4^g (H) or 3^g (T), against ℓ ≥ 9^g. The right tail has the much smaller exponent, so it is heavier, which gives a positive lean growing with g. The sign is set by the exponent split ℓ vs C. N(g) enters only as a prefactor.

N alone is already contradicted. At 729 the leans are H .349, V .277, U .146, but N1 is 16, 64, 48.

Diamond check, with my own recursion (F' = 1-(1-F^s)^b, b branches of s series bonds, level 7). (2,2) gives -0.147, consistent with the -0.104 limit. (3,2) gives -0.336, consistent with -0.37. For these two lattices, wall count far exceeds path count, yet the lean is negative. So "more walls gives positive lean" fails there. (2,3), where ℓ > C but paths outnumber walls, gives -0.45 at level 1 and rises to +0.024 at level 7.

Run 109: gap ordering H > U > V (p .6); Spearman(t, gap) on T about +0.1, since Rényi makes the spacing nearly independent of the max; lean(t2) below lean(t) at 6561.

PREDICTION
SIGN: coefficient of variation of (t2-t)/w on H at 729 above 1 (over-dispersed against the exponential top-spacing value of 1), p=0.8
VALUE: 1.1 to 2.2, 80% interval
NULL: Rényi spacing of the top two points of a Gumbel-domain Poisson process is exponential, so CV = 1.

ASSUMES
Wall and path counts for the diamond variants are my own recursion counts. The tail-exponent argument applies to the bulk third cumulant only through finite-size crossover. The gap behaves as a top-two spacing and not as a bridge-bond artefact.

CHANGES MY MIND
The exact diamond with ℓ > C and N_paths ≫ N_walls (b=2, s=3) settling at a negative limit lean. That would kill the exponent-split rule.

Checker — round 1

15 claims: 11 ok, 0 mismatch, 0 sign conflict, 0 justify, 0 derivation, 4 uncheckable

House 1

CHECKS
none flagged. The table has no MISMATCH, SIGN CONFLICT or JUSTIFY row. R1-1, R1-5, R1-6 and R1-7 are UNCHECKABLE only because the checker could not match the U value to the quoted line. The U row is verified by R1-3 (+0.146 at 729), and the 6561 order H > V > U is read off the table directly (+0.423, +0.335, +0.128). Both claims stand. R1-10 and R1-11 (diamond limits) are OK, and I reproduced them (see FORCED 4).

VERDICT
Identity (2) does not by itself predict a sign [object: exact one-block]. It is the same number as (1), and the Gumbel argument gives +1.14 or -1.14 depending on which side is read as the "many" [object: limit]. The square's zero is a symmetry statement, not an extreme-value one [object: exact one-block]. N(g) does not set the lean. At 729, H (N1=16), U (48) and V (64) lean +0.349, +0.146 and +0.277, which is not monotone in N1 [object: sampled lattice at the sizes measured]. On the diamond, (3,3) has 27 minimum walls against 3 paths and still leans -0.162 [object: limit]. The sign tracks minimum-wall size C against shortest-path length l: C < l gives positive, C >= l gives negative [object: limit]. On the carpets C(g) = 4^g or 3^g and l >= 9^g, which is consistent with every carpet leaning positive. The "turn" on T is a three-size difference and cannot be told from a bump [object: sampled lattice at the sizes measured]. The case's framing, that N counts competitors, is the weakest link. The wall count orders sharpening (1/nu) but not the lean.

DISAGREEMENT
Panel versus case: the case asks whether the number of near-strongest walls sets the lean. Hypothesis and Lateral say no, and the table plus the diamond agree with them. Between the seats, Lateral says the sign comes from the exponent split (C vs l) and Hypothesis says it comes from effective independent paths minus walls. Framing says it is geometry. Lateral and Hypothesis also disagree on the sign of lean(t2) - lean(t) at 6561. The diamond sign data (FORCED 4) favour Lateral's exponent split as a sign rule. No seat has a rule for magnitude. Second-order disagreement: all seats read the run-109 gap as a competition count. Hypothesis alone derived that it is not (FORCED 6), and I side with Hypothesis.

FORCED
1. [object: exact one-block] t = max_W min_{b in W} u_b = min_P max_{b in P} u_b — DERIVATION: bottleneck duality on a planar graph. Hence g1(G) = -g1(dual of G). No sign follows for G itself.
2. [object: exact one-block] A single wall of m >= 2 bonds gives t ~ Beta(1,m), skew > 0. A single path gives the mirror, skew < 0. One wall gives positive skew, so the one-block recursion rising and never turning is explained by having no competitors — DERIVATION: order statistics of m uniforms.
3. [object: limit] The Gumbel limit skew is +-1.14 and is independent of K, the number of competitors. N can enter the lean only through the convergence rate and through dependence — DERIVATION: Fisher-Tippett-Gnedenko.
4. [object: limit] Diamond hierarchical lattice (b branches of s series bonds; recursion F' = 1-(1-F^s)^b, moments by exact Stieltjes sums at level 18, my computation, no fit). Limit leans:
- (2,2): -0.1047 (case: -0.1042).
- (3,2): -0.3716 (case: -0.3717).
- (2,3): +0.111 (-0.032 at level 6, +0.105 at level 12).
- (3,3): -0.162.
- (4,2): -0.507.
The sign is positive only for (2,3), where C=2 < l=3. For (3,3), N_walls=27 against 3 paths gives a negative lean. Hence "more minimum walls gives more positive lean" is false on the diamond — DERIVATION: direct iteration of the exact recursion.
5. [object: sampled lattice at the sizes measured] The T fall pooled at -0.110 ± 0.021 comes from three sets with chi2 = 9.3 on 2 dof. X1 is open. These are inherited from measurement.
6. [object: exact one-block] The second-crossing gap: t2 = max over walls W through the completing bond e of min(W minus e), because walls avoiding e have min <= t. So the gap measures redundancy local to e (the second-weakest link, or the bypass of e), not the global count of competing walls — DERIVATION: deleting e makes W minus e a cut and leaves walls avoiding e unchanged.
7. [object: sampled lattice at the sizes measured] The 2(iii) difference lean(t2) - lean(t) is paired (same samples), so the r8 = 1.42 inflation does not apply and the combined-error rule overstates its error — DERIVATION: shared samples give correlated estimates.

ASSERTED
1. [object: sampled lattice at the sizes measured] Carpet leans are positive because C << l makes the right tail heavier. The sign rule is supported by the diamond (FORCED 4) but not derived for carpets, and it gives no magnitude: T (C=3^g) and H (4^g) are not ordered as the rule would suggest.
2. [object: sampled lattice at the sizes measured] Positive lean means a single dominant wall occasionally yields a high t, which makes Spearman(t, gap) positive.
3. [object: sampled lattice at the sizes measured] The mean gap is ordered H > U > V weakly (p=0.55), because more walls through e shrink the second-weakest link. The overlap structure of H (N=2^C) could reverse this.
4. [object: sampled lattice at the sizes measured] Gap magnitude: mean gap in widths on H is 0.2 to 1.0, heavy-tailed. This is not a fit.

ROAD NOT TAKEN
none — seats did not converge (no three share a direction).

FORECASTS
F1. Order of mean gap (t2-t)/w at 729 across H, U, V.
RESTS ON: ASSERTED 3
SIGN: H > U > V (p=0.55)
VALUE: mean gap on H 0.2 to 1.0 widths, 80% interval; H/V ratio 0.9 to 1.8, 80% interval
NULL: three means equal within bootstrap error.

F2. Spearman(t, gap) within a file on T at 729.
RESTS ON: ASSERTED 2
SIGN: positive (p=0.6)
VALUE: -0.05 to +0.30, 80% interval
NULL: zero.

F3. lean(t2) minus lean(t) on T at 6561. An independent positively skewed gap added to t raises the standardized third cumulant, and a positive gap-t correlation raises it further.
RESTS ON: ASSERTED 2
SIGN: lean(t2) above lean(t) (p=0.7)
VALUE: -0.02 to +0.20 in lean units, 80% interval. Judge it with the paired error, not the combined error.
NULL: equal leans, the gap being small against w.

F4. Diamond (2,3) limit lean.
RESTS ON: FORCED 4
SIGN: positive (p=0.9)
VALUE: +0.10 to +0.12, 90% interval
NULL: zero or negative limit, which would kill the exponent-split sign rule.

TESTS
1. [lab] [cost: free — the code is the same short recursion and the diamond results are already in hand] Done above: diamond (b,s) grid. The sign follows C vs l, not N. This kills "N sets the sign" if (3,3) stays negative at levels beyond 18. Run it for (2,4), (3,4) and (4,3) as well. It predicts positive for C < l and negative for C >= l if the exponent-split rule is right. A mix of signs there means the sign is set by the map's fixed point, not by C vs l.
2. [lab] [cost: free — run 109 outputs] Within-file correlation of the gap with a bridge indicator (the completing bond's removal disconnects a piece of more than x% of the open cluster). It predicts a strong correlation if the gap is bridge-dominated, and a weak one if it measures competition. This resolves the ledger premise that the gap is a monotone competition measure.
3. [lab] [cost: cheap — minutes of compute] Count near-minimum walls (C+1 and C+2 bonds) at generation 1 and 2 for H, U, V, T and the cut-2 patterns. Compute the effective competitor number K_eff from the overlap matrix, for example 1/sum(p_i^2) over wall-minimum winner probabilities by Monte Carlo at gen 1. It predicts K_eff ordered like N1 if N1 is a good proxy (ledger premise), and unordered if it is not.
4. [lab] [cost: cheap — minutes] On diamond (3,2) and (2,3), compute t_min = max over minimum walls only, and its lean against the true lean. It predicts equal sign if minimum walls carry the lean, and different sign if near-minimum walls and dependence carry it.
5. [lab] [cost: free — existing files] Check the square's self-duality premise: does the box have matched boundaries? The measured +0.008, +0.009 and +0.011 are all positive. Test the sign of a pooled mean over the 3 sizes (about +0.009 ± 0.007 on independent files). It predicts zero if the geometry is matched, and a small positive lean if the open-boundary dual is wired.
6. [lab] [cost: free — run 109 on stored samples] The paired lean difference lean(t2) - lean(t) for T, and the same quantity at 729 for H, U, V, X, X1 and the square. Use a paired bootstrap, not the 1.5-combined-errors rule. It predicts t2 leaning more than t if the gap is positively skewed and correlated with t.
7. [lab] [cost: infeasible here — five or more sizes between 81 and 19,683 for each of T and X1 would take days] Whether the T/X1 turn is a trend or a bump. Only a two-parameter-plus-bump correction fit on 5 or more sizes could decide it. A fit on 3 sizes cannot. Predicts a monotone approach under a slow correction and a non-monotone one if the turn is real.
8. [seat] [cost: cheap — one prompt] Task Lateral to state the exponent-split rule for non-hierarchical lattices where C and l do not scale as powers. It predicts a sign rule for all 63 census generators if the rule is sound. Then check the sign of g1 against C vs l using the C values in the record (free).

RESIDUE
The sharper case: replace "the number of near-strongest walls" with "the exponent split C(g) versus l(g), and what fixes the amplitude". Specifically: on the diamond (b,s), is the limit sign exactly the sign of s - b, with equality negative? And can the census predict the sign of g1 for all 63 generators from C(1) and the shortest crossing path, with the amplitude left as a residual? The case's N1 ordering stays as the explanation of 1/nu only. Whether the turn on T and X1 exists remains unresolved until there are at least five sizes.

_Contract check: House FORCED (J1): 5: no derivation sketch — it belongs under ASSERTED; House claims without an object (J2): VERDICT: 2 claim sentence(s) without an object tag, e.g. “On the carpets C(g) = 4^g or 3^g and l >= 9^g, which is consistent with every ca…”._

Checker — House 1

12 claims: 11 ok, 0 mismatch, 0 sign conflict, 0 justify, 0 derivation, 1 uncheckable

BACKPROP (House, closed after House 1 — one short extra call)

resolved-local (the N claim only; the T/X1 turn stays open).

Write back:
- Correction: N(g) and N1 order the sharpening 1/ν only. They do not set the lean. At 729 the lean order H > V > U is not monotone in N1. Diamond (3,3) has 27 walls against 3 paths and leans negative.
- Correction: identity (2) fixes no sign. The sign follows the exponent split, minimum-wall size C against shortest-path length l.
- Correction: run 109's gap measures redundancy local to the completing bond, not a competition count. Judge 2(iii) by a paired bootstrap, not combined errors.
- Probe: the diamond (b,s) grid, comparing lean(t_min) with lean(t).
- Forecasts: H > U > V at p=0.55; Spearman(t, gap) positive at p=0.6.

Next case: does the sign of g₁ follow C(1) versus l across the 63 census generators?

Forecasts (parsed, for scoring)

Found / borrowed tags (codifier, 2026-10-20, §12.8)

PN-009, tagged:
- Ledger, round 1 TAKE: void — "Bookkeeping only. (1) and (2) are one number, so any statement about the lean is a statement about the distribution of a max over walls of a min over bonds, or "
- Framing, round 1 TAKE: borrowed — "The named object is mis-categorised twice. First, "the lean" is not one object. Identity (2) says t is a max over walls of a min over bonds. That is an extreme-"
- Hypothesis, round 1 TAKE: borrowed — "Identity (2) does not predict the sign. It is an equality with (1), and the extreme-value heuristic gives +1.14 or -1.14 depending on which side is read as the "
- Lateral, round 1 TAKE: found — "Known structure: t is the bottleneck (minimax) path value, i.e. the MST/invasion-percolation bottleneck. By max-flow/min-cut for bottleneck capacities, t = max "
- House, FORCED 1 (House 1): void — "[object: exact one-block] t = max_W min_{b in W} u_b = min_P max_{b in P} u_b — DERIVATION: bottleneck duality on a planar graph. Hence g1(G) = -g1(dual of G). "
- House, FORCED 2 (House 1): found — "[object: exact one-block] A single wall of m >= 2 bonds gives t ~ Beta(1,m), skew > 0. A single path gives the mirror, skew < 0. One wall gives positive skew, s"
- House, FORCED 3 (House 1): borrowed — "[object: limit] The Gumbel limit skew is +-1.14 and is independent of K, the number of competitors. N can enter the lean only through the convergence rate and t"
- House, FORCED 4 (House 1): found — "[object: limit] Diamond hierarchical lattice (b branches of s series bonds; recursion F' = 1-(1-F^s)^b, moments by exact Stieltjes sums at level 18, my computat"
- House, FORCED 5 (House 1): procedural — "[object: sampled lattice at the sizes measured] The T fall pooled at -0.110 ± 0.021 comes from three sets with chi2 = 9.3 on 2 dof. X1 is open. These are inheri"
- House, FORCED 6 (House 1): found — "[object: exact one-block] The second-crossing gap: t2 = max over walls W through the completing bond e of min(W minus e), because walls avoiding e have min <= t"
- House, FORCED 7 (House 1): procedural — "[object: sampled lattice at the sizes measured] The 2(iii) difference lean(t2) - lean(t) is paired (same samples), so the r8 = 1.42 inflation does not apply and"
- House, ASSERTED 1 (House 1): found — "[object: sampled lattice at the sizes measured] Carpet leans are positive because C << l makes the right tail heavier. The sign rule is supported by the diamond"
- House, ASSERTED 2 (House 1): found — "[object: sampled lattice at the sizes measured] Positive lean means a single dominant wall occasionally yields a high t, which makes Spearman(t, gap) positive."
- House, ASSERTED 3 (House 1): found — "[object: sampled lattice at the sizes measured] The mean gap is ordered H > U > V weakly (p=0.55), because more walls through e shrink the second-weakest link. "
- House, ASSERTED 4 (House 1): found — "[object: sampled lattice at the sizes measured] Gap magnitude: mean gap in widths on H is 0.2 to 1.0, heavy-tailed. This is not a fit."
- House, TEST 1 (House 1): found — "[lab] [cost: free — the code is the same short recursion and the diamond results are already in hand] Done above: diamond (b,s) grid. The sign follows C vs l, n"
- House, TEST 2 (House 1): found — "[lab] [cost: free — run 109 outputs] Within-file correlation of the gap with a bridge indicator (the completing bond's removal disconnects a piece of more than "
- House, TEST 3 (House 1): found — "[lab] [cost: cheap — minutes of compute] Count near-minimum walls (C+1 and C+2 bonds) at generation 1 and 2 for H, U, V, T and the cut-2 patterns. Compute the e"
- House, TEST 4 (House 1): found — "[lab] [cost: cheap — minutes] On diamond (3,2) and (2,3), compute t_min = max over minimum walls only, and its lean against the true lean. It predicts equal sig"
- House, TEST 5 (House 1): procedural — "[lab] [cost: free — existing files] Check the square's self-duality premise: does the box have matched boundaries? The measured +0.008, +0.009 and +0.011 are al"
- House, TEST 6 (House 1): procedural — "[lab] [cost: free — run 109 on stored samples] The paired lean difference lean(t2) - lean(t) for T, and the same quantity at 729 for H, U, V, X, X1 and the squa"
- House, TEST 7 (House 1): procedural — "[lab] [cost: infeasible here — five or more sizes between 81 and 19,683 for each of T and X1 would take days] Whether the T/X1 turn is a trend or a bump. Only a"
- House, TEST 8 (House 1): void — "[seat] [cost: cheap — one prompt] Task Lateral to state the exponent-split rule for non-hierarchical lattices where C and l do not scale as powers. It predicts "