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

---

type: lab

field: physics

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.

Correction (2026-09-29, after the panel's round 1; the case text above is left as the panel read it). The friction's "the only pair it gets wrong among ten patterns (Y above Z …)" is false. Ordering all ten patterns by y₉ and by measured 1/ν, they disagree on 5 of 45 pairs: H/X (Δy₉ +0.011, measured −0.005), U/Y (−0.0025, +0.011), U/Z (+0.0035, −0.004), B/Y (−0.024, +0.004) and Y/Z (+0.006, −0.015). "Exact" holds for the square lattice and the seven original carpets (all 28 pairs); the three later patterns X, Y, Z carry four of the five disagreements. Only Y/Z exceeds two combined measurement errors (2.7); the other four measured gaps are 0.65–1.9 combined errors. The codifier's error: a claim true of eight lattices was extended to ten without recounting. The premise audit, which would have checked it, failed to run.

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Not part of the case — the codifier's sealed predictions (2026-09-29, before the panel runs and before any twin exponent exists)

The codifier's own lead, withheld from the case text: the ranking survives because the amplitude a and the exponent move together across these patterns (a pattern with a larger exponent also has a larger one-block error), so differences in a/ln L largely cancel within the ranking; where two patterns are close in y (|Δy₉| < 0.005) the cancellation is incomplete and the order is a coin flip, as Y/U shows. For the twins: Z1's measured 1/ν is below Z's by at least 0.008 (Δy₉ = −0.016); X1 and X2 are above X by less than their Δy₉ (so within noise); Y/Y1 is undetermined. These are scored against run 063.

Predictions of what the v5 panel will do: 1. At least two seats tie the exact ranking to a single pattern parameter that sets both the exponent and the size of the one-block error (a co-variation account), and at most one seat treats the ranking as coincidence. 2. Legitimacy cites a pair-slope drift from its record packet (not from the case) in arguing that Y's measured exponent is unconverged. 3. The checker pass flags at least one claim that is wrong on its own source, and the House addresses it by name. 4. The House predicts Z1 below Z, and gives no prediction for Y/Y1 beyond "within noise". Scored against the panel record once it is filed (the next panel number).

The codifier's lead, scored 2026-09-29 against run 063 (cdd879b)

Z1 below Z by at least 0.008: held (0.0385). X1 and X2 above X by less than their Δy₉: failed — both measure below X, by 0.0149 and 0.0298, against Δy₉ = +0.0078 and +0.0101. Y/Y1: −0.0050, within reseed spread (P-048). The co-variation account predicted that y₉ would keep the order when |Δy₉| is not small; on the X twins it does not. With Y/P2, four of five measured twin pairs run against y₉. The friction of this case sharpens: the exact ranking holds across patterns with different cut counts and breaks between patterns with the same ones.

Scored 2026-09-29, after the panel ran (PN-003)

The codifier's sealed panel predictions — 2 of 4 held. 1. Failed. No seat gave a co-variation account (one parameter setting both exponent and error). Lateral read the ranking as a Wegner correction dominated by large gaps; Hypothesis called it "a numerical coincidence in search of a mechanism"; Framing said y ranks a different object; Ledger gave bookkeeping only. 2. Failed. Legitimacy argued from the square-lattice seed spread (LG13–LG28), not from a pair-slope drift, and did not argue about Y's convergence in round 1. 3. Held. The checker flagged Legitimacy's noise floor as wrong on its own source (R1-30, range read as SD); House 1 addressed it by name and recomputed SD = 0.0043. 4. Held. House 1 put Z1 below Z (by ≈ 0.022; measured 0.0385) and called Y/Y1 unresolvable (measured −0.0050, within reseed spread per P-048).

The House's free test, run by the codifier on numbers in hand (RESULTS063.md, cdd879b) — and a sign error in the House's confound. The final House reads the uniform downward shift of all four twins as possibly an estimator confound: "a twin measured mid-transient will read low." The direction is backwards. When a pattern's local slopes fall with L, a three-point fit sits above its last local slope, so a still-sliding pattern's fitted 1/ν reads high against its limit, not low. Test 1 (replace the fit by the 729→6561 local slope) gives, twin minus original, both from run 063: X1 −0.0243, X2 −0.0490, Y1 −0.0141, Z1 −0.0275 — every gap wider than on the fit, all four still below, each ≥ 1.76 × |Δy₉|. That is the outcome the House assigned to Lateral's geometric reading; the confound reading predicted the X gaps would shrink and X/X1's order plausibly restore, and neither happens. Caveats: local slopes carry larger errors than the fit (not yet bootstrapped), the originals Y and Z also fall in run 063 (0.5273 → 0.5076, 0.5269 → 0.5094) while X rises slightly, and this is a post-hoc reading of a House test, not a sealed run.

Case text correction (House BACKPROP i). "Ranks exponents exactly at every block size" overstates the record: y₉ and measured 1/ν disagree on 5 of 45 pairs among the ten patterns (only Y/Z beyond two errors), and three of the four measured twin pairs invert. The ranking holds coarsely — for well-separated patterns — and fails between patterns with matched cut structure.

Checker precision this panel (codifier's reading of every mismatch row): 18 mismatches, 9 real (R1-23 partly, R1-30, H1-15, H1-18, R2-1, R2-13, R2-19, R2-22, R2-29), 9 false. The false ones share four causes: a dispersion statistic compared with one member of the list (H1-5, H2-20); arithmetic not performed (H1-3); a source line truncated mid-sentence (H1-1); a contrast sentence naming both sets read as membership (R1-22, H2-8, H2-13), plus two phrasing or row-count artefacts (H1-26, H1-28). Missed: the House's mid-transient sign error above, which is a directional claim inside an argument, not a quoted number.

Reductio seat (one-off, first of two cases). It found X/X2 contradicting the House's threshold rule and the one-term tension with the non-monotone sequence; Lateral raised the first and House 1's FORCED the second. No contradiction unique to it this case.

Record status, 2026-09-29. The runs this case builds on, as scored: run 061 falsified (its own-fixed-point clause; y₂ and the new rooms), run 063 confirmed (twins' misses differ; clause B inherited), run 066 confirmed (the block sequence turns over on every carpet), run 058 confirmed.

Status, 2026-10-03 (codifier). No run or panel has named this case since PN-003. What has happened to its question — why block estimates rank exponents the same at every block size while their values move — is that the ranking got a predictor: run 080 confirmed run 071's frozen rule out of sample, 1/ν = y₉ (the one-block polynomial's own exponent) plus a class-and-ln N₁ offset, within 2σ on six held-out patterns. A block estimate at any size is a function of the same polynomial, so it ranks as y₉ ranks; the order survives because it is the polynomial's, and the values move because the offset is the lattice's (PG-01, scaffold S3, where the question of what the offset is continues). The case is left open rather than closed, since its must-predict asked for a mechanism and a rule is not one; it is dormant until S3 is decided. Not re-panelled.