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?

PN-005 — 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? · run 2026-10-01, lab roster (physics) · one round, closed after House 1, no lab step · resolved: open

Case id. CASE-28

Case

Type. lab / physics

Presenting 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.

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.

---

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.

Premise ledger

• [measured, load-bearing] The per-sample crossing threshold is the fraction of bonds present when an open left-to-right crossing first appears under uniformly random bond addition.
• [measured, load-bearing] Lattice g1 values are unbiased k-statistics over 8,000 samples per size with bootstrap errors of about ±0.03 per value.
• [derived, load-bearing] The exact one-block recursion's iterated crossing polynomial gives an exact skewness at each depth, rising at every depth on all 63 patterns with no turn.
• [derived, load-bearing] Depth k of the recursion corresponds to lattice size L = 9^k, so k = 2,3,4 are comparable to L = 81, 729, 6561.
• [assumed, load-bearing] Because the one-block recursion shows no turn, the lattice turn must originate in structure the one-block recursion omits (correlations between blocks / multi-block cut configurations).
• [assumed, load-bearing] T's fall (−0.191 ± 0.038) and X1's (−0.116 ± 0.045) are real features rather than sampling flukes, despite each resting on a single sample set.
• [measured, load-bearing] A two-error pattern read from one sample set can fail to replicate; H's apparent alternation (2.1 errors) vanished on fresh seeds at six times the samples.
• [measured, load-bearing] g1 and 1/ν come from the same samples and so are not independent evidence.
• [derived, load-bearing] A falling local slope of the width with L makes an endpoint-slope 1/ν read high, i.e. the tabulated 1/ν is biased upward.
• [derived, load-bearing] None of the tabulated lattice or recursion columns is a limiting (L→∞ or k→∞) value except the stated recursion limit.
• [measured, load-bearing] The square lattice, having no hierarchical levels, serves as a null control with g1 near zero at every size.
• [derived, load-bearing] The sign of g∞ is fixed by the minimum cut against the shortest crossing (C1 ≤ 5 against 8 bonds).
• [assumed, load-bearing] A per-sample auxiliary quantity recordable inside the union–find pass at a few times its cost suffices to resolve the mechanism of the turn.
• [assumed, load-bearing] Splitting samples by such a quantity and recomputing g1 in each part is a valid diagnostic, i.e. the subpopulations are large enough that their g1 errors remain interpretable at 8,000 samples.

Number kinds

• [count] 9×9 grid of cells, 64 kept and 17 removed — structural multiplicity of the generator; 64+17=81 exact
• [count] the eight symmetries of the square
• [count] 63 patterns
• [count] fourteen carpets in the lattice table
• [count] C1, the minimum cut (2, 3 or 4 across the table; C1 ≤ 5 in run 075) — integer fixed by the generator's geometry
• [count] N1, the number of minimum cuts (16, 32, 36, 48, 64, 100, 256) — integer fixed by the generator's geometry
• [count] 8 bonds, the shortest crossing length
• [count] L = 81, 729, 6561 (and 243, 2187; 59049) — powers of 9 (or 3) fixed by the iteration depth
• [count] k = 2, 3, 4; recursion computed to depth 20
• [unclassifiable] 1/ν for the carpets (e.g. 0.3743, 0.5508, 0.2755) — sampled width-estimator value with a stated upward bias; route to a dimensional or computed exponent not nameable from the case
• [dimensional exponent] 1/ν = 0.75 for the square — 2D percolation correlation-length exponent ν = 4/3
• [unclassifiable] y9, the recursion's exponent — named but neither its calculation nor its values are given
• [free, target] g1 lattice values (e.g. T +0.363, X +0.267 at L = 729) — set by measurement; the target asks for g1 in a selected subsample against these
• [computed exponent] g∞, the one-block recursion limit (e.g. T +0.511, U′ +0.706) — exact output of the iterated polynomial, though quoted without error bars
• [free] ±0.03 per lattice g1, ±0.04 for a two-size difference — bootstrap error set by sample count
• [count] 8,000 samples per size; 48,000 for run 079; 2,000 bootstrap replicates
• [free] T's fall −0.191 ± 0.038 (5 errors); X1's −0.116 ± 0.045 (2.6 errors) — measured differences; the 'errors' counts have non-zero spread and one such pattern already failed to replicate
• [free] H steps +0.097, +0.064, +0.062, +0.020 — measured differences between absolute g1 values
• [free] Spearman −0.81 with C1 and −0.77 with y9 — rank correlations set by the measured ensemble of 63
• [free] 100–180 GB for L = 59049 — resource estimate, not structural

Target form. number — Name one per-sample quantity recordable in the union–find pass, and give the g1 it yields in the selected subsample for T and X at L = 729, with a sign plus probability, an interval with coverage, and the null's prediction.. Bound: fired.

Bound note. Bound seat added — no known route for the target (free).

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

Seat record. shown to each seat (its own rows only) · SEAT-RECORD.md · 21 seat rows · SHA-256 65dff31ef486eab39ca99be23ebd08273428ffd14322191f461e51944f96d8a1

Record packet. mode: per-axis (each seat its own; Lateral none; House and checker the full packet) · full: packet-full.txt · 18 facts, cap 40 · SHA-256 b932c55f03179d285baf1ecfff0c0732f2d1918f82662fd5691fcbfad7e36f28

Record packets by seat (one hash per seat)

• Ledger v1: 6 facts · SHA-256 70cb399807cd35a82b066f02a3748059d39a4c2e23a73cfd0fce9fcf345f7df0
• Framing v1: 5 facts · SHA-256 df36003823deae3caea2788a6fe066ec191dca2b2276da2b7bed5c0dcf5330d4
• Hypothesis v1: 3 facts · SHA-256 1590ec1655bc3368d4088664356eff463f6156b08dd316a709b228f950682fd9
• Lateral v1: no packet
• Bound v1: no packet

Record packet — framing packet (RECORD, separate from the case)

`
FR01 [scored] outcome falsified, scored ? — run 073, RESULTS073.md, commit 44c6d3b
FR02 [scored] outcome unanticipated, scored ? — run 074, RESULTS074.md, commit 7d0674e
FR03 [scored] outcome falsified, scored ? — run 079, RESULTS079.md, commit 45833fa
FR04 [scored] outcome falsified, scored ? — run 068, RESULTS068.md, commit 87366cf
FR05 [scored] outcome falsified, scored ? — run 069, RESULTS069.md, commit 4fc965d
`

Record packet — hypothesis packet (RECORD, separate from the case)

`
HY01 [scored] outcome confirmed, scored ? — run 072, RESULTS072.md, commit cbd68f0
HY02 [scored] outcome falsified, scored ? — run 073, RESULTS073.md, commit 44c6d3b
HY03 [scored] outcome confirmed, scored 2026-09-29 — run 058, RESULTS058.md, commit 2313914
`

Record packet — ledger packet (RECORD, separate from the case)

`
LD01 [scored] outcome confirmed, scored ? — run 075, RESULTS075.md, commit ec7a7ad
LD02 [scored] outcome confirmed, scored 2026-09-28 — run 051, RESULTS051.md, commit 43326f5
LD03 [scored] outcome confirmed, scored 2026-09-29 — run 058, RESULTS058.md, commit 2313914
LD04 [scored] outcome confirmed, scored ? — run 063, RESULTS063.md, commit df4dae3
LD05 [scored] outcome falsified, scored ? — run 068, RESULTS068.md, commit 87366cf
LD06 [scored] outcome falsified, scored ? — run 069, RESULTS069.md, commit 4fc965d
`

Record packet — legitimacy packet (RECORD, separate from the case)

`
LG01 [tabulated] carpet=H: 1/ν=0.5468 ± 0.0039, pair slopes=0.560, 0.534 — run 051, RESULTS051.md, commit a1e2ee0
LG02 [tabulated] carpet=U: 1/ν=0.5239 ± 0.0040, pair slopes=0.539, 0.509 — run 051, RESULTS051.md, commit a1e2ee0
LG03 [tabulated] carpet=U′: 1/ν=0.2755 ± 0.0039, pair slopes=0.302, 0.249 — run 051, RESULTS051.md, commit a1e2ee0
LG04 [tabulated] case=Carpet H, full range: pair slopes=0.539, 0.580, 0.529, 0.539 — run 051, RESULTS051.md, commit a1e2ee0
LG05 [tabulated] case=Square lattice: pair slopes=0.746, 0.750, 0.756 — run 051, RESULTS051.md, commit a1e2ee0
LG06 [tabulated] case=Sierpinski gasket: pair slopes=0.263, 0.295, 0.212, 0.289 — run 051, RESULTS051.md, commit a1e2ee0
LG07 [tabulated] pool entry=entry 5: N₁=32, pair slopes=0.5440, 0.5277 — run 063, RESULTS063.md, commit cdd879b
LG08 [tabulated] pool entry=entry 12: N₁=32, pair slopes=0.5390, 0.5030 — run 063, RESULTS063.md, commit cdd879b
LG09 [tabulated] pool entry=entry 10: N₁=48, pair slopes=0.5273, 0.4935 — run 063, RESULTS063.md, commit cdd879b
LG10 [tabulated] pool entry=entry 11: N₁=64, pair slopes=0.4874, 0.4819 — run 063, RESULTS063.md, commit cdd879b
LG11 [tabulated] lattice=X: 1/ν, run 061=0.5522 ± 0.0037, pair slopes=0.5467, 0.5520 — run 063, RESULTS063.md, commit cdd879b
LG12 [tabulated] lattice=Y: 1/ν, run 061=0.5134 ± 0.0036, pair slopes=0.5273, 0.5076 — run 063, RESULTS063.md, commit cdd879b
LG13 [tabulated] lattice=Z: 1/ν, run 061=0.5281 ± 0.0039, pair slopes=0.5269, 0.5094 — run 063, RESULTS063.md, commit cdd879b
LG14 [tabulated] case=Control — square lattice: pair slopes=0.756, 0.742 — run 053, RESULTS053.md, commit bd3e746
LG15 [tabulated] case=Carpet T: pair slopes=0.398, 0.351 — run 053, RESULTS053.md, commit bd3e746
LG16 [tabulated] case=Carpet V: pair slopes=0.494, 0.460 — run 053, RESULTS053.md, commit bd3e746
LG17 [tabulated] case=Control — square lattice: pair slopes=0.723, 0.772 — run 055, RESULTS055.md, commit 2b12735
LG18 [tabulated] case=Carpet A: pair slopes=0.436, 0.423 — run 055, RESULTS055.md, commit 2b12735
LG19 [tabulated] case=Carpet B: pair slopes=0.540, 0.494 — run 055, RESULTS055.md, commit 2b12735
LG20 [tabulated] room=R25: 1/ν=0.5522 ± 0.0037, pair slopes=0.5525, 0.5519 — run 061, RESULTS061.md, commit 465cb8a
LG21 [tabulated] room=R50: 1/ν=0.5134 ± 0.0036, pair slopes=0.5260, 0.5008 — run 061, RESULTS061.md, commit 465cb8a
LG22 [tabulated] room=R75: 1/ν=0.5281 ± 0.0039, pair slopes=0.5254, 0.5308 — run 061, RESULTS061.md, commit 465cb8a
LG23 [tabulated] sample seed=5601: 1/ν=0.7400, pair slopes=0.7430, 0.7324 — run 056, RESULTS056.md, commit fd18f82
LG24 [tabulated] sample seed=5602: 1/ν=0.7368, pair slopes=0.7301, 0.7534 — run 056, RESULTS056.md, commit fd18f82
LG25 [tabulated] sample seed=5603: 1/ν=0.7362, pair slopes=0.7305, 0.7504 — run 056, RESULTS056.md, commit fd18f82
LG26 [tabulated] sample seed=5604: 1/ν=0.7501, pair slopes=0.7527, 0.7436 — run 056, RESULTS056.md, commit fd18f82
LG27 [tabulated] sample seed=5605: 1/ν=0.7405, pair slopes=0.7348, 0.7547 — run 056, RESULTS056.md, commit fd18f82
LG28 [tabulated] sample seed=5606: 1/ν=0.7419, pair slopes=0.7352, 0.7586 — run 056, RESULTS056.md, commit fd18f82
LG29 [tabulated] sample seed=5607: 1/ν=0.7420, pair slopes=0.7347, 0.7600 — run 056, RESULTS056.md, commit fd18f82
LG30 [tabulated] sample seed=5608: 1/ν=0.7416, pair slopes=0.7478, 0.7261 — run 056, RESULTS056.md, commit fd18f82
LG31 [tabulated] window=all four sizes: 1/ν=0.7293 ± 0.0047, pair slopes=0.7149, 0.7478, 0.7192 — run 059, RESULTS059.md, commit ec6e82e
LG32 [tabulated] fit=five sizes, 81 … 6561: 1/ν=0.5504 ± 0.0037, pair slopes=0.5432, 0.5765, 0.5534, 0.5141 — run 059, RESULTS059.md, commit ec6e82e
LG33 [tabulated] lattice=square: p*=0.4849, λ=4.879, y=0.7213, measured 1/ν=0.7411, y − measured=-0.020, measured p∞=0.500 — run 058, RESULTS058.md, commit 2313914
LG34 [tabulated] lattice=H: p*=0.7183, λ=3.454, y=0.5642, measured 1/ν=0.547, y − measured=+0.017, measured p∞=0.684 — run 058, RESULTS058.md, commit 2313914
LG35 [tabulated] lattice=U: p*=0.7803, λ=3.326, y=0.5470, measured 1/ν=0.524, y − measured=+0.023, measured p∞=0.752 — run 058, RESULTS058.md, commit 2313914
LG36 [tabulated] lattice=B: p*=0.7934, λ=3.175, y=0.5258, measured 1/ν=0.5170, y − measured=+0.009, measured p∞=0.733 — run 058, RESULTS058.md, commit 2313914
LG37 [scored] outcome withdrawn, scored 2026-09-28 — run 049, 049-carpet-room-or-dimension.md, commit a95b5aa
LG38 [tabulated] carpet=T: x=0.5, 1/ν=0.3743 ± 0.0039, 1/ν ÷ x=0.749 — run 053, RESULTS053.md, commit 9ca35b8
LG39 [tabulated] carpet=B: s=0.8700, 1/ν=0.5170 ± 0.0039 — run 055, RESULTS055.md, commit 9ca35b8
LG40 [tabulated] seed=1/ν: 5601=0.7400, 5602=0.7368, 5603=0.7362, 5604=0.7501, 5605=0.7405, 5606=0.7419, 5607=0.7420, 5608=0.7416 — run 056, RESULTS056.md, commit 811e657
`

Record packet — full packet — House and checker only (RECORD, separate from the case)

`
F01 [scored] outcome confirmed, scored ? — run 072, RESULTS072.md, commit cbd68f0
F02 [scored] outcome falsified, scored ? — run 073, RESULTS073.md, commit 44c6d3b
F03 [scored] outcome unanticipated, scored ? — run 074, RESULTS074.md, commit 7d0674e
F04 [scored] outcome confirmed, scored ? — run 075, RESULTS075.md, commit ec7a7ad
F05 [scored] outcome falsified, scored ? — run 079, RESULTS079.md, commit 45833fa
F06 [scored] outcome confirmed, scored 2026-09-28 — run 051, RESULTS051.md, commit 43326f5
F07 [scored] outcome confirmed, scored 2026-09-29 — run 058, RESULTS058.md, commit 2313914
F08 [tabulated] lattice=square: p*=0.4849, λ=4.879, y=0.7213, measured 1/ν=0.7411, y − measured=-0.020, measured p∞=0.500 — run 058, RESULTS058.md, commit 2313914
F09 [tabulated] lattice=H: p*=0.7183, λ=3.454, y=0.5642, measured 1/ν=0.547, y − measured=+0.017, measured p∞=0.684 — run 058, RESULTS058.md, commit 2313914
F10 [tabulated] lattice=U: p*=0.7803, λ=3.326, y=0.5470, measured 1/ν=0.524, y − measured=+0.023, measured p∞=0.752 — run 058, RESULTS058.md, commit 2313914
F11 [tabulated] lattice=B: p*=0.7934, λ=3.175, y=0.5258, measured 1/ν=0.5170, y − measured=+0.009, measured p∞=0.733 — run 058, RESULTS058.md, commit 2313914
F12 [tabulated] lattice=V: p*=0.7402, λ=3.122, y=0.5181, measured 1/ν=0.477, y − measured=+0.041, measured p∞=0.697 — run 058, RESULTS058.md, commit 2313914
F13 [tabulated] lattice=A: p*=0.7793, λ=2.874, y=0.4805, measured 1/ν=0.4294, y − measured=+0.051, measured p∞=0.739 — run 058, RESULTS058.md, commit 2313914
F14 [tabulated] lattice=T: p*=0.8322, λ=2.625, y=0.4392, measured 1/ν=0.374, y − measured=+0.065, measured p∞=0.791 — run 058, RESULTS058.md, commit 2313914
F15 [tabulated] lattice=U′: p*=0.9431, λ=2.023, y=0.3206, measured 1/ν=0.276, y − measured=+0.045, measured p∞=0.888 — run 058, RESULTS058.md, commit 2313914
F16 [scored] outcome confirmed, scored ? — run 063, RESULTS063.md, commit df4dae3
F17 [scored] outcome falsified, scored ? — run 068, RESULTS068.md, commit 87366cf
F18 [scored] outcome falsified, scored ? — run 069, RESULTS069.md, commit 4fc965d
`

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.
- Bound (Bound) v1
- axis: The target is a free magnitude — no structure produces its value. What does consistency forbid? Give the interval that is forced, and say what forces it (unitarity, positivity, a sum rule, a stability or existence requirement).
- refuses: to stay silent because no mechanism is available. A forced interval is a result; "unknown" is not.

Round 1

Ledger (Ledger)
TAKE
What's pinned by identity: k=2,3,4 ↔ L=9²,9³,9⁴ is exact; C1, N1 are fixed integers from the generator's geometry, not estimates; g∞'s sign is fixed by C1 against the 8-bond shortest crossing — that's a definition-level fact (run 075), not an inference. The lattice g1 and the recursion g1 are NOT two measurements of the same quantity: the recursion's depth-k polynomial assumes each level's crossing is a function of the sub-block's crossing probability alone (independence across blocks at a level); the lattice threshold carries whatever correlation that assumption discards. So "where the turn comes from" is, by elimination, exactly the quantity the recursion cannot see: inter-block correlation in how the crossing is assembled. That fixes what a diagnostic variable must measure — not inter-block correlation in the abstract, but something recordable per sample that separates "crossing assembled roughly as the one-block recursion would predict" from "crossing assembled by routes the recursion's independence assumption can't represent."

Bookkeeping-computable candidate: at level 1, the 9×9 grid of top-level blocks has its own coarse crossing problem (block occupied/open via its internal crossing). Run two union-finds on the same bond-addition sequence: the full lattice, and the coarse graph of level-1 blocks. Record t_coarse = fraction of bonds present when the coarse block-graph first crosses; record t = k/M, the actual threshold. Δ = t − t_coarse is recordable at ≈2× cost, already available from the same ordered bond stream.

PREDICTION
At L=729, splitting T and X at the median Δ: low-Δ half g1(T) ≈ 0.20 to 0.30; high-Δ half g1(T) ≈ 0.45 to 0.65
low-Δ half g1(X) ≈ 0.15 to 0.25; high-Δ half g1(X) ≈ 0.30 to 0.45
Overall (unsplit, reference): g1(T)=+0.363, g1(X)=+0.267 — high-Δ half carries the larger deviation from the recursion value in both carpets

SIGN: high-Δ half shows larger g1 (more positive, further from recursion k=3 values) than low-Δ half in both T and X, p=0.65
VALUE: Δg1 (high-Δ minus low-Δ) = 0.10 to 0.35, 80% interval
NULL: if Δ carries nothing about skewness, g1 in both halves equals the pooled value within bootstrap error (±0.03·√2 ≈ ±0.042), and no ordering between halves is consistent across T and X.

ASSUMES
- A coarse block-graph union-find on the same stream is a faithful proxy for "recursion-like" assembly (not stated in case; the case only licenses that the recursion omits inter-block structure, not that this specific coarse pass isolates it).
- 4,000-sample half-splits leave g1's bootstrap error small enough to be informative (not verified here; same caveat the case raises about subpopulation size at 8,000 total).

CHANGES MY MIND
If Δ is uncorrelated with the T/X turn — i.e., g1 computed in the high-Δ and low-Δ halves comes out statistically indistinguishable from the pooled value and from each other, for both T and X — then Δ is bookkeeping noise, not the mechanism, and the turn's origin is still unlocated.

CITES
LD01, LD02

Framing (Framing)
TAKE
The question "where does the turn come from" presupposes that g₁(L) at fixed finite L is the object worth chasing across L=81,729,6561. It isn't. g₁ is a shape statistic of a distribution that is itself drifting with L toward a limiting object (g∞, reached only as k→∞ in the recursion) while the lattice's finite-L distribution is contaminated by corrections to scaling that have nothing to do with "turning back" — they're the ordinary non-monotonic approach of a finite-size statistic to its limit, convolved with ±0.03-level sampling noise on an 8,000-sample estimate of a third-cumulant ratio (a notoriously high-variance statistic). The case already half-admits this: T's fall is 5 "errors" from one sample set, X1's is 2.6, and H's apparent alternation (2.1 errors) vanished on reseed. Two of three "turning" carpets are below or near the replication threshold that already failed once in this very dataset. So "the turn" as named is not a stable feature of carpets T and X1; it's an artifact of reading a single noisy finite-size point as if it were a trajectory. The actual object this case should be about is the finite-size correction term to g₁(L) — i.e., g₁(L) − g∞ as a function of 1/L (or of the block-block correlation correction), not "g₁ turns back or doesn't." Asked straight, "does g₁ turn back" is asking whether a scaling correction changes sign, which is a legitimate but differently-shaped question than the one posed.

PREDICTION
SIGN: no reproducible sign change in T's g1(L=729→6561) difference on fresh independent samples, p=0.7
VALUE: -0.08 to +0.08, 80% interval (resampled T g1 difference, fresh seeds, 8000 samples)
NULL: resampling with independent seeds at same L reproduces the −0.191 fall within bootstrap error (i.e. null = turn is real and stable)

SIGN: no reproducible sign change in T's g1(L=729→6561) difference on fresh independent samples, p=0.7
VALUE: -0.08 to +0.08, 80% interval
NULL: fall replicates at ~same magnitude under reseed

ASSUMES
That an independent-reseed replication run (same design as run 079's test of H) is the right falsification probe for T and X1's claimed turn; none stated beyond that.

CHANGES MY MIND
If a fresh-seed run at L=729 and 6561 (matched sample count, not just 8,000) reproduces T's fall and X1's fall within ~1 error bar on independent data, I retract — the turn is then a real lattice-vs-recursion discrepancy, not a framing artifact of undersampled skewness.

CITES
F03

Hypothesis (Hypothesis)
TOOL
One-block hierarchical recursion (R(p) iterated to give exact depth-k skewness)
REQUIRES
(1) Crossing of the depth-k lattice is a function only of crossing/non-crossing states of depth-(k−1) blocks, with no additional information propagating across block boundaries. (2) The blocks composing a given level, conditioned on the level below, are statistically independent inputs to R. (3) No shared bonds or boundary sites link adjacent blocks in a way that correlates their crossing events beyond what R encodes.
SATISFIED
(1) not checkable from the case — the case does not demonstrate this decomposition holds for the drawn lattice, only asserts it defines R.
(2) no — the case's own load-bearing premise is that the lattice turn "must originate in structure the one-block recursion omits (correlations between blocks)." That is an admission the independence precondition is not satisfied, inferred rather than directly tested.
(3) not checkable from the case — no measurement of inter-block boundary correlation is reported; this is exactly the open question the required prediction is meant to probe.

TOOL
k-statistic skewness estimator (third cumulant / variance^1.5) with bootstrap SE, used to assert "the turn" and "the fall" at L=729
REQUIRES
Samples iid; the statistic's sampling distribution reasonably summarized by a bootstrap SE at the quoted replicate count; sample size large enough relative to the true skewness for third-cumulant estimates to be stable (skewness estimators converge more slowly than means).
SATISFIED
Iid samples — yes (independent percolation realizations, stated in provenance).
Bootstrap SE validity for a third-moment statistic at N=8,000 — not checkable from the case; no diagnostic (e.g., bootstrap distribution shape, N-dependence of the SE) is given, only a flat "±0.03 per value" quoted uniformly across all 14 carpets regardless of each carpet's own variance/kurtosis.
Stability of the estimator at this N — no: the case itself reports H's apparent 2.1-error alternation "vanished on fresh seeds at six times the samples," i.e. the same nominal SE under-covered the true sampling variability for at least one carpet. That is a direct, in-case failure of this precondition, not a hypothetical one.

TOOL
Implicit significance test ("T's fall" at 5 errors, "X1's fall" at 2.6 errors) treated as a real feature to be explained
REQUIRES
The quoted "error" must be the correct SE for the specific contrast being tested (a difference between two sizes, stated as ±0.04, not ±0.03); no uncorrected multiple comparisons across the 14 carpets tested simultaneously; the single sample set must not itself be an outlier draw.
SATISFIED
Correct SE for the contrast — partially: the case does use ±0.04 for the two-size difference, so this part is satisfied.
No multiple-comparison inflation — no. Fourteen carpets are read at once; under 14 independent-ish tests, a ≥2.6σ excursion on at least one is non-negligibly likely (p≈1−(1−2·Φ(−2.6))^14 ≈ 0.1–0.15 even with no real effect), and X1's 2.6-error fall sits right in that regime.
Not an outlier single draw — no, explicitly flagged in the case as resting on "a single sample set," and the ledger lists this as an assumed (not measured) load-bearing premise. The precondition is stated as unmet by the case's own audit.

TAKE
Every quantitative claim this case leans on — the turn being real, the recursion's lack-of-turn being the explanatory baseline, the two-vs-twelve carpet split — ultimately rests on two unchecked preconditions: that the one-block recursion's independence assumption is a good approximation at these depths (never directly tested, only inferred by elimination), and that a 2–5 "error" deviation read from one 8,000-sample set at one of fourteen simultaneously-tested carpets is a real effect rather than the multiple-comparisons tail the case's own H-episode already demonstrated can happen. I do not get to assume the recursion's omission is the explanation just because it's the only documented difference between the two models; "the recursion has no turn" is necessary but not sufficient to prove inter-block correlation is the mechanism, and the case concedes this with the word "assumed." The honest state is: tool preconditions are unmet or unverified at the two places the argument needs them most.

PREDICTION
SIGN: no clean separation (split g₁ differs from whole-sample g₁ by less than its own SE), p=0.55
VALUE: -0.05 to 0.05 (difference in g₁ between split and whole sample, T and X combined), 80% interval
NULL: splitting by the recordable quantity shifts g₁ by an amount consistent with zero, i.e. within ±0.03–0.04 of the whole-sample value, for both T and X

SIGN: deficit (below the parental mean), p=0.8 — VALUE: low to high — n/a (quantity not committed as primary; see first SIGN line as primary call)
VALUE: n/a
NULL: n/a

ASSUMES
That the quoted ±0.03/±0.04 error bars, demonstrated already to under-cover at least once (H), are similarly underestimated for T and X; that 14 simultaneous carpet comparisons were not corrected for multiplicity.

CHANGES MY MIND
A direct per-sample test showing the recursion's independence assumption holds at depth 2–3 for T and X specifically (e.g., measured inter-block crossing correlation statistically indistinguishable from zero) would validate the recursion's precondition and let the "omitted correlation" explanation stand on checked ground rather than elimination; conversely, replication of T's fall on an independent fresh-seed 48,000-sample run (as was done for H) surviving at the same magnitude would satisfy my significance-tool precondition and I'd accept the fall as real.

CITES
HY01, HY02, HY03

Lateral (Lateral)
TAKE
The lattice-vs-recursion split is not a new effect; it is the standard real-space-RG confluent-singularity picture, and the two share Wegner's correction-to-scaling theorem (1972): an observable approaching a fixed-point value does so as O(L) = O_∞ + A L^{-ω} + B L^{-ω'} + …, where the one-block recursion computes only the leading, exactly-self-similar channel (it has no boundary/cross-block term by construction — it is a pure decimation map, structurally the hierarchical-lattice analogue of Migdal–Kadanoff RG, which is exact only because the one-block map treats each block as independent). The lattice realizes the carpet on an actual plane, where blocks are not independent: there exist additional, irrelevant operators carrying crossing information across block boundaries (cut-sharing between adjacent level-1 blocks), i.e., a second scaling field with opposite-sign amplitude at intermediate L. A turn is exactly what a two-term Wegner expansion produces when the irrelevant correction's amplitude has the opposite sign to the leading approach and a comparable exponent at these L — nothing more exotic is required. I will not call this a "resemblance" to RG without that theorem: the shared result is specifically the confluent (two-exponent) correction-to-scaling expansion, not merely "finite-size effects." The quantity that should carry this is the one that measures whether a sample's spanning cluster used only the minimal, block-local cut-path or recruited neighboring block columns — i.e., whether it sampled the leading or the irrelevant channel.

PREDICTION
SIGN: T's lateral-crossing subsample has lower g1 than T's direct-crossing subsample (deficit), p=0.75
VALUE: g1(direct) − g1(lateral) for T at L=729: 0.15 to 0.45, 80% interval
NULL: no difference between direct and lateral subsamples' g1 (quantity carries nothing), predicted gap ≈ 0 ± 0.05 for both T and X

NULL
(quantity carries nothing about skewness): direct- and lateral-footprint subsamples of both T and X have equal g1 within bootstrap error (~0.04) at L=729, and X shows no such gap either.

ASSUMES
- "Lateral footprint" (number of distinct level-1 block columns touched by the spanning cluster at threshold) is a legitimate proxy for the irrelevant/cross-block scaling channel; this is not stated in the case.
- The two-exponent Wegner form applies to a discrete threshold-skewness observable in the same way it applies to continuous order parameters.
- The sign imbalance (T falls, X doesn't) reflects a genuine amplitude-sign difference between carpets rather than C1/N1-driven noise alone.

CHANGES MY MIND
If splitting T and X by block-footprint at L=729 shows equal g1 in both parts (gap within ±0.04, consistent with null), the turn is not a competing-correction effect and the Wegner/confluent-singularity identification is wrong — I'd withdraw the shared-theorem claim entirely rather than weaken it.

Bound (Bound)
TAKE

My axis is consistency: what interval is forced on a free magnitude, and by what. Here the target quantity is g₁ in a subpopulation selected by an auxiliary per-sample quantity — a free magnitude with no mechanism given in advance. But it is not unconstrained. The subpopulation g₁'s are tied by a sum rule: the full-sample third cumulant decomposes exactly as a weighted combination of the sub-sample third cumulants plus a between-group term from the mean-threshold offset (law of total cumulance, third order). That forces: if the sample splits into two parts of comparable size with different sub-means, the whole-sample skewness is bounded by a convex combination of the parts' skewness plus a term fixed by their mean separation and the overall variance — it cannot exceed what the parts and the mean-split jointly allow. So whatever quantity is chosen, the two sub-g₁'s cannot both sit on the same side of the whole-sample value with the same magnitude as the whole; at least one part must move further from zero than the pooled value, in the direction consistent with the sign of the mean-shift it carries. This is forced by algebra, not physics — it holds regardless of which event-quantity is chosen, provided the split is on something correlated with threshold value at all. If the chosen quantity is uncorrelated with the threshold's upper tail, the split must leave both parts statistically indistinguishable from the pooled g₁ within error.

PREDICTION
SIGN: for T, the late-crossing-time subsample (top half by number of bonds added before crossing) shows higher g₁ than the pooled +0.363; for X, the split produces no significant difference, p=0.65
VALUE: T late-half g₁: +0.42 to +0.65, 80% interval; X late-half g₁: +0.20 to +0.35, 80% interval
NULL: no effect predicts both sub-g₁'s equal to the pooled value within ±0.03–0.05 (the single-sample-set error scale), with no systematic ordering by the auxiliary quantity

ASSUMES
The auxiliary quantity is "bonds added before first crossing" (a count, cheap to log in the union–find pass); that it is correlated with where the crossing cluster sits relative to the minimum cut; that subsample sizes (~4,000) keep bootstrap error near ±0.04–0.05.

CHANGES MY MIND
If the late/early split on T shows sub-g₁'s statistically indistinguishable from each other and from pooled (within ±0.04), the sum-rule constraint would still hold trivially but would carry no diagnostic content — meaning the turn is not visible in any single-event split and must come from higher-order (multi-sample, correlated-block) structure outside what a per-sample quantity can expose.

Checker — round 1

29 claims: 5 ok, 9 mismatch, 1 sign conflict, 2 justify, 0 derivation, 12 uncheckable; 2 extracted claim(s) dropped because their quote is not in the checked text

House 1

CHECKS

[R1-3] Ledger's "g∞'s sign is fixed by C1 against the 8-bond shortest crossing." The flag is a paraphrase artefact: the checker found the source's members ([minimum cut, shortest crossing, C₁ ≤ 5, 8 bonds]) unsegregated and the text naming two of them. The source sentence says exactly this. Claim survives, but I do not rest anything on it: g∞'s sign concerns [object: limit] of the recursion and is not in dispute.

[R1-4] "k=2,3,4 ↔ L=9²,9³,9⁴." Flagged because the text writes powers where the source writes 81/729/6561. 9²=81, 9³=729, 9⁴=6561. Survives, arithmetic identity.

[R1-6], [R1-7], [R1-13], [R1-14], [R1-17], [R1-18] are all set-membership paraphrase flags of the same kind — the checker could not match a restated list against the source's members. [R1-6] (T and X1 fall by 5 and 2.6 errors), [R1-7]/[R1-14] (H's 2.1-error alternation vanished at 48,000) and [R1-17] (fourteen carpets) are verbatim in the case. Survive. [R1-13] does not survive as stated: Hypothesis attributes to the case the premise "the turn must originate in structure the recursion omits." The case's own FRICTION text says only "the lattice turn is not in the one-block recursion"; the causal premise lives in the ledger as assumed, not in the case's framing. The distinction matters and I use it in DISAGREEMENT.

[R1-12] and [R1-24] "T falls." The checker read the full series +0.129 → +0.363 → +0.172 and returned "mixed." The flag is wrong because it tests the wrong window: both seats claim a fall over 729→6561 only, and 0.363 → 0.172 falls. Survives for that window; the three-point series is indeed non-monotone, which is the whole case.

[R1-19] SIGN CONFLICT on X1's "2.6 errors": recomputed −0.116/0.045 = −2.58. The flag is correct as to sign and I accept it — the signed z is negative (a fall); "2.6 errors" is a magnitude. No claim of mine rests on the sign convention, and the magnitude 2.58 (not 2.6) is below the threshold that already failed to replicate once.

[R1-20] JUSTIFY — "the nominal SE under-covered for at least one carpet." Mechanism: a bootstrap SE on a third-cumulant ratio at N = 8,000 is itself estimated from sixth moments of a heavy-right-tailed threshold distribution; its sampling error is large and biased low when the tail is under-resolved, so ±0.03 reads low (too tight). Direction in the source series: H at 48,000 per size runs +0.161, +0.258, +0.322, +0.384, +0.404 — monotone rising, no alternation, i.e. the 2.1-error structure seen at 8,000 (run 074) was absent once the tail was resolved. The series supports under-coverage. Claim survives, with the direction stated: quoted errors read low.

[R1-21] JUSTIFY — same mechanism extended to T and X. The source series cannot confirm it for T and X (no 48,000 run exists for them), so it survives only as a transfer from H, not as a measurement. I treat it as plausible, not established, and it is the reason T's nominal 5.0 errors should be read as fewer than 5.

Nothing in my verdict rests on [R1-13] as Hypothesis stated it.

VERDICT

The weight falls on the significance question, not the mechanism question. [object: sampled lattice at the sizes measured] T's fall of −0.191 ± 0.038 and X1's −0.116 ± 0.045 are single sample sets, read off fourteen carpets at once, with an error scale whose under-coverage was demonstrated in-house on H at this exact sample count. Hypothesis names the unmet precondition — bootstrap SE validity for a third cumulant at N = 8,000 — and that tool is load-bearing for the only claim actually in dispute, namely that there is a turn. That outranks every mechanism proposal here. [object: exact one-block] the recursion rises monotonically at k = 2,3,4 on all 63 patterns; [object: limit] g∞ is positive throughout. The gap between them is real at k = 4 for T (0.434 vs 0.172) and large enough that noise alone strains, so I do not dismiss the turn — I decline to explain it before it is replicated. Bound's chosen quantity is the threshold itself and is inadmissible.

DISAGREEMENT

Two conflicts, ranked.

First, and decisive: Framing and Hypothesis say the explanandum may not exist — T's and X1's turns are single-sample-set excursions among fourteen simultaneous comparisons, with an error bar already caught reading low once on H. Ledger, Lateral and Bound take the turn as given and race to a mechanism. Hypothesis names an unmet precondition of the k-statistic/bootstrap significance tool, and that tool is load-bearing for the disputed claim itself, not for some side branch. It outranks the mechanism dispute.

Second, and subordinate: the case's own framing is in conflict with the panel. The case asks "where does the turn come from," presupposing it comes from somewhere. The ledger's causal premise — that the turn must live in inter-block correlation because the recursion omits it — is marked assumed, and it is an argument from elimination over a single documented difference between two models. The recursion's independence precondition is indeed unsatisfied (Hypothesis, item 2), but nobody on the panel contests that; it sits in an uncontested part of the argument. I note it and rank it below the replication question.

Third, methodological: Bound splits on "bonds added before crossing," which is M·threshold. Conditioning on the target by construction partitions the distribution into its own halves; any g₁ difference there is algebra, not physics.

FORCED

1. [object: sampled lattice at the sizes measured] Splitting samples on bonds-added-before-crossing is degenerate — DERIVATION: threshold ≡ k/M where k is that count; the split is on the target variable, so sub-g₁'s are forced by truncation of a known distribution and carry zero information about mechanism.
2. [object: sampled lattice at the sizes measured] For any split into parts, the pooled third cumulant equals the weighted sum of part third cumulants plus between-group terms in the sub-mean offsets — DERIVATION: law of total cumulance at third order; hence both parts cannot sit on the same side of the pooled g₁ at equal magnitude unless sub-means coincide.
3. [object: sampled lattice at the sizes measured] A full-lattice crossing can occur before the coarse top-level block graph crosses — DERIVATION: a spanning path may enter and leave a level-1 block through top/bottom edges, so the block contributes to the crossing without being internally left–right crossed; the coarse open-rule is strictly stronger than the lattice requirement. This is exactly the channel the one-block map cannot represent.
4. [object: sampled lattice at the sizes measured] Under fourteen simultaneous two-sided comparisons, P(at least one |z| ≥ 2.58 under the null) ≈ 0.13 — DERIVATION: 1 − (1 − 2Φ(−2.58))^14. X1 sits inside that envelope; T at 5.0 nominal errors does not.
5. [object: exact one-block] T: k = 2,3,4 → +0.044, +0.316, +0.434 — inherited from a measurement-free exact computation (run 075, depth 20), no error bar quoted.
6. [object: sampled lattice at the sizes measured] ±0.03 per g₁ and ±0.04 per two-size difference — inherited from bootstrap, 2,000 replicates; shown above to read low.
7. [object: sampled lattice at the sizes measured] 1/ν is a model artefact to the extent flagged: a falling width local slope makes the endpoint-slope estimator read high, and it shares samples with g₁.

ASSERTED

1. [object: sampled lattice at the sizes measured] The early-crossing channel of FORCED 3 grows in weight with L faster on T (C₁ = 3, the smallest cut in the table) than on X (C₁ = 4), and early crossings truncate the upper tail of the threshold distribution, lowering g₁.
2. [object: sampled lattice at the sizes measured] Part of T's 729→6561 drop is real and part is sampling excursion; I cannot apportion them without a reseed.

ROAD NOT TAKEN

none — seats did not converge.

FORECASTS

F1. Replication of T's fall.
RESTS ON: FORCED 4
SIGN: T's 729→6561 difference remains negative on fresh seeds at 48,000 samples, but smaller in magnitude than −0.191 (p=0.60)
VALUE: −0.16 to −0.02, 80% interval
NULL: if the fall is pure sampling excursion, the reseeded difference is 0.00 ± 0.015 and the turn evaporates as H's alternation did.

F2. X1's fall does not replicate.
RESTS ON: FORCED 4
SIGN: |reseeded X1 difference| < 0.05 (p=0.65)
VALUE: −0.07 to +0.03, 80% interval
NULL: no effect also predicts ≈ 0; the discriminating alternative is replication near −0.116.

F3. The diagnostic quantity. Record, in the same union–find pass, a second union–find over the coarse graph of the 81 top-level blocks, a block declared open when it is internally left–right crossed; let t_coarse be the bond fraction at coarse crossing and Δ = t − t_coarse. Split T and X at L = 729 by sign(Δ). Δ < 0 is the "recursion-illegal" subpopulation of FORCED 3.
RESTS ON: ASSERTED 1
SIGN: for T, g₁(Δ<0) < g₁(Δ≥0) (p=0.62); for X the gap is smaller and may not clear error (p=0.50 for the same ordering)
VALUE: T — g₁(Δ<0) = +0.15 to +0.34, g₁(Δ≥0) = +0.36 to +0.52, 80% interval; X — g₁(Δ<0) = +0.18 to +0.31, g₁(Δ≥0) = +0.24 to +0.36, 80% interval. Pooled references: T +0.363, X +0.267.
NULL: if Δ carries nothing about skewness, both parts equal the pooled value within ±0.045 (half-sample bootstrap) and the ordering is inconsistent between T and X.

F4. Weight of the illegal channel.
RESTS ON: FORCED 3
SIGN: the fraction of samples with Δ < 0 is larger for T than for X at L = 729 (p=0.60)
VALUE: T 0.15 to 0.45, X 0.08 to 0.35, 80% interval
NULL: equal fractions within 0.03, which would remove the asymmetry ASSERTED 1 needs.

TESTS

1. [lab] [cost: free — per-sample thresholds from runs 072/073 are already stored] Recompute T's and X1's g₁ errors by subsampling the existing 8,000 thresholds at N = 1,000/2,000/4,000 and checking the SE's N-scaling against 1/√N, plus the bootstrap distribution's own skewness. Predicts SE inflating faster than 1/√N and a skewed bootstrap distribution if the errors read low (Hypothesis); clean 1/√N and near-symmetric replicates if ±0.03 is honest (Ledger, Lateral, Bound).
2. [seat] [cost: free — no new compute] Task Hypothesis: produce the multiplicity-corrected significance for the fourteen-carpet scan at 729→6561, with and without the H-calibrated SE inflation. Predicts X1 falls below threshold under either correction; T survives only if the inflation factor is under ~1.6.
3. [seat] [cost: cheap — definitional, minutes] Task Ledger: pin the coarse open-rule and the interface convention for the block graph, and state whether Δ < 0 is possible under that rule. Predicts Δ < 0 reachable if blocks connect through top/bottom edges (FORCED 3); if the rule forbids it, Δ is one-sided and F3's split must be re-cut at the median.
4. [lab] [cost: hours — full resample of two carpets at two sizes] Fresh-seed run of T and X1 at L = 729 and 6561, 48,000 samples per size, the run 079 design. Predicts a diminished but negative T difference if F1 holds; a null difference if Framing is right.
5. [lab] [cost: hours — dual union–find, ~2× the pass] The F3 run: T and X at L = 729, 8,000 samples minimum (16,000 preferred so each half is ~4,000+), recording t, t_coarse, Δ, and the count of top-level blocks the spanning path traverses without being internally crossed. Predicts the F3 split if ASSERTED 1 holds; flat halves if the turn is not in any per-sample quantity.
6. [lab] [cost: infeasible here — L = 59049 needs 100–180 GB, a size far beyond the largest run so far] A fourth lattice point to settle whether T's g₁ resumes rising. Not available; this is why tests 1 and 4 must carry the weight.

RESIDUE

Not closed. The sharper case: run the F3 split at L = 81, 729 and 6561 on T, and ask whether the weight of the Δ < 0 subpopulation grows monotonically with L while the Δ ≥ 0 subpopulation's g₁ tracks the exact one-block values (+0.044, +0.316, +0.434). If it does, the turn is a mixture effect with a named, recordable channel and the recursion is recovered as the conditional law on the legal half. If the Δ ≥ 0 half still fails to track the recursion, then the omitted structure is not the coarse-crossing channel and no per-sample quantity will expose it — the next object would be a two-sample correlation across neighbouring blocks, which no union–find pass can record.

_Contract check: House adopted a flagged claim: H1-10 (from Hypothesis), H1-25 (from Ledger); House FORCED (J1): 5: no derivation sketch — it belongs under ASSERTED; 6: no derivation sketch — it belongs under ASSERTED; 7: no derivation sketch — it belongs under ASSERTED; House claims without an object (J2): VERDICT: 2 claim sentence(s) without an object tag, e.g. “Hypothesis names the unmet precondition — bootstrap SE validity for a third cumu…”._

Checker — House 1

31 claims: 12 ok, 7 mismatch, 1 sign conflict, 2 justify, 1 derivation, 8 uncheckable

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

open. Write back: a probe, not an edge — the significance question is upstream of the mechanism question and isn't settled. Record for the panel: T's and X1's "falls" rest on single 8,000-sample sets, under an error scale demonstrated in-house to under-cover (H's alternation vanished at 6× samples). No mechanism proposal (Δ-split, footprint-split, bond-count-split) should be scored until that's checked. Correction to the ledger: the recursion's independence precondition being unmet is uncontested, not the live dispute — don't let it stand in for "the turn is real."

Next case: fresh-seed 48,000-sample reruns of T and X1 at L=729,6561 (run-079 design). If the fall survives shrunk, open the F3 dual-union-find split (Δ = t − t_coarse) as the mechanism probe. If it doesn't survive, the case closes as a multiple-comparisons artifact and the panel should drop it.

Independence — record citations

• round 1: Ledger LD01, LD02 · Framing F03 · Hypothesis HY01, HY02, HY03 · Lateral none · Bound none

Forecasts (parsed, for scoring)

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Codifier's note on this record (2026-10-01)

• Settings. Record loaded, with per-axis packets: Ledger, Framing and Hypothesis received their own, and Lateral none, by design. Bound received no packet, because the tool's per-axis mode does not give Bound one. The full packet went to the House and the checker only. Legitimacy's packet was loaded but Legitimacy did not sit. The seat record was loaded. One round and the House (S2: no lab step). Reductio was deselected, since S1 needs a second round.
• Bound was added by the audit, not by hand: "no known route for the target (free)". A first, hand-pasted Bound seat raised a "stance text differs from version 1" warning. It was removed before the run, and the seat that sat is the tool's stock Bound v1.
• Premise ledger, retagged by the user on the codifier's advice: "depth k ↔ L = 9ᵏ" and "a falling local slope makes an endpoint-slope 1/ν read high" were moved from ASSUMED to DERIVED. The first is the construction and the second an established estimator rule, so neither is a test to order.
• Case text. Before the panel ran, "map" was renamed "one-block recursion" throughout (commit 12ff592). The tool's file-reference check read "the map" as a pointer to a file. The object is the same and no number changed.
• One premise the case did not carry, and the House got wrong. The House (adopting Hypothesis, R1-20 and FORCED 6) states that the ±0.03 error "reads low (too tight)", taking run 079's vanished alternation as its evidence. Run 079's own six-block calibration, which was not in the case, says the opposite. On H at 48,000 samples per size, the bootstrap error on the Δ⁴g₁ contrast (0.130) exceeded the six-block spread over √6 (0.090). The bootstrap error was not too small there; it may be conservative. A two-error excursion failing to replicate is what honest errors do about 5% of the time. The instrument notes should have said so. This is scored in SEAT-RECORD and recorded as a gap in the case.
• The checker caught one real slip. H1-25: the House's coarse graph has "81 top-level blocks", but a pattern keeps 64 of its 81 cells, so the coarse graph has 64 open-able blocks. The House's other flags are paraphrase artefacts, as it said.
• Export defect (for the Differential). The Download button's label did not update when the button became usable. The button worked.