Continuation of PN-005 (docs/v5-plan.md §12.7): the same five seats, cold to the framework, given the case, their own round 1 and House 1, and the lab report of runs 081, 083, 089 and 092 (numbers only), plus a one-off Reductio seat; one round and a House final. Body below is the tool's export, unedited beyond the id.

Continuation of PN-005 (docs/v5-plan.md §12.7): the same five seats, cold to the framework, given the case, their own round 1 and House 1, and the lab report of runs 081, 083, 089 and 092 (numbers only), plus a one-off Reductio seat; one round and a House final. Body below is the tool's export, unedited beyond the id.

PN-006 — 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) + Reductio (one-off, round 2) · two rounds, 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 · Reductio 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
• Reductio 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.
- Reductio (Reductio) v1
- axis: Assume the House's leading account is exactly true — what else must then hold, and does the case or the lab report already contradict it?
- refuses: to propose an account of its own. It derives the consequences of House's account and checks them against the case and lab report; it never offers a rival.

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

Lab report

LAB REPORT — since round 1 and House 1 (numbers only)

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

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

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

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

MUST PREDICT (round 2)

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

Round 2

Ledger
TAKE
Pin what's now exact: t ≤ t_coarse always, since coarse-graph crossing (every block on some left–right path internally crossed) is sufficient but not necessary for the lattice to cross — the lab report's three-way exhaustive partition (t<t_coarse, t=t_coarse, t>t_coarse absent) confirms this ordering, not an artifact of the split. The pooled-file bookkeeping also settles what round 1 left open: T's fall is no longer a single-sample-set read. Eleven files pool to g₁ = +0.254 ± 0.011 at L=6561, three files pool to +0.345 ± 0.019 at L=729, difference −0.091 ± 0.022, cross-machine bit-exact on the underlying thresholds, bootstrap error matching between-file spread at ratio 0.90–1.00. That is a quantity now computed two independent ways and in agreement — it is real, not noise. The t≥t_coarse / t<t_coarse split already localizes it: T's fall sits almost entirely in the t≥t_coarse part (−0.168 ± 0.100 vs −0.036 ± 0.038 in t<t_coarse); X's t≥t_coarse part rises instead (+0.213→+0.403). The open-block count n_open (37 of 64 in t<t_coarse, 49 T / 45 X in t≥t_coarse) is reported only at L=729 — the same count at L=6561 is the next pinned quantity to obtain, and by the ordering identity it is bounded above by 64 and below by whatever supports a left-right path.

PREDICTION
mean n_open (t≥t_coarse part, T) at L=6561 exceeds its L=729 value of 49
mean n_open (t≥t_coarse part, X) at L=6561 is within ±2 of its L=729 value of 45
g1(t≥t_coarse) − g1(t<t_coarse) at L=6561 is larger in magnitude for T than for X

SIGN: T's gap (g1 high-n_open minus low) widens 729→6561 while X's stays roughly flat, p=0.55
VALUE: T gap at 6561 = 0.15 to 0.45 (vs +0.241±0.062 at 729); X gap at 6561 = −0.05 to +0.15 (vs +0.053±0.060 at 729), 80% interval each
NULL: if n_open carries nothing about skewness beyond t≥t_coarse, both gaps hold constant (within ±0.06) from 729 to 6561, and the t≥t_coarse/t<t_coarse asymmetry already measured does not deepen with L.

ASSUMES
- n_open (count of the 64 top-level blocks internally crossed at the moment of global crossing) is recordable in the union-find pass at a few times its cost, as the case's instrument notes require; not separately verified here.
- The t≥t_coarse vs t<t_coarse asymmetry already measured at L=81,729,6561 extrapolates monotonically rather than saturating.

CHANGES MY MIND
If n_open at L=6561 in the t≥t_coarse part is statistically indistinguishable between T and X (both near 47–51, gap difference < 0.06), or if T's within-part gap stops widening, the n_open count carries nothing extra beyond the already-measured t≥t_coarse split, and the mechanism claim reduces to "the coarse-crossing identity partitions the data" with no further bookkeeping traction.

CITES
LD01

Framing
TAKE
Round 1 I called the turn an artifact of under-covered bootstrap SEs on a high-variance third-cumulant statistic. The lab report falsifies that for T specifically: eleven files across two machines, byte-for-byte reproducible samples (ruling out a sampler bug), pooled fall −0.091 ± 0.022 — four errors, not the "single sample set, 5 errors" I leaned on. T's decline is real. But the report also hands me a sharper framing error than my first one. T and X1 were named together as instances of "the turn" by the case's own question. They are not the same object. X1's two independent sample sets give −0.116 ± 0.045 and +0.117 ± 0.035 — opposite signs, each individually "significant," together incoherent. That is not a turn with noisy magnitude; that is an estimator with no stable sign for X1 at all. The case's framing error wasn't "is the turn real," it was treating "T turns back" and "X1 turns back" as one phenomenon with one cause. They need separate verdicts: T has a real, replicated, negative g₁(729→6561); X1 has no replicated sign and should be dropped from "the turn" entirely, not explained. Separately, the coarse-split result cuts against FORCED 3's named mechanism: the fall for T sits mainly in the t ≥ t_coarse ("recursion-legal") part (−0.168 ± 0.100), not the "illegal" early-crossing channel (−0.036 ± 0.038) — so the object Ledger/Hypothesis named (leakage through top/bottom block edges) isn't where T's real effect lives either.

PREDICTION
SIGN: X1's pooled 729→6561 difference over ≥6 independent files will straddle zero, no stable sign, p=0.65
VALUE: -0.05 to +0.05, 80% interval
NULL: no effect predicts 0.00 ± (pooled SE), indistinguishable from a stable sign at this precision — same prediction as mine, so only a tight nonzero pooled value would refute me

SIGN: X1's pooled 729→6561 difference over ≥6 independent files will straddle zero, no stable sign, p=0.65
VALUE: -0.05 to +0.05, 80% interval
NULL: pooled difference centers on 0.00 within ±0.02

SIGN: T's pooled difference will continue to replicate negative on a further independent file set, p=0.85
VALUE: -0.14 to -0.04, 80% interval
NULL: no effect predicts 0.00 ± pooled SE (~0.02), which the existing 11-file pool already excludes

ASSUMES
That "the turn" as a single named object should be split into per-carpet verdicts rather than explained once; the case's phrasing ("where does the turn come from") presumes one mechanism for T, X1 together, which I now hold is the wrong unit of analysis.

CHANGES MY MIND
If a further ≥6-file pool for X1 converges to a stable nonzero sign (either direction) with pooled SE well under the magnitude, I retract — X1 would then be a second real instance of the same phenomenon as T, and the case's original single-object framing would be vindicated.

CITES
none

Hypothesis
TOOL
Inverse-variance pooling of T's three independent 729→6561 differences (−0.191±0.038, −0.111±0.031, −0.022±0.040) into a single −0.110±0.021
REQUIRES
(1) The three estimates are independent draws from the same underlying quantity (no batch/machine/version drift). (2) Each quoted SE correctly reflects that estimate's sampling variance, so inverse-variance weights are valid. (3) The resulting χ² about the pooled mean is consistent with the stated SEs (i.e., no excess scatter).
SATISFIED
(1) partially yes — determinism tests show the sampler reproduces bit-for-bit across machines from the same seed, ruling out an RNG/hardware bug as the source of any drift; but the three sets are not stated to be matched in all other respects (file provenance differs: one explicitly machine-A, two machine-B).
(2) no — directly contradicted: χ² about the pooled mean is 9.3 on 2 df (p≈0.01). If the quoted SEs were correct, χ² should average ≈2. This is a measured excess of ≈4.65× in variance, i.e. the true SE on each difference is roughly √4.65≈2.16× the quoted bootstrap SE.
(3) no, as just shown — the pooling tool's own internal consistency check fails. The pooled value −0.110±0.021 should be read as −0.110±0.045 once corrected, which is still a real (~2.4σ) effect but far weaker than the −0.191±0.038 (5σ-looking) number the case opened with.

TOOL
Bootstrap SE validity for the k-statistic, checked by within-file block agreement ("ratio 1.00 over ten files") versus between-file (independent reseed) agreement
REQUIRES
Within-file block resampling must be a faithful proxy for between-file (fully independent) variability, i.e. no source of variance that only appears across independent draws (batch effects, insufficiently long tail resolution per file, etc.).
SATISFIED
No — the two checks disagree. Within one file, bootstrap matches block spread exactly (ratio 1.00). Across eight independent fresh-seed files of T at 6561 (same machine, same protocol), spread/RMS(error) = 0.90 — still fine. But across the three-file comparison of T's 729→6561 difference and across the eleven-file pool, excess scatter appears (χ²=9.3/2df, and three g1(6561) values 0.172/0.282/0.248 spanning more than their quoted errors cover once the −0.191 file is included). This means the extra variance is not a generic bootstrap failure (contradicts my round-1 claim that the H episode generalizes cleanly) — it is specific to whichever files differ in provenance, most plausibly the original −0.191 file. The precondition fails, but the failure mode is narrower than I asserted in round 1: it looks like one discordant file, not systematic bootstrap under-coverage across the board.

TOOL
Coarse-crossing split (F3: t vs t_coarse) as a mechanism-revealing per-sample quantity
REQUIRES
(1) The split must not be degenerate with the target statistic (unlike bonds-added-before-crossing, which is algebraically identical to t). (2) The two parts must be large enough, and their own g1 estimates stable enough, to support the comparison. (3) The direction and rough magnitude predicted from the geometry (ASSERTED 1 / FORCED 3) should hold if the mechanism is real.
SATISFIED
(1) yes — t_coarse is computed from a distinct (coarse block) union–find, not derivable from t alone; not degenerate.
(2) partially — parts are large (12–40% and 60–88% of 8,000+ samples), but the within-part g1 errors are not quoted in the lab report (only the cross-part difference, ±0.062 at 729), so I cannot independently re-check their bootstrap validity; same unresolved precondition as above, one level down, not checkable from what's given.
(3) yes, directionally — at 729, g1(t≥t_coarse) − g1(t<t_coarse): T +0.241±0.062 (≈3.9σ), X +0.053±0.060 (≈0.9σ, not significant). This matches my F3 prediction's sign and falls inside my quoted 80% intervals for both parts of T. The split is informative, not degenerate — but note the fall itself (T, 729→6561) lives almost entirely in the t≥t_coarse part (−0.168±0.100) while the t<t_coarse part is flat (−0.036±0.038); that is the opposite assignment from what ASSERTED 1 guessed (I had expected the "illegal early-crossing" channel, t<t_coarse, to carry the fall). The split works as a tool; my stated mechanism attached to it does not survive unmodified.

TAKE
Partial correction, not reversal. The headline significance machinery is weaker than quoted but the effect is not zero: pooling T's three independent 729→6561 differences gives −0.110±0.021 nominally, but the pool's own χ²=9.3/2df shows the quoted SEs are collectively too tight by a factor ≈2.16, which is exactly the kind of precondition failure I flagged in round 1 — now measured, not inferred. Corrected, the pooled fall is roughly −0.11±0.045, a real but far more modest (~2.4σ) effect, not the 5σ-looking single-file number the case opened with. Separately, the within-file bootstrap check (ratio 1.00) and the eight-fresh-seed spread (ratio 0.90) show the tool is fine in the well-controlled regime, so the failure is localized — most plausibly to the original −0.191 file — not a blanket indictment of the estimator. The coarse-crossing split (F3) passed its own precondition checks and reproduced my predicted sign and magnitude at 729, which is the first tool in this case whose preconditions I can actually verify as met. But it relocated the fall to the t≥t_coarse part, the "recursion-like" channel, not the "recursion-illegal" one I'd guessed — so the split is a working instrument, but my mechanism attached to it (ASSERTED 1) is wrong as stated.

PREDICTION
SIGN: mean number of top-level blocks open at the moment of crossing, B, rises more (or falls less) for T than for X between L=729 and L=6561, p=0.55
VALUE: ΔB(T, 6561 minus 729) = +2 to +8 blocks (out of 64); ΔB(X, 6561 minus 729) = −2 to +2 blocks, 80% interval
NULL: if B carries nothing about skewness, ΔB(T) ≈ ΔB(X) ≈ 0 ± 1 block and neither correlates with the within-size g1 split already measured (T's +0.241±0.062 gap, X's +0.053±0.060 gap)

ASSUMES
That the demonstrated 2.16× SE-inflation found for T's pooled 729→6561 difference generalizes to the per-part (t≥t_coarse vs t<t_coarse) g1 comparisons, which have not themselves been cross-checked against independent reseeds; this is an extrapolation, not a measurement.

CHANGES MY MIND
A repeat of the F3 split on a second, independently reseeded 8,000+-sample file of T and X at 729 and 6561, reproducing both the +0.241±0.062 within-size gap and the concentration of the fall in the t≥t_coarse part at the same magnitude, would settle that the split's result (not just its sign) is real and not another discordant-file artifact like the original −0.191.

CITES
HY01, HY02, HY03

Lateral
TAKE
The F3 split I called for was run, and it hits my sign prediction but breaks my mechanism as stated. At L=729 the coarse/non-coarse gap is T +0.241±0.062 vs X +0.053±0.060 — exactly the asymmetry I predicted, inside my stated interval. But the within-part breakdown reverses my attribution: T's 729→6561 fall sits almost entirely in the t≥t_coarse part (−0.168±0.100), while the t<t_coarse ("early/illegal") part is nearly flat (−0.036±0.038), and the weight of the illegal part barely moves (0.854→0.875). So this is not a mixture-weight effect of a growing illegal channel, as I had it. It is a Wegner confluent-correction effect living inside the coarse-respecting population: that is where the irrelevant cross-block field evidently acts, not in the short-circuited paths. The early-crossing part looks closer to the pure leading (one-block) channel and stays put, consistent with carrying little of the correction term. I keep the theorem — two-exponent correction-to-scaling, Wegner (1972) — but reassign which subpopulation carries the subleading amplitude. Separately, χ²=9.3 on 2 df across T's three independent files confirms Hypothesis's under-coverage claim from round 1: the quoted bootstrap SEs are too tight: I concede that point fully now.

PREDICTION
SIGN: within the t≥t_coarse subsample, the SD of individual top-level block crossing-times (among blocks open at the moment of lattice crossing) decreases from L=729 to 6561 for T; it does not decrease for X. p=0.6
VALUE: ratio σ_block(6561)/σ_block(729) for T: 0.60 to 0.95, 80% interval; for X: 0.90 to 1.25, 80% interval
NULL: both ratios equal 1.00 ± 0.10, with no asymmetry between T and X (quantity carries nothing about skewness)

SIGN: deficit (below the parental mean), p=0.6
VALUE: ratio σ_block(6561)/σ_block(729), T: 0.60–0.95, 80% interval
NULL: ratio = 1.00 ± 0.10

ASSUMES
- Block-crossing-time spread among open blocks is a legitimate proxy for the amplitude of the cross-block (irrelevant) scaling field, distinct from both the threshold itself and from t<t_coarse status.
- The t≥t_coarse / t<t_coarse split's within-part heterogeneity found in the lab report is the right axis to re-cut on, rather than noise from pooling non-independent files (T's own across-file χ² is large).

CHANGES MY MIND
If σ_block shows no L-dependence difference between T and X within the t≥t_coarse part (ratios both within 1.00±0.10), I withdraw the channel-reassignment and treat the t≥t_coarse/t<t_coarse gap itself as a static geometric artifact of C₁, not a signature of a second scaling field — at that point I have no further mechanism to offer under the Wegner theorem and would default to Framing's position that the fall is not yet established as a physical feature.

Bound
TAKE

The lab report changes what's forced, not the axis. Pooled over eleven files, T's fall is −0.091 ± 0.022 (χ² = 9.3/2 against the inverse-variance mean, which is itself suspicious — the files disagree more than their stated errors allow, but the pooled estimate is still ~4σ from zero). The bootstrap error is validated independently (within-file spread matches it, ratio 1.00; exact bitwise reproducibility across machines). So the free magnitude I press on is no longer "is there a fall" — that's now a measured quantity with a forced-nonzero sign. What's still free is where the fall lives, and the sum rule still has purchase: the total third cumulant of the pooled sample is an exact weighted combination of the sub-sample cumulants plus a between-group term in the sub-mean offset. The F3 data already shows this operating — T's 729→6561 change splits as −0.036 ± 0.038 (t < t_coarse) versus −0.168 ± 0.100 (t ≥ t_coarse): the whole fall is forced into the minority tail (12.5%→14.3% share) because the majority part is flat. That's not optional; given the pooled change and the majority part's near-zero change, the minority part's change is algebraically determined up to the weight's own error. I extend this to a genuine multi-block statistic, since round 2 asks for one.

PREDICTION
SIGN: K = number of top-level blocks internally crossed at the moment of global crossing (over all 64 blocks); at L=729, g1(K≥median) − g1(K<median) is positive and larger for T than for X, p=0.70
VALUE: T gap: +0.14 to +0.30 at 729, +0.06 to +0.20 at 6561, 80% interval each; X gap: −0.02 to +0.12 at both sizes, 80% interval
NULL: no effect predicts both gaps ≈ 0 (within ±0.04) at every size, with T and X indistinguishable from each other

ASSUMES
K correlates with the t≥t_coarse indicator already measured (I use that correlation, not an independent claim); subsample sizes near 2,000–4,000 after an 8,000–16,000 sample run keep bootstrap error near ±0.04.

CHANGES MY MIND
If the T gap at 6561 collapses to ≈0 while the pooled fall persists at −0.09, the sum-rule decomposition would have to be carried by a part my K-statistic doesn't select — meaning the forced constraint is real but my chosen quantity isn't the channel, and a finer multi-block statistic (e.g., margin between the two earliest-completing disjoint coarse paths) would be needed.

Reductio
TAKE
Testing House's verdict (decline to explain the turn as unreplicated noise, nominal SE reads low) against the lab report:

Limits/Invariance: House's mechanism needs the bootstrap SE to under-cover generally. Lab report directly checks this — "bootstrap error matches block spread, ratio 1.00 over ten files" — contradicted. Pooled T: 729 +0.345±0.019, 6561 +0.254±0.011, fall −0.091±0.022, a 4.1σ effect across eleven files. The turn has now replicated; House's "decline to explain until replicated" condition is met and should be retracted for T.

Twins: T and X1 were treated as the same phenomenon (two falling carpets). X1's fresh set gives +0.117±0.035, opposite sign to the round-1 −0.116±0.045 — the twin diverges. This vindicates House's multiplicity/significance worry specifically for X1, contradicting ASSERTED 1's implicit symmetry and F2's value interval (−0.07 to +0.03, p=0.65): actual +0.117 lies outside it, sign flipped.

Neighbours: applying the artefact hypothesis to the coarse-crossing split (nearest untested class), T shows a real, significant channel signal (t≥t_coarse gap +0.241±0.062) that X lacks (+0.053±0.060, consistent with zero) — undermines "decline to explain," since a mechanism-relevant per-sample split already discriminates T from X.

Joint consistency: χ²=9.3/2df across T's three independent sample sets exceeds nominal errors, but same-seed redraws reproduce exactly — rules out RNG/machine bugs, leaving genuine between-file scatter larger than quoted SE for T specifically, not a blanket "errors read low."

PREDICTION
House's account predicts T's gap should keep shrinking toward zero as more files are pooled (artefact decaying with N). Test: pool further independent files at both sizes; if the account holds, the 729−6561 difference should trend from −0.091 toward 0 as file count grows, rather than stabilizing near −0.09–0.11.

SIGN: fall persists, does not shrink to zero, p=0.65
VALUE: −0.13 to −0.04, 80% interval (pooled 729−6561 gap after further files)
NULL: no real effect predicts the pooled gap converges to 0.00 ± 0.015 as more files accumulate, mirroring H's vanished alternation.

ASSUMES
That further pooled files are drawn with the same protocol (independent seeds, same machine-reproducibility check) as those already reported; the case does not state this continuation will occur.

CHANGES MY MIND
If a further round of ~5 independent files at L=729 and 6561 (≥40,000 samples/size total) drives the pooled T gap toward 0.00±0.02 while the coarse-split signal (g₁(t≥t_coarse)−g₁(t<t_coarse)) also collapses toward zero at 729, House's "artefact, not yet replicated" verdict survives intact and the mechanism proposals (ASSERTED 1, FORCED 3) lose their target.

Checker — round 2

30 claims: 17 ok, 1 mismatch, 0 sign conflict, 1 justify, 0 derivation, 11 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.

House final

CHECKS

[R2-5] MISMATCH — Ledger: "the lab report's three-way exhaustive partition (t<t_coarse, t=t_coarse, t>t_coarse absent) confirms t ≤ t_coarse always." The flag is right and the claim does not survive. Arithmetic on the report's own shares: T at 729, 0.854 (t<) + 0.102 (t=) = 0.956, leaving 0.044 with t > t_coarse; T at 81, 0.764 + 0.124 = 0.888 → 0.112; T at 6561 → 0.023; X → 0.212, 0.093, 0.058. So coarse crossing is not sufficient for lattice crossing (blocks are not glued by shared boundary sites alone), and the "identity" Ledger built its axis on is false in a few to twenty percent of samples. Nothing of mine rests on it; I use the residual as data instead.

[R2-21] JUSTIFY — Hypothesis: quoted bootstrap SE on T's 729→6561 difference reads low across independent files. Mechanism: bootstrap resamples within a file and so cannot see between-file variation (seed-block, machine-batch, tail resolution of a third cumulant); χ² = 9.3 on 2 df is a measured 4.65× variance excess. Direction in the source series: −0.191 → −0.111 → −0.022 drifts toward zero in draw order, i.e. the first file is the outlier and the inflation is concentrated there, not spread evenly. Counter-evidence in the same report: within-file bootstrap matches block spread (ratio 1.00, ten files) and eight fresh seeds at 6561 give spread/RMS(error) = 0.90, and four other carpets give pairwise rms 0.86 errors. So the estimator is sound in the controlled regime; the excess is file-specific. Claim survives in attenuated form: inflate by ~1.5, not 2.16, giving pooled fall −0.091 ± 0.033 (~2.8σ). I adopt the attenuated version, not Hypothesis's 2.16.

[H1-6] MISMATCH — my round-1 line "the three-point series is non-monotone" was recomputed as "mixed" against a claimed "rises". The flag is a tag collision, not an error: mixed is non-monotone. Survives.

[H1-8] SIGN CONFLICT — "T at 5.0 nominal errors": recomputed −5.03. The signed z is negative (a fall); "5.0 errors" was a magnitude. Flag correct as to convention; nothing rests on the sign of that quotient. Survives as magnitude only.

[H1-10] JUSTIFY — "±0.03 reads low (too tight)." Mechanism as above. Source series: H at 48,000 rises monotonically (+0.161 … +0.404) with the 8,000-sample alternation gone, supporting under-coverage at 8,000. Now partly overturned by the new report's within-file and eight-seed checks: the blanket version is wrong, the file-specific version survives. I adopt only the file-specific version, as in [R2-21].

[H1-22] JUSTIFY — endpoint-slope 1/ν reads high when the width's local slope falls with L. Direction confirmed by the fresh samples: T local slope 0.349 vs tabulated 0.3743, X1 0.532 vs 0.5359 — both tabulated values sit above the fresh local slope, as the mechanism predicts. Survives; the tabulated 1/ν column is biased upward and is not independent evidence (shared samples).

[H1-19], [H1-20], [H1-28], [H1-29] MISMATCH — all four are list-paraphrase collisions (run 075's "exact skewness … depth 20"; the two-part error scale; run 079's "fresh seeds, 48,000 per size"; "100–180 GB"). Each restates the source faithfully. Survive.

[H1-24] MISMATCH — I wrote "T (C₁ = 3, the smallest cut in the table)". The flag is correct: U′ has C₁ = 2. Does not survive. ASSERTED 1 of round 1 rested on T having the smallest cut; it does not. The claim is demoted and does not carry weight below.

[H1-25] MISMATCH — "the coarse graph of the 81 top-level blocks". Correct: 64 blocks are kept, 17 removed. Does not survive as written; all coarse-graph statements below say 64. The lab report used 64.

[H1-d1] DERIVATION — law of total cumulance at third order, not recomputed. I re-derive it in FORCED 2 and do not lean on it for a sign.

FORECAST SCORE

S-Ledger | median-Δ split at 729: high-Δ half carries larger g₁ in both T and X; T 0.20–0.30 / 0.45–0.65, X 0.15–0.25 / 0.30–0.45 | sign split: T +0.157 (t<t_coarse) vs +0.399 (t≥); X +0.159 vs +0.213 | HIT on sign; every quoted interval except X's low half missed
S-Framing | no reproducible sign change in T's 729→6561 difference; −0.08 to +0.08 | fresh 16,000-sample set gives −0.111 ± 0.031; eleven-file pool −0.091 ± 0.022 | MISS
S-Hypothesis | no clean separation; split g₁ within ±0.05 of pooled | T gap +0.241 ± 0.062 (3.9σ) at 729 | MISS
S-Lateral | T's lateral subsample lower g₁ than direct; gap 0.15–0.45 at 729 | nearest realised proxy g₁(t≥t_coarse) − g₁(t<t_coarse) = +0.241 for T, +0.053 for X | HIT (sign and value, on the proxy mapping; "lateral footprint" itself was never recorded)
S-Bound | split on bonds-added-before-crossing: T late half +0.42 to +0.65 | not run; the quantity is M·threshold and was ruled degenerate | UNSCORABLE
H1-F1 | T's fall replicates negative but smaller than −0.191; −0.16 to −0.02 | fresh set −0.111 ± 0.031; pooled −0.091 ± 0.022 | HIT
H1-F2 | |X1 reseeded difference| < 0.05; −0.07 to +0.03 | +0.117 ± 0.035 | MISS
H1-F3 | T: g₁(Δ<0) = +0.15 to +0.34, g₁(Δ≥0) = +0.36 to +0.52, ordering Δ<0 below Δ≥0 | T +0.157 and +0.399, ordering as predicted; X +0.159 / +0.213, both below my X intervals | HIT for T; X intervals missed
H1-F4 | fraction with Δ<0 larger for T than X at 729; T 0.15–0.45, X 0.08–0.35 | T 0.854, X 0.852 — equal, and both far above my intervals | MISS

VERDICT

T's fall is real [object: sampled lattice at the sizes measured]: eleven files pool to −0.091 ± 0.022, bit-exact cross-machine redraws exclude a sampler artefact, eight fresh seeds give spread/RMS(error) 0.90, and excess kurtosis collapses +0.311 → +0.006 — an independent shape change skewness alone need not have shown. Inflated for the χ² excess it is still ≈2.8σ. X1 is not a second instance [object: sampled lattice at the sizes measured]: −0.116 then +0.117. Both halves of the case's framing fail. Worse for the ledger's assumed premise: the recursion-illegal early-crossing part (t<t_coarse) grows with L, 0.764→0.854→0.875, yet T's g₁ there is flat (−0.036 ± 0.038); the fall sits in the shrinking coarse-respecting minority (0.236→0.146→0.125), −0.168 ± 0.100. [object: exact one-block] the recursion rises monotonically; [object: limit] g∞ is positive. The omitted channel exists and grows but does not carry the turn.

DISAGREEMENT

Named conflict, ranked.

First, and decisive: the case's own framing is the thing in conflict. The case asks "where does the turn come from," presupposing one phenomenon on two carpets caused by what the one-block recursion omits. The lab report breaks it in two places at once. X1 has no stable sign, so there is no two-carpet phenomenon; and the coarse split puts T's fall in the recursion-legal subpopulation while the illegal early-crossing channel grows in weight and stays flat in skewness. The ledger's load-bearing assumed premise — "the turn must originate in structure the one-block recursion omits" — was an argument from elimination over the single documented model difference, and the first per-sample probe of that difference assigned the effect elsewhere. Ledger, Lateral and Bound all built on the premise; Lateral at least conceded the reassignment.

Second: Framing versus everyone on X1 — Framing is right, and its round-1 position on T is falsified by its own criterion.

Third, the unmet-precondition claim: Hypothesis names a real failure of inverse-variance pooling (χ² = 9.3 on 2 df violates its own consistency precondition), and that tool is load-bearing for the disputed claim — how big is T's fall. It therefore outranks Lateral's and Ledger's mechanism dispute on the size question. But it does not reach zero: the report's within-file and eight-seed checks localise the excess to one file. Hypothesis's 2.16× blanket inflation over-corrects; ~1.5× is what the evidence supports.

Fourth, uncontested and ranked last: the recursion's independence precondition is unsatisfied by construction. Nobody disputes this; it sits in a part of the argument no seat contests, and it is no longer the explanation.

FORCED

1. [object: sampled lattice at the sizes measured] Coarse crossing is not sufficient for lattice crossing: t > t_coarse in 11.2%, 4.4%, 2.3% of T samples at L = 81, 729, 6561 (X: 21.2%, 9.3%, 5.8%) — DERIVATION: 1 − share(t<t_coarse) − share(t=t_coarse) from the report's own two share lines; the residual is positive, so blocks must be joined by linking bonds not contained in any block's internal crossing.
2. [object: sampled lattice at the sizes measured] For any split, pooled κ₃ = Σ wᵢκ₃⁽ⁱ⁾ + 3Σ wᵢ(μᵢ−μ)σᵢ² + Σ wᵢ(μᵢ−μ)³ — DERIVATION: law of total cumulance at third order; hence a split into parts with equal sub-means cannot move κ₃ except through the parts' own κ₃.
3. [object: sampled lattice at the sizes measured] T's fall is not a mixture-weight effect: the heavier-skew minority (t≥t_coarse) loses weight with L for both T and X (T 0.236→0.146→0.125; X 0.279→0.148→0.111), yet X rises — DERIVATION: apply 2 with the measured weights and part-wise g₁'s; X's within-part g₁ rises in both parts (+0.153→+0.213→+0.403 and +0.016→+0.159→+0.269), T's does not (+0.094→+0.399→+0.230 and +0.129→+0.157→+0.121).
4. [object: sampled lattice at the sizes measured] The lattice is not one map applied repeatedly at these sizes — DERIVATION: for X, g₈₁ and g₇₂₉ differ at 19 quantile nodes with rms 3.09 errors against 1.06 for two halves of one file, all 19 of one sign and ≈7% of the 729 width; a single quantile map would give within-file-scale scatter.
5. [object: sampled lattice at the sizes measured] Splitting on bonds-added-before-crossing is degenerate — DERIVATION: that count is M·t exactly, so the split is on the target.
6. Inherited from measurement: pooled T g₁ +0.345 ± 0.019 (729) and +0.254 ± 0.011 (6561) [object: sampled lattice at the sizes measured]; T excess kurtosis +0.311 ± 0.069 → +0.006 ± 0.052 [same object].
7. Inherited, measurement-free: recursion T +0.044, +0.316, +0.434 and g∞ +0.511 [object: exact one-block / limit], no error bar.
8. Model artefact: tabulated 1/ν (T 0.3743 vs fresh local slope 0.349) reads high by the falling-local-slope mechanism and shares samples with g₁ [object: sampled lattice at the sizes measured].
9. Fitted: the SE inflation factor ≈1.5 adopted for T's pooled fall — fit on 3 points (the three independent 729→6561 differences via their χ²).

ASSERTED

1. [object: sampled lattice at the sizes measured] T's fall is gated by single laggard blocks: skewness in the threshold comes from samples where one block on the completing coarse path finishes far after the rest, and that gating weakens with L on T but not on X.
2. [object: sampled lattice at the sizes measured] The −0.191 file is a legitimate but outlying draw; the honest central value is the eleven-file pool.
3. [object: limit] Whether lattice g₁ for T converges to the recursion's g∞ = +0.511 is untestable from three sizes and I take no position.

ROAD NOT TAKEN

none — seats did not converge.

FORECASTS

F1. The multi-block statistic. For each sample record every kept top-level block's internal crossing time τᵢ (bond fraction at which block i is first crossed side to side) and the blocks open at global crossing. Let P* be the first left–right block path to complete; define the laggard gap G = [τ_max(P*) − τ_2nd-max(P*)] / SD(t), averaged over samples. It is a statistic of ≥2 blocks, recordable in the same pass. Rationale: a long right tail in t is produced by samples waiting on one straggler block; G measures how much of the crossing time is owed to a single block rather than to the collective.
RESTS ON: ASSERTED 1
SIGN: G(T) − G(X) is negative at 6561 and less negative (or positive) at 729 — i.e. T loses laggard-gating between the sizes while X does not (p=0.62)
VALUE: G(T): 0.30 to 0.65 at 729, 0.12 to 0.42 at 6561; G(X): 0.25 to 0.60 at 729, 0.25 to 0.65 at 6561, 80% interval each
NULL: G carries nothing about skewness — G(T) and G(X) are equal within ±0.05 at both sizes, show no L-trend, and splitting each carpet at median G gives sub-g₁'s equal to pooled within ±0.045.

F2. T's fall persists under further pooling.
RESTS ON: FORCED 6
SIGN: negative (p=0.85)
VALUE: −0.13 to −0.04, 80% interval, for the 729→6561 difference after ≥5 further independent files
NULL: a decaying artefact predicts drift to 0.00 ± 0.02 as files accumulate, as H's alternation did.

F3. X1 has no stable sign.
RESTS ON: FORCED 6 (by contrast)
SIGN: pooled X1 difference straddles zero (p=0.70)
VALUE: −0.06 to +0.07, 80% interval over ≥6 files
NULL: indistinguishable from mine at low precision; only a pooled |value| > 0.08 with SE < 0.03 refutes me.

F4. The t > t_coarse residual keeps shrinking.
RESTS ON: FORCED 1
SIGN: decreasing with L (p=0.75)
VALUE: T at L = 729 on fresh seeds 0.02 to 0.07 — fit on 3 points (the 0.112 / 0.044 / 0.023 series), 80% interval
NULL: no L-dependence predicts a flat ≈0.04 at every size.

TESTS

1. [lab] [cost: free — the F3/coarse-split files already hold per-sample part labels] Recompute T's and X's part-wise g₁ with their own bootstrap errors at all three sizes, which the report withheld. Predicts the t≥t_coarse fall survives at ≥2σ if FORCED 3 holds; if its error is ±0.15 or worse the −0.168 ± 0.100 line carries nothing and the localisation collapses.
2. [lab] [cost: free — stored thresholds] Recompute the eleven-file T pool with the −0.191 file held out, and with a jackknife over files. Predicts a fall near −0.07 to −0.09 with χ² near 1 per df if ASSERTED 2 holds; a fall unchanged at −0.11 if the scatter is generic.
3. [seat] [cost: free — no new compute] Task Framing: state the stopping rule for declaring X1's sign unstable, so the two-file result is not itself a multiplicity read. Predicts a pre-registered ≥6-file criterion; if none can be stated, the X1 verdict is a post-hoc read like the one it criticises.
4. [seat] [cost: cheap — definitional] Task Ledger: retract the t ≤ t_coarse identity given FORCED 1 and re-state the block-joining convention used by the lab. Predicts the residual is linking bonds between edge-disjoint blocks; if instead it is an equality-rounding artefact, FORCED 1 is void and the partition is after all two-way.
5. [lab] [cost: hours — per-block τ recording, roughly 2–3× the pass, two carpets at two sizes] The F1 run: T and X at L = 729 and 6561, ≥16,000 samples, recording all 64 τᵢ and the open set at crossing. Predicts F1's sign and the median-G split separating g₁ if ASSERTED 1 holds; flat G and flat split if the turn is not a per-sample block-timing effect.
6. [lab] [cost: hours — same pass] Extend the quantile-map comparison of FORCED 4 to T at 729 vs 6561, with attention to the lowest quantile node where T already showed 7.4 errors. Predicts the fall is a left-tail lengthening (early crossings becoming commoner) if ASSERTED 1 holds; a right-tail shortening if the gating story is wrong.
7. [lab] [cost: infeasible here — L = 59049 needs 100–180 GB, far beyond the largest run] A fourth size to see whether T resumes rising. Unavailable; tests 1, 2 and 5 must carry it.

RESIDUE

Not closed. The sharper case: run the F1 block-timing record on T at all three sizes and ask whether ⟨τ_max(P*) − τ_2nd-max(P*)⟩, normalised by the threshold width, declines on T while staying flat on X, and whether the median-G high part reproduces the exact one-block values (+0.044, +0.316, +0.434) while the low part falls away. If it does, the turn is a laggard-gating loss with a recordable per-sample signature and the recursion is the conditional law on the gated half. If G is flat and the parts track each other, no per-sample quantity holds it, and the next object is the lattice's quantile map itself — FORCED 4 already shows that map is not size-independent at 3σ for X, which is a cleaner handle than any subsample split.

BACKPROP

open.

Write back three things. (i) An edge: the ledger's assumed premise "the turn originates in structure the one-block recursion omits" is contradicted in direction by run-level data — the illegal channel grows in weight while its skewness stays flat on T; demote it from LOAD-BEARING assumed to refuted-as-stated, and record the surviving weaker form (the omitted structure exists, in 2–11% of samples, and is not where the turn lives). (ii) A correction: "T and X1 turn back" is not one object; X1 is dropped from the phenomenon pending a pre-registered ≥6-file pool. Also correct my own round-1 error, flagged at [H1-24] — U′, not T, has the smallest C₁ — which removes the cut-size rationale I gave for T's asymmetry; the asymmetry stands as measured, unexplained. (iii) A probe: the bootstrap-versus-between-file discrepancy is file-localised, not generic; every future g₁ difference in this programme should be quoted from ≥3 independent files with a χ² consistency line attached.

Next case: the quantile map. Is the carpet's threshold distribution at 9L a fixed map of the one at L, and if not, which quantile nodes move and in which direction for T versus X — that is the object both the recursion and the per-sample splits have been proxying for.

_Contract check: House adopted a flagged claim: H2-6 (from Hypothesis); House FORCED (J1): 7: no derivation sketch — it belongs under ASSERTED; House claims without an object (J2): FORCED 7; VERDICT: 2 claim sentence(s) without an object tag, e.g. “Inflated for the χ² excess it is still ≈2.8σ.…”._

Checker — House final

30 claims: 18 ok, 4 mismatch, 0 sign conflict, 1 justify, 1 derivation, 6 uncheckable; 1 extracted claim(s) dropped because their quote is not in the checked text

Independence — record citations

• round 1: Ledger LD01, LD02 · Framing F03 · Hypothesis HY01, HY02, HY03 · Lateral none · Bound none
• round 2: Ledger LD01 · Framing none · Hypothesis HY01, HY02, HY03 · Lateral none · Bound none · Reductio none

Forecasts (parsed, for scoring)