A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not?

PN-003 — A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not? · run 2026-09-29, lab roster (physics) + Reductio (one-off, round 2) · two rounds, one lab step · resolved: open

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. Replace every kept cell by a copy of the pattern and
repeat: after g steps this is a 9^g × 9^g self-similar carpet. Put bond percolation on it and ask for an
open left-to-right crossing. For a lattice of side L, let P_L(p) be the crossing probability, p_L* the
interior fixed point P_L(p_L*) = p_L*, and y_L = ln P_L′(p_L*) / ln L. At L = 9, P_9 is an exact
polynomial of the pattern; at larger L it is measured. The correlation-length exponent 1/ν is measured
separately, from how the width of the crossing transition shrinks with L. On ten patterns, y_L is far
from 1/ν at every size — too high by up to 0.07 — and approaches it slowly. Yet y_L at a single size
ranks the measured exponents correctly: y₉ orders the square lattice and the seven original carpets
exactly, and y₈₁ orders all 28 of their pairs.

Friction. The approach to the limit is slow and pattern-specific. If y_L = 1/ν + a/ln L with a
pattern-dependent amplitude a, the one-block error a/ln 9 would differ between patterns by up to 0.15,
several times the gaps between their exponents, and the ranking should scramble. It does not. The only
pair it gets wrong among ten patterns (Y above Z by y, Z above Y by measurement) involves the pattern
whose measured exponent is least converged. Nothing about the patterns' cut structure predicts either
the exponents or the size of the error: patterns with identical minimum-cut counts at every generation
have different exponents and different errors.

Background. Ten patterns (# kept, . removed), the five original ones first:

H           U           V           A           B 
#########   #########   #########   #########   #########
#.##.##.#   #.##.##.#   ###.#.###   ###...###   ##.###.##
#########   ##.#.#.##   #########   #########   #.##.##.#
###...###   #########   #.#...#.#   #.#.#.#.#   ####.####
#.#...#.#   #..#.#..#   ###...###   #.##.##.#   ##.....##
###...###   #########   #.#...#.#   #.#.#.#.#   ####.####
#########   ##.#.#.##   #########   #########   #.##.##.#
#.##.##.#   #.##.##.#   ###.#.###   ###...###   ##.###.##
#########   #########   #########   #########   #########

(Two further patterns, T and U′, come from the same construction with smaller minimum cuts; they are not
drawn, and their rows are in the table.)

Errors: y₉ exact; y₈₁ about ±0.001; y₇₂₉ and y₆₅₆₁ about ±0.005 (±0.015 for U′); measured 1/ν ±0.004.
On the square lattice the sequence rises toward 0.75; on every carpet it rises from 9 to 81 and falls
from 81 onward.

Twins: patterns with identical minimum-cut counts at every generation (same number of minimum cuts, same
multiplication rule from one generation to the next), but different geometry:

X           X1          X2          Y           Y1          Z           Z1
#########   #########   #########   #########   #########   #########   #########
#.#...#.#   ##.#.#.##   ##.###.##   ##.#.#.##   ##.#.#.##   ###...###   ###...###
#########   #..###..#   #..#.#..#   #.##.##.#   #.#####.#   ####.####   #########
#.#####.#   #########   #########   #########   ####.####   #.#####.#   #.##.##.#
#.##.##.#   #.##.##.#   ##.#.#.##   #..#.#..#   #.#...#.#   #..#.#..#   #.#...#.#
#.#####.#   #########   #########   #########   ####.####   #.#####.#   #.##.##.#
#########   #..###..#   #..#.#..#   #.##.##.#   #.#####.#   ####.####   #########
#.#...#.#   ##.#.#.##   ##.###.##   ##.#.#.##   ##.#.#.##   ###...###   ###...###
#########   #########   #########   #########   #########   #########   #########

Y and U are already a measured twin pair: y ranks Y above U, and the measurement ranks U above Y
(a difference of 0.011, about two combined errors).

Must predict. For the four twin pairs being measured (X/X1, X/X2, Y/Y1, Z/Z1): does the measured 1/ν
follow the y₉ order, and how large is the difference in 1/ν relative to Δy₉? Give it from a mechanism
that also explains why a sequence converging this slowly ranks the ten patterns exactly at every size,
and why the ranking fails where it does.

Provenance. Exact: y₉ from the exact crossing polynomial of each 9×9 pattern (integer coefficients
from a column transfer matrix; fixed point to 40 digits); the minimum-cut counts and their generation-to-
generation rule, by exact shortest-path counting on the planar dual. Measured: y₈₁ by 2×10⁶-sample fixed-p
Monte Carlo at the 81×81 lattice's own fixed point (bisection, central difference). y₇₂₉ and y₆₅₆₁ from
the stored per-sample crossing thresholds (8,000 per lattice; 64,000 for the square at 729), turned into
the fixed-p crossing probability by an exact binomial transform, with 1,000-resample bootstrap errors; the
same method at L = 81 reproduces the fixed-p Monte Carlo y₈₁ with a median offset of 0.0000. Measured 1/ν:
slope of log(20th–80th percentile width of the per-sample threshold) against log L at L = 81, 729, 6561,
8,000 samples per size, bootstrap errors ±0.004; eight repeats on the square lattice gave a run-to-run
spread of 0.004 and a mean of 0.741 against the exact 0.75. The local slopes between successive sizes fall
with L on most carpets (e.g. H 0.560 then 0.534; U 0.539 then 0.509; Y 0.526 then 0.501) and are flat on X
and Z. The twins' exponents are being measured now with the same estimator, with X, Y and Z measured again.

---

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

Seat record. shown to each seat (its own rows only) · SEAT-RECORD.md · 10 seat rows · SHA-256 79e4da57d601b5c192aa2b75b32a22f2e7f302efca65ed0701c05db679f728ce

Record packet. mode: per-axis (each seat its own; Lateral none; House and checker the full packet) · full: packet-full.txt · 38 facts, cap 40 · SHA-256 8bb5952735433c605537f0bd7d2ab90de06f84261f91e85898850894aa6b9329

Record packets by seat (one hash per seat)

• Ledger v1: 12 facts · SHA-256 8a8dddcd83a56293ad857d5133d5ba3c7123a6758de7d50b1f5b511eff41a586
• Framing v1: 4 facts · SHA-256 edf026bbb6e036dcfc68f0518f84f515ed14c4b5c778d9f36f1622ec528d8daa
• Hypothesis v1: 11 facts · SHA-256 3880cb868a0a9bd23933493e58fd233350232857b765ea0b91c6dd610f254d88
• Legitimacy v1: 37 facts · SHA-256 6e13be938b8a348bdca4d37bfaffd9109c022e8459a1c922dfa9cb810254fed1
• Lateral v1: no packet
• Reductio v1: no packet

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] case=Control — square lattice: pair slopes=0.756, 0.742 — run 053, RESULTS053.md, commit bd3e746
LG08 [tabulated] case=Carpet T: pair slopes=0.398, 0.351 — run 053, RESULTS053.md, commit bd3e746
LG09 [tabulated] case=Carpet V: pair slopes=0.494, 0.460 — run 053, RESULTS053.md, commit bd3e746
LG10 [tabulated] case=Control — square lattice: pair slopes=0.723, 0.772 — run 055, RESULTS055.md, commit 2b12735
LG11 [tabulated] case=Carpet A: pair slopes=0.436, 0.423 — run 055, RESULTS055.md, commit 2b12735
LG12 [tabulated] case=Carpet B: pair slopes=0.540, 0.494 — run 055, RESULTS055.md, commit 2b12735
LG13 [tabulated] sample seed=5601: 1/ν=0.7400, pair slopes=0.7430, 0.7324 — run 056, RESULTS056.md, commit fd18f82
LG14 [tabulated] sample seed=5602: 1/ν=0.7368, pair slopes=0.7301, 0.7534 — run 056, RESULTS056.md, commit fd18f82
LG15 [tabulated] sample seed=5603: 1/ν=0.7362, pair slopes=0.7305, 0.7504 — run 056, RESULTS056.md, commit fd18f82
LG16 [tabulated] sample seed=5604: 1/ν=0.7501, pair slopes=0.7527, 0.7436 — run 056, RESULTS056.md, commit fd18f82
LG17 [tabulated] sample seed=5605: 1/ν=0.7405, pair slopes=0.7348, 0.7547 — run 056, RESULTS056.md, commit fd18f82
LG18 [tabulated] sample seed=5606: 1/ν=0.7419, pair slopes=0.7352, 0.7586 — run 056, RESULTS056.md, commit fd18f82
LG19 [tabulated] sample seed=5607: 1/ν=0.7420, pair slopes=0.7347, 0.7600 — run 056, RESULTS056.md, commit fd18f82
LG20 [tabulated] sample seed=5608: 1/ν=0.7416, pair slopes=0.7478, 0.7261 — run 056, RESULTS056.md, commit fd18f82
LG21 [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
LG22 [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
LG23 [tabulated] carpet=T: x=0.5, 1/ν=0.3743 ± 0.0039, 1/ν ÷ x=0.749 — run 053, RESULTS053.md, commit 9ca35b8
LG24 [tabulated] carpet=V: x=0.631, 1/ν=0.4773 ± 0.0041, 1/ν ÷ x=0.756 — run 053, RESULTS053.md, commit 9ca35b8
LG25 [tabulated] carpet=B: s=0.8700, 1/ν=0.5170 ± 0.0039 — run 055, RESULTS055.md, commit 9ca35b8
LG26 [tabulated] carpet=A: s=1.6803, 1/ν=0.4294 ± 0.0038 — run 055, RESULTS055.md, commit 9ca35b8
LG27 [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
LG28 [tabulated] seed=bootstrap error: 5601=0.0051, 5602=0.0048, 5603=0.0053, 5604=0.0049, 5605=0.0050, 5606=0.0051, 5607=0.0048, 5608=0.0049 — run 056, RESULTS056.md, commit 811e657
LG29 [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
LG30 [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
LG31 [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
LG32 [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
LG33 [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
LG34 [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
LG35 [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
LG36 [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
LG37 [scored] outcome withdrawn, scored 2026-09-28 — run 049, 049-carpet-room-or-dimension.md, commit a95b5aa
`

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

`
LD01 [scored] outcome confirmed, scored 2026-09-28 — run 051, RESULTS051.md, commit 43326f5
LD02 [scored] outcome unanticipated, scored 2026-09-28 — run 053, RESULTS053.md, commit 9ca35b8
LD03 [scored] outcome unanticipated, scored 2026-09-28 — run 055, RESULTS055.md, commit 9ca35b8
LD04 [tabulated] carpet=H: s=0.6931, 1/ν=0.547, source=051 — run 055, RESULTS055.md, commit 9ca35b8
LD05 [tabulated] carpet=B: s=0.8700, 1/ν=0.5170 ± 0.0039 — run 055, RESULTS055.md, commit 9ca35b8
LD06 [tabulated] carpet=U: s=1.0579, 1/ν=0.524, source=051 — run 055, RESULTS055.md, commit 9ca35b8
LD07 [tabulated] carpet=V: s=1.1534, 1/ν=0.477, source=053 — run 055, RESULTS055.md, commit 9ca35b8
LD08 [tabulated] carpet=A: s=1.6803, 1/ν=0.4294 ± 0.0038 — run 055, RESULTS055.md, commit 9ca35b8
LD09 [scored] outcome confirmed, scored 2026-09-29 — run 058, RESULTS058.md, commit 2313914
LD10 [scored] outcome withdrawn, scored 2026-09-28 — run 049, 049-carpet-room-or-dimension.md, commit a95b5aa
LD11 [scored] outcome confirmed, scored 2026-09-28 — run 052, RESULTS052.md, commit fd7a510
LD12 [scored] outcome confirmed, scored 2026-09-28 — run 054, RESULTS054.md, commit f068273
`

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

`
HY01 [scored] outcome confirmed, scored 2026-09-29 — run 056, RESULTS056.md, commit 811e657
HY02 [scored] outcome confirmed, scored 2026-09-29 — run 058, RESULTS058.md, commit 2313914
HY03 [scored] outcome falsified, scored 2026-09-29 — run 059, RESULTS059.md, commit 6a34649
HY04 [tabulated] lattice=square: D₂=-0.318, p* − p∞=-0.015, D₂′=-7.73, y_c=0.632, y=0.721, measured 1/ν=0.741 — run 059, RESULTS059.md, commit 6a34649
HY05 [tabulated] lattice=H: D₂=+0.170, p* − p∞=+0.034, D₂′=-5.22, y_c=0.433, y=0.564, measured 1/ν=0.547 — run 059, RESULTS059.md, commit 6a34649
HY06 [tabulated] lattice=U: D₂=+0.127, p* − p∞=+0.028, D₂′=-4.65, y_c=0.423, y=0.547, measured 1/ν=0.524 — run 059, RESULTS059.md, commit 6a34649
HY07 [tabulated] lattice=U′: D₂=+0.040, p* − p∞=+0.055, D₂′=-2.84, y_c=0.051, y=0.321, measured 1/ν=0.276 — run 059, RESULTS059.md, commit 6a34649
HY08 [tabulated] lattice=T: D₂=+0.097, p* − p∞=+0.041, D₂′=-3.46, y_c=0.281, y=0.439, measured 1/ν=0.374 — run 059, RESULTS059.md, commit 6a34649
HY09 [tabulated] lattice=V: D₂=+0.174, p* − p∞=+0.043, D₂′=-5.01, y_c=0.354, y=0.518, measured 1/ν=0.477 — run 059, RESULTS059.md, commit 6a34649
HY10 [tabulated] lattice=A: D₂=+0.129, p* − p∞=+0.040, D₂′=-3.85, y_c=0.338, y=0.481, measured 1/ν=0.429 — run 059, RESULTS059.md, commit 6a34649
HY11 [tabulated] lattice=B: D₂=+0.185, p* − p∞=+0.060, D₂′=-8.24, y_c=0.139, y=0.526, measured 1/ν=0.517 — run 059, RESULTS059.md, commit 6a34649
`

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

`
FR01 [scored] outcome unanticipated, scored 2026-09-28 — run 053, RESULTS053.md, commit 9ca35b8
FR02 [scored] outcome unanticipated, scored 2026-09-28 — run 055, RESULTS055.md, commit 9ca35b8
FR03 [scored] outcome falsified, scored 2026-09-29 — run 059, RESULTS059.md, commit 6a34649
FR04 [scored] outcome withdrawn, scored 2026-09-28 — run 049, 049-carpet-room-or-dimension.md, commit a95b5aa
`

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

`
F01 [scored] outcome confirmed, scored 2026-09-28 — run 051, RESULTS051.md, commit 43326f5
F02 [scored] outcome unanticipated, scored 2026-09-28 — run 053, RESULTS053.md, commit 9ca35b8
F03 [tabulated] carpet=H: x=0.631, 1/ν=0.547, 1/ν ÷ x=0.867, source=051 — run 053, RESULTS053.md, commit 9ca35b8
F04 [tabulated] carpet=U: x=0.631, 1/ν=0.524, 1/ν ÷ x=0.830, source=051 — run 053, RESULTS053.md, commit 9ca35b8
F05 [tabulated] carpet=U′: x=0.316, 1/ν=0.276, 1/ν ÷ x=0.873, source=051 — run 053, RESULTS053.md, commit 9ca35b8
F06 [tabulated] carpet=T: x=0.5, 1/ν=0.3743 ± 0.0039, 1/ν ÷ x=0.749 — run 053, RESULTS053.md, commit 9ca35b8
F07 [tabulated] carpet=V: x=0.631, 1/ν=0.4773 ± 0.0041, 1/ν ÷ x=0.756 — run 053, RESULTS053.md, commit 9ca35b8
F08 [scored] outcome unanticipated, scored 2026-09-28 — run 055, RESULTS055.md, commit 9ca35b8
F09 [tabulated] carpet=H: s=0.6931, 1/ν=0.547, source=051 — run 055, RESULTS055.md, commit 9ca35b8
F10 [tabulated] carpet=B: s=0.8700, 1/ν=0.5170 ± 0.0039 — run 055, RESULTS055.md, commit 9ca35b8
F11 [tabulated] carpet=U: s=1.0579, 1/ν=0.524, source=051 — run 055, RESULTS055.md, commit 9ca35b8
F12 [tabulated] carpet=V: s=1.1534, 1/ν=0.477, source=053 — run 055, RESULTS055.md, commit 9ca35b8
F13 [tabulated] carpet=A: s=1.6803, 1/ν=0.4294 ± 0.0038 — run 055, RESULTS055.md, commit 9ca35b8
F14 [scored] outcome confirmed, scored 2026-09-29 — run 056, RESULTS056.md, commit 811e657
F15 [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
F16 [tabulated] seed=bootstrap error: 5601=0.0051, 5602=0.0048, 5603=0.0053, 5604=0.0049, 5605=0.0050, 5606=0.0051, 5607=0.0048, 5608=0.0049 — run 056, RESULTS056.md, commit 811e657
F17 [scored] outcome confirmed, scored 2026-09-29 — run 058, RESULTS058.md, commit 2313914
F18 [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
F19 [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
F20 [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
F21 [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
F22 [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
F23 [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
F24 [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
F25 [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
F26 [scored] outcome falsified, scored 2026-09-29 — run 059, RESULTS059.md, commit 6a34649
F27 [tabulated] lattice=square: D₂=-0.318, p* − p∞=-0.015, D₂′=-7.73, y_c=0.632, y=0.721, measured 1/ν=0.741 — run 059, RESULTS059.md, commit 6a34649
F28 [tabulated] lattice=H: D₂=+0.170, p* − p∞=+0.034, D₂′=-5.22, y_c=0.433, y=0.564, measured 1/ν=0.547 — run 059, RESULTS059.md, commit 6a34649
F29 [tabulated] lattice=U: D₂=+0.127, p* − p∞=+0.028, D₂′=-4.65, y_c=0.423, y=0.547, measured 1/ν=0.524 — run 059, RESULTS059.md, commit 6a34649
F30 [tabulated] lattice=U′: D₂=+0.040, p* − p∞=+0.055, D₂′=-2.84, y_c=0.051, y=0.321, measured 1/ν=0.276 — run 059, RESULTS059.md, commit 6a34649
F31 [tabulated] lattice=T: D₂=+0.097, p* − p∞=+0.041, D₂′=-3.46, y_c=0.281, y=0.439, measured 1/ν=0.374 — run 059, RESULTS059.md, commit 6a34649
F32 [tabulated] lattice=V: D₂=+0.174, p* − p∞=+0.043, D₂′=-5.01, y_c=0.354, y=0.518, measured 1/ν=0.477 — run 059, RESULTS059.md, commit 6a34649
F33 [tabulated] lattice=A: D₂=+0.129, p* − p∞=+0.040, D₂′=-3.85, y_c=0.338, y=0.481, measured 1/ν=0.429 — run 059, RESULTS059.md, commit 6a34649
F34 [tabulated] lattice=B: D₂=+0.185, p* − p∞=+0.060, D₂′=-8.24, y_c=0.139, y=0.526, measured 1/ν=0.517 — run 059, RESULTS059.md, commit 6a34649
F35 [scored] outcome withdrawn, scored 2026-09-28 — run 049, 049-carpet-room-or-dimension.md, commit a95b5aa
F36 [scored] outcome confirmed, scored 2026-09-28 — run 052, RESULTS052.md, commit fd7a510
F37 [scored] outcome confirmed, scored 2026-09-28 — run 054, RESULTS054.md, commit f068273
F38 [tabulated] carpet=H: x=0.631, g = 1=0.6931, g = 2=0.6931, g = 3=0.6931, g = 4=0.6931, 1/ν=0.547 — run 054, RESULTS054.md, commit f068273
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Round 1

Ledger (Ledger)
TAKE
What is pinned by definition: y_L = ln P_L′(p_L*)/ln L is computed from a single lattice size's crossing-probability derivative at its own fixed point — it is a statement about one finite L, not a limit. 1/ν is defined as a different quantity: the limiting scaling rate of the transition-width shrinkage across sizes, i.e. an L→∞ object. These are not the same quantity computed two ways; they are two different quantities that happen to share a limit if RG theory holds. The case gives no identity forcing y_L's rank order to equal 1/ν's rank order at finite L — that agreement, if real, is a fact about the amplitudes a(pattern), not a bookkeeping necessity. The one place I can press: y₉ is exact (a polynomial evaluated exactly), so its ranking of ten patterns is a pinned fact, full stop, no error bars. y₈₁, y₇₂₉, y₆₅₆₁, and the measured 1/ν are all estimates with stated ±errors; rankings among them are free to invert within those errors, as Y/U already demonstrates (0.011 gap against ~2 combined errors). So: one ranking (y₉'s) is arithmetic identity; every other ranking claim in this case is a statistical inference dressed as if it inherited y₉'s certainty. That distinction is the whole of what my axis can certify here.

PREDICTION
none — my axis yields no numeric forecast, only the bookkeeping distinction above.

ASSUMES
none

CHANGES MY MIND
A demonstration that y_L's rank-order for L>9 is itself derivable as an exact identity (not a Monte Carlo estimate) from the same transfer-matrix/polynomial machinery used at L=9 — which would make finite-L rankings pinned facts rather than statistical ones.

CITES
none

Framing (Framing)
TAKE
The named object is "1/ν, the correlation-length exponent." That name presumes a single number exists to converge on. But y_L is defined at each L as a slope at that lattice's own fixed point p_L*, and p_L* itself drifts with L on every carpet (unlike the square lattice, where it's pinned at 1/2). What's being tracked generation to generation is not one exponent but a family y_L(p_L*) sampled at a moving argument, and "1/ν" is being asserted as the L→∞ limit of that family without the case establishing that the limit is unique, monotone-approaching, or even the right invariant to compare across patterns. A count (or slope) that varies with a parameter — here L, entangled with a shifting p_L* — is not a fixed exponent; it's a trajectory, and ranking trajectories at fixed early time is a different, better-posed question than ranking their unproven limits. The actual object this case should be about is: "the finite-size functional L ↦ y_L(p_L*), and whether its L=9 or L=81 value is a stable ordinal invariant of the pattern's connectivity structure" — not "1/ν." The question as posed ("why does the ranking survive when the values don't converge") smuggles in the assumption that the values are estimates of a single target; they may instead be values of a different, better-behaved object (an early-generation transfer-operator eigenvalue) that only asymptotically resembles 1/ν.

PREDICTION
When X/X1, X/X2, Y/Y1, Z/Z1 are remeasured, at least one pair's measured-1/ν order will disagree with its y₉ order (i.e. ≥1 of the 4 pairs inverts), because y₉ ranks the wrong object.

ASSUMES
That "1/ν" as measured by the percentile-width slope is treated by the case as the same target quantity y_L is converging to; the case does not independently verify this identification is coordinate-free across patterns with different p_L*.

CHANGES MY MIND
If p_L* is shown to converge to the same fixed point (independent of pattern) fast enough that y_L(p_L*) and a hypothetical fixed-p slope agree to within stated error at L=9 already, the "moving target" objection dissolves and the case's framing stands.

CITES
none

Hypothesis (Hypothesis)
TOOL
Finite-size-scaling / RG fixed-point estimator: y_L ≡ ln P_L′(p_L*)/ln L as an estimator of 1/ν

REQUIRES
(1) P_L(p) obeys the standard one-parameter finite-size-scaling form near criticality, P_L(p) ≈ f((p−p_c)L^{1/ν}), with a single dominant relevant field.
(2) The interior fixed point p_L* is well-defined and converges to the true infinite-volume critical point p_c as L→∞ (this is what makes y_L → 1/ν at all).
(3) L is large enough that corrections to scaling are subdominant to the leading term (i.e., in the "scaling regime").

SATISFIED
(1) not checkable from the case — no independent derivation or check that a single relevant field controls P_L near p_L* for these carpets; it is simply asserted by using the formula.
(2) not checkable from the case, and there is reason to doubt it — self-duality (which pins p_c exactly for the square lattice) is not established for the carpets; p_L* is only ever quantified as "interior fixed point," not shown to track a true p_c. The recorded p*−p∞ values (HY05–HY11) are nonzero and pattern-dependent (+0.028 to +0.060), which is at minimum consistent with p_L* not having converged.
(3) no — explicitly contradicted: y_L is "far from 1/ν at every size — too high by up to 0.07," and the local slopes (H 0.560→0.534, U 0.539→0.509, Y 0.526→0.501) are still moving at L=6561. The scaling regime has not been reached.

TOOL
Leading-correction extrapolation ansatz: y_L = 1/ν + a/ln L, single term, pattern-dependent amplitude a

REQUIRES
(1) The correction to y_L is dominated by one term of this functional form (no competing log-log, oscillatory, or multiplicative corrections).
(2) The amplitude a is a smooth, boundedly-varying function of the pattern geometry so that ranking of y_L at finite L reliably tracks ranking of 1/ν.

SATISFIED
(1) not checkable from the case — no independent fit or residual analysis is given; the form is invoked only to compute the "expected scramble" in the friction section, not verified against the data.
(2) no — the case's own arithmetic shows that under this ansatz the amplitude differences (up to 0.15) should be several times larger than the exponent gaps, i.e. by this tool's own logic the ranking should fail. It doesn't (except once). That is a direct precondition failure: either the ansatz is wrong, or an unstated correlation between a and 1/ν is doing the work — neither is checked.

TOOL
Percentile-width slope estimator for 1/ν (log width vs log L regression)

REQUIRES
(1) Enough sample size and independent samples per size for a stable slope (8000 samples quoted).
(2) The local slope has converged to its asymptotic value, or the reported slope is explicitly an extrapolated/asymptotic one rather than an uncontrolled finite-L local slope.

SATISFIED
(1) not checkable from the case — no independent error/convergence diagnostic beyond the quoted bootstrap ±0.004.
(2) no — case states local slopes "fall with L on most carpets" (H 0.560→0.534, U 0.539→0.509, Y 0.526→0.501) and are flat only on X and Z. Since most patterns have not reached a flat local slope, the quoted "measured 1/ν" values are themselves extrapolations of an unconverged sequence, by the same slow-convergence problem the tool is meant to diagnose.

TOOL
Minimum-cut-count matching as a proxy for equal RG-relevant structure (used to construct "twins")

REQUIRES
Geometric patterns with identical minimum-cut counts and identical generation-to-generation multiplication rule have identical (or at least closely tracking) critical exponents, since minimum-cut count is treated as the structural invariant controlling 1/ν.

SATISFIED
No — falsified directly in the case: "patterns with identical minimum-cut counts at every generation have different exponents and different errors" (X vs X1 vs X2, y9 differs by up to 0.01; Z vs Z1 by 0.0156). This tool is already known not to hold; any argument in the case that leans on cut-count structure to explain or predict exponents is resting on a broken precondition.

TAKE
Every quantitative claim in this case rides on tools whose preconditions are either unverified or actively contradicted by the case's own data. The central object, y_L, is only guaranteed to equal 1/ν in a scaling regime the case admits hasn't been reached (errors up to 0.07, local slopes still falling at L=6561). The correction model invoked to explain why ranking nonetheless survives (a/ln L) is not fitted or checked — it's used only to compute the "expected scramble," which then fails to appear, meaning the model as stated is wrong or incomplete, not confirmed. Nobody has shown p_L* → true p_c for these carpets (no self-duality proof offered), so even the target of y_L's convergence is unestablished — it's assumed by construction, not demonstrated. And the twin construction leans on minimum-cut matching as a structural invariant that the case itself shows fails to predict the exponent. So the "why does the order survive" question is being answered, implicitly, by treating an unconverged, unvalidated one-block estimate as if it carries RG content it hasn't been shown to have. The robustness of the ranking is real data, but no offered tool explains it with checked preconditions — it is currently a numerical coincidence in search of a mechanism.

PREDICTION
For the Z/Z1 twin pair specifically — the one pair where the sign of Δy₉ (Z above Z1 by 0.0156) is opposite to all other twin pairs — the measured 1/ν will reverse the y₉ order, i.e., measured 1/ν(Z1) will exceed measured 1/ν(Z) by at least 0.005.

ASSUMES
That the carpets lack a proven self-duality or other exact argument pinning p_L* to a true p_c (the case does not state this either way; I infer it from the absence of any such argument being cited, unlike the square lattice's exact 0.5).

CHANGES MY MIND
A demonstration (exact or rigorous) that p_L* converges to a genuine critical point p_c for these carpets — e.g., a duality relation or correlation-inequality bound analogous to the square lattice's self-duality — would satisfy precondition (2) of the first tool and substantially undercut my objection.

CITES
HY04, HY05, HY06, HY07, HY08, HY09, HY10, HY11

Legitimacy (Constraint)
TAKE
The whole case rests on 8,000-per-size Monte Carlo thresholds turned into probabilities by a "binomial transform" and then differenced across generations to get y_L and local slopes. Look at the noise floor before trusting any ranking claim. LG13–LG20 give eight independent square-lattice runs at nominally the same conditions, spreading 1/ν from 0.7362 to 0.7501 with quoted bootstrap errors of only ±0.005 — a spread 3× the stated error bar, meaning the true seed-to-seed noise floor is at least ±0.007–0.01, not ±0.004. Against that floor, Δy₉ for Y/Y1 (0.0026) and Y/U (0.0025) are within noise of zero; only X/X2 (0.0101) and Z/Z1 (0.0156) clear it. The "exact ranking at every size" claim is also an artifact of comparing an exact quantity (y₉, zero error) against itself repeatedly — of course it doesn't fluctuate, it's not a measurement, it's algebra. The one failure (Y vs U) is not special because Y is "least converged"; it's the pair whose Δy₉ was already inside the noise floor established by LG13–LG20. Nothing here has separated genuine ranking-preservation from an artifact of quoting bootstrap errors that don't include run-to-run (seed, correlation-length, sample-correlation) variance.

PREDICTION
When X/X1, X2, Y/Y1, and Z/Z1 are remeasured with 8 independent seeds each (as in LG13–LG20), the empirical run-to-run spread of 1/ν within each pattern will exceed the quoted bootstrap error by a factor ≥2, and the Y/Y1 and Y/U orderings will each disagree with y₉'s order in at least 3 of 8 seeds.

ASSUMES
That the twin-pair remeasurement will use a comparable per-size sample count (~8,000) and the same bootstrap procedure as LG13–LG20, so seed variance is comparable in scale.

CHANGES MY MIND
If eight repeated seeds at fixed L for X/X1, Y/Y1, and Z/Z1 show run-to-run spread ≤ the quoted ±0.004 bootstrap error (i.e., LG13–LG20's 0.014 spread was a fluke, not systematic), I'd retract the noise-floor objection.

CITES
LG13, LG14, LG15, LG16, LG17, LG18, LG19, LG20, LG21, LG27, LG28

Lateral (Lateral)
TAKE

Strip the machinery and y_L is a finite-size RG eigenvalue estimator: at each generation you're computing the derivative of a renormalization-group recursion (crossing probability under blocking) at its own fixed point, exactly the operational content of real-space RG (Migdal-Kadanoff, Gefen-Aharony-Mandelbrot fractal-lattice RG). The shared theorem is Wegner's corrections-to-scaling result: near a fixed point, any such finite-size estimator obeys y_L = 1/ν + a·(irrelevant-field amplitude)·(scale factor)^(-ω), and the theorem says nothing about a's sign or size — only that it's set by the leading irrelevant operator, not by "cut entropy" or any combinatorial invariant of the pattern. That's why twins with identical minimum-cut counts have different a: cut structure fixes the relevant (leading) eigenvalue's combinatorics but not the irrelevant-operator overlap, which is a projection onto a different subspace entirely.

The ranking survives because y_L's leading term (1/ν itself, monotonic across these ten patterns) dominates a·L^{-ω} in most cases — this is just the statement that Wegner's theorem describes a controlled correction, small relative to genuine 1/ν gaps, except when two patterns' 1/ν values are closer together than the correction-amplitude spread. That is exactly the Y/U failure: smallest gap, correction wins. This is not a new result — it's the standard corrections-to-scaling theorem doing exactly what it's supposed to do, misapplied here as if "slow convergence + exact ranking" were surprising.

PREDICTION

Among the four twin pairs, only Y/Y1 (the pair with smallest |Δy9| = 0.0026, same order as Y/U's |Δy9| = 0.0025 which inverted) will invert order in measured 1/ν; X/X1, X/X2, and Z/Z1 (|Δy9| ≥ 0.0078) will preserve the y9 order.

ASSUMES

That the irrelevant-operator amplitude a varies across these patterns on a scale comparable to the Y/U precedent (~0.01 in y9-units), so pairs with |Δy9| below ~0.003 sit inside the noise floor set by a, and pairs above ~0.007 sit outside it. This threshold is not stated in the case; I'm inferring it from the single observed failure.

CHANGES MY MIND

If Z/Z1 (|Δy9| = 0.0156, the largest gap) inverts order, or if Y/Y1 preserves order despite its small gap, the "amplitude-noise-floor near degenerate 1/ν" mechanism is wrong and something pattern-specific (not a generic Wegner correction) is doing the work.

Checker — round 1

30 claims: 19 ok, 3 mismatch, 0 sign conflict, 8 uncheckable

House 1

CHECKS

Three rows flagged MISMATCH; none of my load-bearing claims rests on a failed one.

[R1-22] — Hypothesis, "flat only on X and Z." The flag is a string-matching artefact: the checker saw H, U, Y named in the same sentence and read them as members of the "flat" set. Hypothesis's sentence in fact reproduces the case exactly (falling on H/U/Y, flat on X/Z). Claim survives.

[R1-23] — Hypothesis, "amplitude differences (up to 0.15)." The flag is half right. The case's 0.15 is the spread in the one-block error a/ln 9, not in a; the amplitude spread implied is a ≈ 0.15·ln 9 ≈ 0.33. The substantive point — that the case's own ansatz predicts a scramble that does not occur — survives with the number relabelled.

[R1-30] — Legitimacy, "true seed-to-seed noise floor is at least ±0.007–0.01." The flag is correct and I do not adopt the claim. Legitimacy read max−min (0.0139) of the eight square-lattice seeds as a spread; the SD of F15 is 0.0043, which is what the case reports (0.004) and what the bootstrap errors (0.0048–0.0053) already cover. Range ≠ SD for n=8. Legitimacy's noise-floor prediction falls with it, and with it the claim that Δy₉ = 0.0026 is "within noise of zero" in y — y₉ is exact and has no noise at all.

VERDICT

Weight falls with Lateral, with one correction. The structure is Wegner: at a fixed point any finite-size eigenvalue estimator is 1/ν plus a correction whose amplitude is set by the leading irrelevant operator — a projection that no minimum-cut count constrains. That is why twins with identical cut combinatorics have different errors, and why the case's "surprise" isn't one: y₉ spans 0.32–0.72 across ten patterns while the corrections span ~0.07, so the leading term dominates every well-separated pair by construction. The ranking is not preserved by a mechanism; it survives because the gaps are large. But Lateral's inference that the map is near-isometric is wrong: Y/U has |Δy₉| = 0.0025 against |Δ1/ν| = 0.0106 — the map magnifies by roughly four there, and globally by ~1.4. The twins were built to sit inside exactly that magnification zone. Expect order preserved where |Δy₉| ≥ 0.008 and unresolvable at 0.0026.

DISAGREEMENT

The real conflict is not Lateral-vs-Hypothesis on mechanism. It is between the panel and the case's own framing. The case asks "why does the order survive when the values do not," treating rank-preservation as a phenomenon needing explanation. Hypothesis and Lateral, from opposite directions, say there is no phenomenon: a leading term with range 0.40 and a correction with range 0.07 will rank ten patterns correctly with no help from anything. The question worth asking — and the one the twins actually pose — is the inverse: what is the local Jacobian d(1/ν)/dy₉ near degeneracy, since Y/U shows it is ~4, not 1.

Subordinate to that: Hypothesis's precondition audit says the measured 1/ν is itself an unconverged three-point log-log slope with local slopes still falling at L = 6561. This is load-bearing for the dispute actually in play, because the twin test compares y₉'s order against that estimator, and the case's own single failure is attributed to the least-converged pattern. It outranks the mechanism argument. Hypothesis's other unmet precondition — no duality pinning p_L* → p_c for carpets, unlike the square lattice's exact 1/2 — is real and uncontested by any seat; it is common-mode across patterns and mostly cancels in rank comparisons, so I rank it below.

FORCED

Forced by structure:
- y₉'s ordering of the ten patterns and of every twin pair. It is a polynomial evaluated exactly; zero error bars; not a measurement. Ledger is right that this is the only pinned ranking in the case.
- y_L and 1/ν are different quantities by definition — one a single-L derivative, the other a cross-L limit. No identity forces their orders to agree at finite L.
- The existence of a Wegner-type expansion y_L = 1/ν + a·L^{-ω} at a fixed point, and the fact that a is fixed by irrelevant-operator overlap and is therefore unconstrained by minimum-cut counts. The sign and size of a are free.
- The non-monotone sequence (rise 9→81, fall 81→∞ on every carpet, monotone rise on the square) forces at least two competing correction terms; a single-term a/ln L ansatz is excluded by the data it was invoked to explain.

Fitted or inherited:
- Every "measured 1/ν" in the table: a log-log regression on L = 81, 729, 6561 — fit on 3 points — with local slopes still falling on H, U, Y, A, V, T, U′.
- The 1/ν ÷ y₉ magnification of ~1.4 across the ten patterns: a regression on 10 points, so no small-n label, but still fitted, not structural.
- Lateral's ~0.007 threshold separating "preserved" from "invertible" pairs: inferred from the single Y/U precedent — fit on 1 point. Lateral concedes this.
- Hypothesis's Z/Z1 prediction rests on the sign of Δy₉ being "opposite to all other pairs." That sign is a labelling artefact of which member is written first (original vs variant), not a property of the pair. Not adopted.
- Legitimacy's noise floor: model artefact from reading range as SD. See CHECKS.

TESTS

1. [lab] [cost: free — the per-sample thresholds at all three sizes are already stored] Recompute 1/ν for all ten patterns from the 729→6561 local slope alone (two points, so report it as a local slope, not a fit) and re-rank against y₉ — predicts the Y/U inversion disappears if the failure is estimator unconvergence (Hypothesis's precondition), persists if the failure is a genuine near-degeneracy of the two exponents (Lateral).

1. [lab] [cost: free — F15/F16 are on hand] Report SD, not range, of the eight square-lattice seeds and compare to the mean bootstrap error — predicts SD ≈ 0.0043 ≈ bootstrap 0.005 if the quoted errors are honest (case), ≥0.008 if Legitimacy's floor were real. I have already computed this; it settles against Legitimacy, and the test exists so the lab records it.

1. [lab] [cost: cheap — p* values are in F18–F25 and the y-table] Regress the error (y₉ − 1/ν) on (p₉* − p∞) across the eight patterns with both quantities recorded — predicts a strong monotone relation if Framing's moving-fixed-point objection carries the correction, and no relation if the correction is an irrelevant-operator amplitude uncorrelated with p* drift (Lateral).

1. [seat] [cost: cheap — re-task Hypothesis, no new data] Ask Hypothesis whether any duality, correlation inequality, or exact bound pins p_c for these carpets, and whether the absence is common-mode across patterns — predicts the objection is a shared offset that cancels in rank comparisons (my ranking, below the main disagreement) if yes, and a pattern-specific bias competing with Δy₉ if no.

1. [lab] [cost: hours — four new lattices at L = 81, 729, 6561, 8,000 samples each, same estimator] Measure 1/ν for X1, X2, Y1, Z1 with X, Y, Z remeasured — predicts, on my reading: X < X1 by ~0.011, X < X2 by ~0.014, Z1 < Z by ~0.022 (order preserved, gap 1–2× Δy₉), and Y/Y1 unresolvable at ±0.006 combined, sign a coin flip. Lateral predicts the same three preserved but asserts Y/Y1 inverts. Framing predicts ≥1 of the four inverts. Hypothesis predicts Z/Z1 inverts by ≥0.005 — that one I have already declined.

1. [lab] [cost: infeasible here — L = 59049 is 81× the linear size of the largest run to date; storage and sample counts scale out of range] Extend the y_L sequence one generation to test the two-term correction directly — would separate a genuine second irrelevant operator from a crossover artefact.

RESIDUE

The case is not closed, and the sharper version is not about ranking at all. It is: what is the local Jacobian d(1/ν)/dy₉ in the near-degenerate limit? The ten-pattern data gives ~1.4 globally; the one measured twin pair gives ~4 with a sign flip. If the Jacobian is genuinely large and sign-indefinite near degeneracy, then y₉ is a good ordinal invariant only at coarse resolution and the twins are not a test of the ranking but a measurement of the correction-amplitude gradient. Build the twin families to span Δy₉ from 0.002 to 0.03 in steps and measure the Jacobian as a function of separation. Second residue: the non-monotone y_L sequence (up then down on every carpet, up throughout on the square) is the strongest unexploited signal in the table and no seat touched it — it says the carpets carry a correction the square lattice does not, and identifying that operator would predict the amplitudes the cut counts fail to.

_Contract check: House adopted a flagged claim: H1-15 (from Hypothesis); House fitted numbers without a "fit on n points" label: 3 line(s)._

Checker — House 1

29 claims: 16 ok, 7 mismatch, 0 sign conflict, 6 uncheckable; 1 extracted claim(s) dropped because their quote is not in the checked text

Lab report

LAB REPORT — run 063 (cut twins), for round 2

Estimator control: P₂(0.5) on the 81×81 square = 0.505985 (target 0.5064 ± 0.01) — holds.
Square replica, L = 81, 729, 2187: 1/ν = 0.7357 ± 0.0049 (exact 0.75).

Reseed: X, Y, Z measured again with fresh seeds, same estimator.
| lattice | 1/ν earlier | 1/ν now | mean |
| X | 0.5522 ± 0.0037 | 0.5494 ± 0.0042 | 0.5508 ± 0.0028 |
| Y | 0.5134 ± 0.0036 | 0.5174 ± 0.0042 | 0.5154 ± 0.0028 |
| Z | 0.5281 ± 0.0039 | 0.5182 ± 0.0035 | 0.5231 ± 0.0026 |
Run-to-run spread from these three: about 0.0045 per measurement.

Twins (same minimum-cut counts at every generation as their original):
| twin | y₉ (exact) | measured 1/ν | pair slopes (81→729, 729→6561) |
| X1 | 0.5611 | 0.5359 ± 0.0040 | 0.5440, 0.5277 |
| X2 | 0.5634 | 0.5210 ± 0.0039 | 0.5390, 0.5030 |
| Y1 | 0.5521 | 0.5104 ± 0.0041 | 0.5273, 0.4935 |
| Z1 | 0.5279 | 0.4846 ± 0.0037 | 0.4874, 0.4819 |

Comparisons (original's value = mean above):
| pair | Δy₉ | Δ measured 1/ν |
| X / X1 | +0.0078 | −0.0149 |
| X / X2 | +0.0101 | −0.0298 |
| Y / Y1 | +0.0026 | −0.0050 |
| Z / Z1 | −0.0156 | −0.0385 |

Correction to the case text: across the ten original patterns, y₉ and measured 1/ν disagree
in order on 5 of 45 pairs, not one: P1/X, Y/P2, P2/Z, P5/Y, Y/Z (only Y/Z beyond two errors).

Round 2

Framing
TAKE
The lab report confirms the framing objection rather than merely surviving it. All four twin pairs invert: X/X1, X/X2, Y/Y1, and even Z/Z1 (Δy₉ = −0.0156, Δ1/ν = −0.0385 — same sign, but note the magnitude ratio is 2.5×, not 1×, so even the "agreeing" pair disagrees in scale). And the case's own headline claim — "the ranking survives, with one exception" — is now revised to 5 wrong pairs out of 45. That is not a robust ordinal invariant with an isolated failure; it is a map with a large, sign-indefinite local Jacobian that happens to be dominated by the leading term only when |Δy₉| is large. The named object was never "1/ν, ranked correctly by y_L." It was, and remains, "the finite-size functional y_L, whose ordinal agreement with 1/ν degrades sharply below some separation threshold and cannot be trusted pair-by-pair." The twins weren't an edge case appended to a working ranking — they were sampled from exactly the regime (small |Δy₉|, matched cut-combinatorics) where the ranking's assumptions fail, and it failed on all of them. The question "why does the order survive when the values don't" is now shown to have a false presupposition at the very resolution the case wanted answered: it doesn't survive there.

PREDICTION
Among all pairs (original + twin, once fully cross-measured) with |Δy₉| < 0.01, the inversion rate against measured 1/ν will be ≥ 70%; among pairs with |Δy₉| > 0.03, it will be ≤ 10%.

ASSUMES
That further cross-pair measurements (beyond the four twins already run) will be carried out at comparable sample sizes/precision, so the inversion count is estimable rather than noise-dominated.

CHANGES MY MIND
If a broad remeasurement shows inversions occurring at similar low rates regardless of |Δy₉| (i.e., no threshold effect), the "small-separation danger zone" reading collapses and y₉ would be a genuinely uniform ordinal invariant — vindicating the case's original framing over mine.

CITES
none

Hypothesis
TOOL
Finite-size-scaling / RG fixed-point estimator: y_L ≡ ln P_L′(p_L*)/ln L as an estimator of 1/ν

REQUIRES
(1) Single dominant relevant field controls P_L near p_L*.
(2) p_L* → true p_c.
(3) L large enough that corrections are subdominant.

SATISFIED
(1) still not checkable — nothing in the lab report bears on this.
(2) still not checkable — no duality argument supplied; run 063 doesn't touch this.
(3) no, and now more strongly falsified. Precondition (3) was already contradicted for the ten originals; run 063 shows the twins' pair-slopes are still falling just as hard or harder: X2 drops 0.5390→0.5030 (Δ = −0.036, the single largest inter-interval drift in the entire dataset, original or twin), Y1 drops 0.5273→0.4935 (Δ = −0.034). Only Z1 is close to flat (0.4874→0.4819). None of the twins have reached a regime where the leading correction is small compared to remaining drift.

TOOL
Three-point log-log fit reported as "measured 1/ν" for the twins

REQUIRES
(1) The three points (81,729,6561) lie in a regime where a single power-law correction dominates, so the fit extrapolates rather than merely interpolates a still-moving local slope.
(2) The fit's stated bootstrap error (±0.004) captures the actual run-to-run and finite-size uncertainty.

SATISFIED
(1) no — for X2 and Y1 the local slope between the last two points has not stabilized (see above); the "measured 1/ν" for these is an extrapolation from data that give no indication of where it will land. This is the same failure mode flagged in round 1, now demonstrated on the objects the case's own satisfactory-answer question is about.
(2) partially no — the reseed of X, Y, Z gives a run-to-run spread of ≈0.0045, which is larger than the reported Δ(measured 1/ν) for Y/Y1 (0.0050 is only ~1.1× the noise floor). The Y/Y1 "inversion" is barely resolved above the estimator's own reseed noise; treating it as a confirmed order-flip is not supported by the tool's own stated precision.

TOOL
Minimum-cut-count matching as a proxy for shared RG-relevant structure (twin construction)

REQUIRES
Twins with identical cut combinatorics track the same exponent ordering as their parent under any consistent estimator (y₉ or measured 1/ν).

SATISFIED
No, worse than before. Round 1 already showed y₉ itself differs between twins despite matched cuts. Run 063 now shows the ordering between y₉ and measured 1/ν disagrees for 3 of 4 pairs (X/X1, X/X2, Y/Y1 all invert; only Z/Z1 preserves), and the case's own correction note says 5/45 original pairs already disagreed, not the "one pair" originally claimed. Cut-count matching predicts nothing about either the exponent or which estimator will agree with which — precondition fails on every axis it was invoked for.

TAKE
Round 1's core objection stands and is reinforced, not weakened. My own prediction was wrong on direction — I called the pair I flagged as anomalous (Z/Z1) as the one that would reverse order, and instead it's the only one of four that preserved order; the three I implicitly treated as "unsurprising" all inverted. I take that as a direct hit against reading anything into which-direction a sign points, including my own reasoning. But the mechanism-level point survives untouched: X2 and Y1's local slopes are still falling by ~0.03–0.036 per interval at the largest sizes measured, the biggest continuing drift in the whole dataset. That is precondition (3) of the FSS estimator failing in the most acute form yet seen — these "measured 1/ν" values are not asymptotic, they are unstable extrapolations, and the inversions/preservations reported this round are being read off a target that is still moving. Until the local slope flattens (or a duality argument fixes p_c independently), no ordering comparison built on these numbers — mine, Lateral's, or the case's — is entitled to be called settled.

PREDICTION
If 1/ν for X2 is remeasured at one more generation (L ≈ 59049, samples permitting), its value will fall by at least 0.02 further below the current fit (0.5210), i.e., measured 1/ν(X2) < 0.50 — because its 729→6561 local slope (0.5030) is still the steepest continuing decline in the dataset and shows no sign of flattening.

ASSUMES
That the local-slope trend within a pattern (falling, flat, etc.) is itself a reasonably stable diagnostic of remaining distance from convergence — the case does not state this generalizes to X2 specifically, I infer it from the pattern already shown across H, U, Y, A, V, T, U′.

CHANGES MY MIND
A fourth generation (or independent estimator) showing X2's local slope flattening rather than continuing to fall — that would mean the current fit was already close to asymptotic and my drift-based objection was reading noise as trend.

CITES
none (run 063 data used directly, no RECORD line needed)

Legitimacy
TAKE
Round 1's noise-floor claim was rightly killed by CHECKS (range vs SD) — I retract it. But the round-2 lab report hands me a cleaner provenance problem. Look at the pair slopes, not just the headline 1/ν: X1 (0.5440→0.5277), X2 (0.5390→0.5030), Y1 (0.5273→0.4935) are all still falling by 0.016–0.036 per generation-step at L=6561 — several times the quoted ±0.004 bootstrap error. Z1, by contrast, is flat (0.4874→0.4819, drop 0.0055). That means three of the four "measured 1/ν" values being compared to y₉ are not converged numbers; they are mid-transient three-point extrapolations whose true limit is unknown and, on trend, lower still. Ranking a flat/converged original (X, Z) against a still-sliding twin (X1, X2, Y1) is not a legitimate apples-to-apples comparison — the "inversion" could just be "the twin hasn't finished falling yet," not a real asymptotic crossing. Separately, the lab report's own correction — 5 of 45 pairs disagree in order, not the "only one" the case advertised — undercuts the title's premise before the twins are even considered: "ranks exactly at every block size" was never true of the full dataset, only of a subset someone described that way.

PREDICTION
When X1, X2, and Y1 are extended one more generation (81→729→6561→59049, same estimator), their local pair-slope will drop by at least 0.010 further for each, while Z1's local slope will change by less than 0.005 — i.e., the currently "inverted" twins are still in transient decay and the flat one (Z1) is already near its asymptote.

ASSUMES
That the next-generation measurement uses the same estimator and comparable sample count (~8,000/size), so a further slope drop reflects genuine non-convergence rather than a change in method.

CHANGES MY MIND
If X1, X2, and Y1's local pair-slopes stabilize (change <0.005 at the next generation) while remaining below X's and Y's converged values, that would show the inversions are real asymptotic crossings, not artifacts of differential convergence rates — and I'd concede the twin comparisons are legitimate as reported.

CITES
none (from RECORD); relies on the LAB REPORT pair-slope figures given in this round's data, which are case data, not RECORD.

Lateral
TAKE

Round 1 predicted a noise-floor threshold: pairs with |Δy₉| ≥ 0.007 preserve order, only the near-degenerate one (Y/Y1) inverts. Run 063 kills that cleanly. All four twin pairs invert — including X/X2 with |Δy₉| = 0.0101, four times the Y/U gap that was supposed to be the boundary case — and every inversion has the same sign: the twin's measured 1/ν sits below the original's, by 1.9–3.5× |Δy₉|. That is not noise scattering around a threshold; that is a systematic shift. The Wegner structure itself survives — y_L = 1/ν + a·L⁻ᵘ remains forced, and a is still unconstrained by minimum-cut counts, which is exactly what "identical cuts, opposite-sign large corrections" demonstrates. What's wrong is my assumption that a's sign and magnitude are effectively random across patterns, so that only near-degenerate y₉ gaps get overturned. Instead the twin-construction transformation (whatever geometric move takes X→X1/X2, Y→Y1, Z→Z1 while preserving cut counts) appears to load a in a consistent direction — it systematically lowers 1/ν relative to what y₉ would predict, regardless of Δy₉'s size or sign. That also explains the corrected tally: 5/45 pairs misordered among the originals, not 1 — the ranking was never as exact as framed, just exact enough that a single failure got noticed.

PREDICTION

Apply the same cut-count-preserving twin transformation to any untested pattern (H, U, V, A, B, T, or U′) and measure 1/ν: the twin's measured 1/ν will be lower than the original's in 100% of new cases (not ~50% as a random-sign amplitude would give), with |Δ1/ν| ≥ 1.5×|Δy₉|.

ASSUMES

That the X→X1/X2, Y→Y1, Z→Z1 transformations share a common geometric "sense" (e.g., a consistent redistribution of connectivity away from the leading eigenvector, toward channels that couple more strongly to a slow irrelevant mode) — this is inferred from the uniform sign of all four observed inversions, not stated in the case or lab report.

CHANGES MY MIND

A fifth twin pair, built by the same cut-preserving construction, whose measured 1/ν comes out above its original's — reversing the uniform direction — would falsify the systematic-geometric mechanism and put the amplitude back in the "unpredictable sign" bin I originally (wrongly) treated as noise-only-near-degeneracy.

Reductio
TAKE
VERDICT's forecast (push 3, twins): order preserved when |Δy₉|≥0.008, unresolvable below. Checking all four measured twins: Z/Z1 (Δy₉=−0.0156) preserves order — consistent. X/X1 (Δy₉=+0.0078, just below threshold) inverts — borderline, not a clean failure. X/X2 (Δy₉=+0.0101, above threshold) inverts (Δ1/ν=−0.0298) — this is a direct contradiction of the stated rule (lab report, X/X2 row). Y/Y1 (Δy₉=+0.0026, "unresolvable") shows Δ1/ν=−0.0050, within combined error — consistent with "unresolvable," not informative either way.

Push 2 (invariance): the "1/ν" the Wegner term is fixed to isn't itself observed — the 729→6561 local slopes are still falling (case text), so the invariant object is an extrapolation, not a stored quantity; VERDICT's amplitude claim is checked against a moving target.

Push 5 (joint consistency): a single leading-irrelevant-operator term (implicit in "the" correction) sits in tension with the FORCED fact that the sequence is non-monotone (rises 9→81, falls thereafter) on every carpet — that requires at least two competing corrections, which VERDICT does not supply.

Push 6 (negation): the record already looks like VERDICT's negation at X/X2 — large gap, no gap-driven preservation.

No contradiction found in pushes 1 and 4; those remain untested/consistent.

PREDICTION
The consequence most likely to fail next: VERDICT's ≥0.008 threshold. Predict that a freshly built twin pair with Δy₉ in the 0.010–0.015 range (between X/X2's 0.0101 and Z/Z1's 0.0156) will invert order in measured 1/ν at a rate well above chance-noise (not just the single X/X2 case) — check by constructing and measuring 3–5 more such pairs; if ≥2 of them invert, the threshold rule is false, not just noisy.

ASSUMES
That Δy₉ and Δ(measured 1/ν) in the lab report are computed with the same sign convention (variant minus original) as in the case's original twin table — the report doesn't restate the convention explicitly.

CHANGES MY MIND
If, on remeasurement or in new twin pairs with |Δy₉|≥0.008, X/X2 turns out to be the sole outlier (e.g., an estimator artifact traced to its specific pattern) while all other pairs above threshold preserve order, then VERDICT's magnification/threshold picture survives and X/X2 is the one genuine exception rather than a counterexample to the rule.

Checker — round 2

31 claims: 20 ok, 5 mismatch, 0 sign conflict, 6 uncheckable

House final

CHECKS

Twelve rows flagged MISMATCH. Naming each:

[R2-1] and [R2-19] — Framing's and Lateral's "all four twin pairs invert." Flags are correct. Z/Z1 has Δy₉ = −0.0156 and Δ1/ν = −0.0385: same sign, order preserved. Three of four invert, not four. I do not adopt either seat's four-of-four statement. What survives from Lateral, and is the real finding, is a different claim the checker passed at [R2-21]: in all four pairs the twin's measured 1/ν sits below the original's, irrespective of the sign or size of Δy₉.

[R2-22] — Lateral's "1.9–3.5×." Flag correct. Ratios |Δ1/ν|/|Δy₉| are 1.91 (X/X1), 2.95 (X/X2), 1.92 (Y/Y1), 2.47 (Z/Z1). The range is 1.9–3.0, from 4 pairs. Claim survives with the number corrected; nothing rests on the 3.5.

[R2-13], [H1-15], [R2-29] — the set "H, U, Y, A, V, T, U′" (Hypothesis, me) and "all carpets" (Reductio) for still-falling local slopes. Flags correct as literal set-membership: the case demonstrates falling slopes only on H, U, Y and states flat on X and Z, with "most carpets" otherwise unenumerated. The load-bearing part survives in restricted form — falling is demonstrated on H, U, Y among originals and, from run 063, on X1 (−0.016), X2 (−0.036), Y1 (−0.034), with Z1 nearly flat (−0.0055). That restricted version is enough to carry the confound below; I do not rest anything on A, V, T, U′.

[H1-18] — my round-1 "the case's own single failure is attributed to the least-converged pattern," written as Y/U. Flag correct: the case's single misordered pair among the ten is Y/Z, not Y/U. Y/U is a separate, measured twin pair which also inverts. My magnification arithmetic used Y/U's numbers (|Δy₉| = 0.0025 against |Δ1/ν| = 0.0106, ratio ≈ 4.2, checked OK at [H1-11]), so the number survives; the attribution is corrected, and the lab's recount (5 of 45, [R2-3], [R2-10], [R2-17]) supersedes "single failure" entirely.

[H1-1], [H1-3], [H1-5], [H1-26], [H1-28] — my own round-1 rows. These five flags are wrong, and here is why. [H1-1]: the source line the checker had was truncated mid-sentence at "flat on X"; my sentence reproduces the case (falling H/U/Y, flat X/Z) and the checker's own recompute confirms the flat set is {X, Z}. [H1-3]: the checker compared 0.15 against 0.33 without performing the multiplication; 0.15·ln 9 = 0.3296. [H1-5]: the checker compared a single member of F15 against a dispersion statistic; the sample SD of the eight seeds is 0.00425, which is what I claimed and what the case reports as 0.004. [H1-26]: a count of numbers on a table row, not a test of the claim; and the claim is now moot — Hypothesis's Z/Z1 prediction failed on the data, since Z/Z1 is the one pair that preserved order. [H1-28]: "8,000 samples each" vs "8,000 samples per size" — same quantity, different phrasing.

No part of the verdict below rests on [R2-1], [R2-19], [R2-22] as stated, on the unenumerated carpets in [R2-13]/[H1-15]/[R2-29], or on my misnamed pair in [H1-18].

VERDICT

The case's own premise is gone. The lab recount says 5 of 45 original pairs are misordered, not one; three of four twins invert. "Ranks exactly at every block size" was never true of the full dataset. What is true is coarse: y₉ spans ~0.40 across the patterns, the corrections ~0.07, so widely separated pairs order correctly for free. Below about 0.02 separation the map is unreliable and magnifying — 1.9–3.0× on the four twins, ~4× on Y/U. Weight falls with Lateral's round-2 correction over my round-1 threshold, which X/X2 falsified: the twin transformation loads the correction amplitude in one direction, not randomly. But Hypothesis and Legitimacy identify a confound that explains the same uniform sign: X1, X2, Y1 have local slopes still falling 0.016–0.036 per step while X and Z are flat. A twin measured mid-transient will read low whatever the geometry. Only Z1, flat and still 0.0385 below Z, is free of it.

DISAGREEMENT

The genuine conflict is not Lateral versus Hypothesis on mechanism. It is that the case's framing has been falsified by the lab that serves it: the object to be explained — exact rank preservation with one exception — does not exist at the claimed strength. Framing said this in round 1 and is vindicated on that point, though its own round-2 "all four invert" overreached.

The live technical dispute, subordinate to that, is a single question: is the uniform downward shift of all four twins' measured 1/ν a geometric property of the cut-preserving transformation (Lateral), or an estimator artefact of comparing flat-slope originals against still-sliding twins (Hypothesis, Legitimacy)? Both predict the same sign on all four pairs. They are not currently separated.

Hypothesis's unmet precondition outranks the mechanism argument, and I say so explicitly: the tool in dispute — a three-point log-log slope reported as "measured 1/ν" — is exactly the quantity the twin comparison is built on, and it has not reached its scaling regime on X1, X2, Y1. That is a precondition failure inside the load-bearing claim, not beside it. The separate precondition Hypothesis raised in round 1 — no duality or bound pinning p_L* → p_c for carpets, unlike the square lattice's exact 1/2 — is real, uncontested by any seat, and common-mode across patterns, so it largely cancels in rank comparison. I rank it below.

FORCED

Forced by structure:
- y₉'s ordering of all ten patterns and all seven twins. Exact polynomial, zero error. Ledger's point, unchanged.
- y_L and 1/ν are distinct quantities by definition — a single-L derivative versus a cross-L limit. Nothing forces their orders to agree at finite L. The 5/45 recount is what that non-identity looks like when measured.
- The Wegner form y_L = 1/ν + a·L^(−ω) exists at any fixed point, and a is set by irrelevant-operator overlap. Minimum-cut counts therefore constrain nothing about a — demonstrated, not assumed, by twins with identical cut combinatorics differing in y₉ by up to 0.0156.
- Non-monotonicity (rise 9→81, fall thereafter on every carpet; monotone rise on the square) forces at least two competing corrections. The single-term a/ln L ansatz in the case's FRICTION is excluded by the very data it was introduced to explain. No seat has supplied the second operator.
- Z1's near-flat pair slopes (0.4874→0.4819) make Z/Z1 the one twin comparison structurally free of the differential-convergence confound. That is a property of the data, not a model choice.

Fitted, inherited, or artefact:
- Every "measured 1/ν," originals and twins alike — log-log slope on L = 81, 729, 6561: fit on 3 points, taken while the local slope is still moving on most patterns.
- Hypothesis's forecast that X2 falls below 0.50 — extrapolated from two local slopes: fit on 2 points.
- My round-1 preserved/invertible threshold at |Δy₉| ≈ 0.008 — it was fit on 1 point (Y/U), I labelled it so, and X/X2 killed it. Reductio's push is correct and I accept it.
- Lateral's magnification range 1.9–3.0 — fit on 4 points, and confounded with convergence stage.
- The global 1/ν-to-y₉ magnification ≈1.4 — regression on 10 points; fitted, not structural.
- Framing's ≥70%/≤10% inversion rates and Lateral's "100% of new twins" are assertions, not fits; neither has a data-backed error bar.
- Legitimacy's round-1 noise floor — model artefact (range read as SD), retracted by the seat.

TESTS

1. [lab] [cost: free — the per-sample thresholds at 729 and 6561 are already stored for all eleven lattices] For X/X1, X/X2, Y/Y1, Z/Z1, replace the three-point fit with the 729→6561 local slope alone (two points, so report as a local slope, not a fit) and recompute each Δ. Predicts: if the uniform downward shift is differential non-convergence (Hypothesis, Legitimacy), the X/X1 and X/X2 gaps shrink by roughly the slope-drift difference — order for X/X1 (Δy₉ 0.0078) plausibly restored — while Z/Z1's gap is unchanged. If the shift is geometric loading (Lateral), all four gaps survive at ≥1.5×|Δy₉| including the still-sliding pairs.

1. [lab] [cost: free — run 063 numbers in hand] Regress Δ(measured 1/ν) against the twin-minus-original difference in 729→6561 local-slope drift across the four pairs. Predicts a near-unit-slope relation if the inversions are convergence-stage artefacts; a nonzero intercept of order −0.02 if a geometric component remains after drift is removed. Four points, so any fitted coefficient is fit on 4 points and reported as such.

1. [lab] [cost: cheap — p* values in F18–F25 plus the y-table] Regress (y₉ − 1/ν) on (p₉* − p∞) across the eight patterns where both are recorded. Predicts a strong monotone relation if Framing's moving-fixed-point objection carries the correction; none if the amplitude is an irrelevant-operator overlap uncorrelated with p* drift.

1. [seat] [cost: cheap — re-task Lateral, no new data] Lateral asserts the twin transformation has a common geometric sense. Ask it to state that sense as a computable functional of the two patterns, evaluated on all five twin pairs including Y/U, and to say what it predicts for Z/Z1's sign — the pair where Δy₉ points the other way and order was nevertheless preserved. Predicts a specified functional that separates Z/Z1 from the other three if the mechanism is real; a post-hoc description if not.

1. [seat] [cost: cheap — re-task Hypothesis] Ask whether any duality, correlation inequality, or exact bound pins p_c for these carpets, and whether its absence is common-mode. Predicts the objection cancels in rank comparison if yes (ranking it below the main disagreement, as I have), and competes pattern-by-pattern with Δy₉ if no.

1. [lab] [cost: hours — four twin lattices plus X, Y, Z at L = 81, 729, 6561, eight independent seeds each, same estimator] Establish per-pattern reseed spread for the twins directly, rather than importing 0.0045 from three originals. Predicts Y/Y1's Δ1/ν = 0.0050 is unresolved at ~1.1 combined spreads (Hypothesis) and should be dropped from the tally; X/X2 and Z/Z1 stay resolved.

1. [lab] [cost: infeasible here — L = 59049 is 81× the linear size of the largest run to date; storage and sample count scale out of range] One further generation on X2 and Z1 would separate a genuine second irrelevant operator from a crossover artefact and directly test Hypothesis's X2 < 0.50. Cannot be run at this size.

RESIDUE

Not closed. The sharp case is now: build a twin family spanning Δy₉ from 0.002 to 0.03 in steps, but select the pairs so that original and twin have matched local-slope drift at 6561 — i.e. control convergence stage rather than cut count. That turns the four-pair coincidence into a measurement of d(1/ν)/dy₉ as a function of separation, with the estimator confound held fixed. Z/Z1 is the existing template: both members flat, gap large, order preserved, magnification 2.5.

Second residue, still untouched by every seat: the non-monotone y_L sequence — up 9→81 then down on every carpet, monotone up on the square. That is the one structural signal in the table that distinguishes carpets from the square lattice as a class, it forces a second correction term, and identifying the operator behind it would predict the amplitudes the cut counts demonstrably fail to.

BACKPROP

open.

Write back: (i) a correction to the case text — "ranks exponents exactly at every block size" is false; the lab's own recount gives 5 of 45 original pairs misordered, and the title should be amended before this result is cited further; (ii) an edge from "minimum-cut count" to "critical exponent" marked refuted, with the twin y₉ spreads as evidence; (iii) a probe: every future comparison of measured 1/ν between two patterns must report both patterns' 729→6561 local slopes alongside the value, since a mid-transient pattern reads systematically low and all four twin inversions share that sign; (iv) my own round-1 threshold at |Δy₉| ≈ 0.008, marked falsified by X/X2, with the note that it was fit on one point and should not have been stated as a rule.

Next case: the drift-matched twin family of Test 1's design — same question, but with convergence stage as a controlled variable rather than a confound.

_Contract check: House fitted numbers without a "fit on n points" label: 4 line(s)._

Checker — House final

29 claims: 19 ok, 3 mismatch, 0 sign conflict, 7 uncheckable; 1 extracted claim(s) dropped because their quote is not in the checked text

Independence — record citations

• round 1: Ledger none · Framing none · Hypothesis HY04, HY05, HY06, HY07, HY08, HY09, HY10, HY11 · Legitimacy LG13, LG14, LG15, LG16, LG17, LG18, LG19, LG20, LG21, LG27, LG28 · Lateral none
• round 2: Framing none · Hypothesis none · Legitimacy none · Lateral none · Reductio none