run 109 — Lab

Math Mapgenerated 2026-10-25 · a354234

PUBLIC BUILD

slug: second-crossing-gap
state: run
created: 2026-10-20
sealed: 2026-10-20 (committed before the reading exists in any program and before any computation)
builds_on: [107, 092, 081, 073, 054]
concepts: [correction-to-scaling, min-cut]
programme: PG-01
line: L1, the hump
from: CASE-31 (cases/CASE-31-best-wall-weakest-link.md), the question of 2026-10-19/20: the threshold is the weakest link of the strongest wall (bottleneck duality), and whether the number of near-strongest walls sets the lean of the threshold pile. This run defines the reading the case's must-predict names; a panel on the case may still shape what is sealed after it, not this.
statistic: R27_tree_groups/carpet_perc/. A new flag --second in perc_cluster.c (perc.c not edited; perc_cluster.c's md5 before and after recorded; the same per-sample seeding, so sample index i under seed s adds the same bonds in the same order as perc.c, run 092's check): at the first crossing with k bonds added, record k and the completing bond b*, then continue adding bonds in the same order with b* removed from the union-find (rebuild the forest without b* from the first k − 1 bonds, then continue from bond k + 1) until left and right are joined again at k₂ bonds; write "k p_c k2". t = k/M, t₂ = k₂/M, w = q₀.₈ − q₀.₂ of t in that file, gap = (t₂ − t)/w per sample. Files, re-derived from their original perc seeds (declared reuse; the t column must reproduce the stored file bit for bit, control (i)): H6 (L = 729) and U3 from samples051 (seed 51); T3, T4 (L = 6561) and V3 from samples053 (seed 53); C3 (X) seed 6315 and F3 (X1) seed 6303 from samples063 (run063.sh's S2 and S1 streams); sq729_s5601 from samples056 (seed 5601). 8,000 samples each, 8 threads, into samples109/. Per file: mean and median of gap with bootstrap errors (1,000 resamples; numpy default_rng seed 159001 + file rank in the order listed), Spearman(t, gap), and g₁ of t and of t₂ as k-statistics with bootstrap errors; the difference g₁(t₂) − g₁(t) with its error from the same resamples. Scored quantities: (A) the mean gap at 729 on H, U, V; (B) Spearman(t, gap) on T3; (C) g₁(t₂) − g₁(t) on T4.
controls: (i) the "k p_c" columns of every re-derived file equal the stored file's bit for bit (every line); (ii) k₂ ≥ k on every sample, and on a hand-made graph of two disjoint left–right paths of lengths 3 and 5 the gap is reproduced exactly by enumeration (every bond order, 8!/… small enough to enumerate: the lab enumerates all orders of the 8 bonds and compares perc_cluster's per-order k, k₂ to the enumeration); (iii) the square at 729 (sq729_s5601): g₁(t) within 2 errors of 0, reproducing the record; g₁(t₂) and the mean gap reported; (iv) the 2 × 2 all-bonds case of the cluster instrument's check 4 gives k₂ = k + 1 when the completing bond is one of two parallel spanning bonds (reported).
reported: the gap's full distribution per file (deciles); Spearman(t, gap) on every file; g₁(t₂) on every file; for X1 (F3) and X (C3) the same three quantities as T3; the correlation between gap and the cluster mass at the first crossing where perc_cluster already records it (mass column present when run with the crossing readings on); wall time per file.
pre_seal_glimpse: none. No second crossing has been computed on any graph. The stored t values and their leans (the table in CASE-31) are the record and were read before sealing.
scored: 2026-10-20
outcome: unanticipated
outcome_note: all three clauses in the seal's unanticipated bands and no falsifier fired — (A) mean gap H 0.3934 ± 0.0060, U 0.3826 ± 0.0056, V 0.3853 ± 0.0061 (order H > V > U; H − V +0.96 combined errors against the 3 required); (B) Spearman(t, gap) on T3 −0.1493 ± 0.0112 against the −0.2 line; (C) g₁(t₂) − g₁(t) on T4 −0.0135 ± 0.0261. Controls (i), (ii) pass; (iii) fails as written on one square file (g₁(t) +0.1035 ± 0.0341, 3.04 errors from 0) and compares a file to a pooled record; (iv) holds in 12 of 24 orders — both are the seal's wording faults, neither a withdrawal condition.
source: ../lab/runs/109-second-crossing-gap.md

Prediction(A) mean gap orders H > U > V (fewer minimum walls, larger gap), with H − V above 3 combined errors; (B) Spearman(t, gap) on T at 729 is below −0.2; (C) on T at 6561, g₁(t₂) − g₁(t) is above +2 combined errors. Three clauses; confirmed only if all three hold.

Falsifier(A) V's mean gap above H's by more than 2 combined errors, or (B) Spearman above +0.2, or (C) g₁(t₂) − g₁(t) below −2 combined errors. Unanticipated: any clause neither confirmed nor falsified (A: order not H > U > V but no reversal beyond 2 errors, or H − V between 0 and 3 errors; B: Spearman in [−0.2, +0.2]; C: the difference within ±2 errors). Withdrawn if control (i) or (ii) fails.

Sealed 2026-10-20

As written. The lab chat reads this file above the heading "Not part of the seal" only. The case CASE-31 names this run as its pending run; the panel, if it runs, shapes what follows 109, not 109.

Scored 2026-10-20 — `unanticipated`

Scored by a blind reader given the seal above the "Not part of the seal" heading and RESULTS109.md. Controls: (i) pass (8 files, 0 mismatching lines, each checked before the next draw); (ii) pass (k₂ ≥ k everywhere; the two-path graph's 40,320 orders enumerated, 0 mismatches; 200,000 sampler draws agree with the enumeration of their own order); (iii) fails as written — the square file sq729_s5601 has g₁(t) = +0.1035 ± 0.0341, 3.04 errors from 0, against the record's pooled +0.0088 ± 0.0111 over eight files; the control compared one file to a pool and never computed a single file's spread (the codifier's fault, OPS); the t column is the stored file's bit for bit, so this is a property of the stored file; (iv) the statement "k₂ = k + 1" holds in 12 of 24 orders (the other spanning bond need not be next) — a check written without computing the trivial case (OPS); the reported half is met. Neither (iii) nor (iv) withdraws (the seal names only (i) and (ii)).

clause	sealed	measured	reading
(A) mean gap, 729	H > U > V, H − V > 3 errors	H 0.3934 ± 0.0060, U 0.3826 ± 0.0056, V 0.3853 ± 0.0061	order H > V > U; H − V +0.96 errors; U − V −0.32: no order resolved
(B) Spearman(t, gap), T3	< −0.2	−0.1493 ± 0.0112	negative, short of the line
(C) g₁(t₂) − g₁(t), T4	> +2 errors	−0.0135 ± 0.0261	zero within error

Reading (the codifier's, after the score): the wall count sets neither the gap nor the lean, as the House said; the gap's mean is 0.35–0.39 widths on every file including the square, and Spearman(t, gap) is negative on every file (square −0.289, carpets −0.14 to −0.24), so the anti-correlation is generic to the crossing process, not to carpets. Two reported numbers stand out: on T at 729 the second crossing leans less than the first by −0.103 ± 0.023 (4.4 errors), while at 6561 the two leans agree — T's extra lean at 729 (its hump) belongs to the first crossing and not to the second; and the one square file at +0.10 is the House's TEST 5 (is the box self-dual as wired?) with a number already attached. The sealed margins were the codifier's and were too bold in all three clauses; the signs sealed in (B) held and in (C) did not.

Faults in the seal's wording (reader's list, kept): (iii) packs two tests and compares unlike quantities; (iv) a claim with no pass criterion that enumeration refutes in half the orders; (ii) a truncated phrase; (B) a bare threshold; "combined error" undefined for the paired difference (the House's paired-bootstrap correction was right); the bootstrap spec left rank base, w handling, quantile method and the g₁ estimator to the lab, which fixed all four before computing and said so.

The codifier’s lead is kept for the private site.

generated 2026-10-25the lab index

Sealed 2026-10-20

Scored 2026-10-20 — unanticipated

Scored 2026-10-20 — `unanticipated`