A one-block polynomial ranks eight lattices' percolation exponents exactly and misses every value — what sets the size of the miss?

For the panel, paste only the seven fields below (type · field · object · friction · background · provenance · must predict), or use the tool's Load case file, which reads exactly these and ignores the last section. The last section is not part of the case.

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type: lab

field: physics

object. Take a 9×9 grid of cells, keep 64 of them and remove 17, keeping the whole 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 lattice of kept cells, a self-similar carpet of dimension ln 64 / ln 9 ≈ 1.893. Put bond percolation on it: each bond between neighbouring kept cells is open with probability p, and ask whether an open path crosses from the left edge to the right. For one 9×9 pattern this crossing probability R(p) is an exact polynomial in p. Iterating p ↦ R(p) is a renormalization: its interior fixed point p* approximates the threshold, and y = ln R′(p*) / ln 9 approximates the correlation-length exponent 1/ν. On five such patterns, all with the same minimum left–right cut (4 bonds at the first step, 4^g at step g), y orders the five measured exponents exactly, but every value is too high, by amounts from 0.009 to 0.051 that follow no obvious feature of the pattern.

friction. Three natural accounts each fail against the numbers. (1) "The miss is the finite-cell error of a small block and grows with how badly one block approximates the whole": the pattern whose number of minimum cuts falls furthest short of what independent blocks would give has the smallest miss. (2) "The threshold and the exponent are one quantity seen twice, so the pattern whose p* is furthest from the true threshold has the largest exponent miss": the pattern with the largest threshold gap (0.060) has the smallest exponent miss (0.009). (3) "The exponent is set by the zeros of R(p) nearest p*, and the miss is those zeros being in the wrong place": the logarithmic derivative λ = ℓ + p*·Σ 1/(p* − p_j) over the zeros p_j is a cancellation — the nearest four zeros contribute more than the whole sum, and the far zeros subtract.

background.

The five known patterns (# kept, . removed):

G1          G2          G3          G4          G5
#########   #########   #########   #########   #########
#.##.##.#   #.##.##.#   ###.#.###   ###...###   ##.###.##
#########   ##.#.#.##   #########   #########   #.##.##.#
###...###   #########   #.#...#.#   #.#.#.#.#   ####.####
#.#...#.#   #..#.#..#   ###...###   #.##.##.#   ##.....##
###...###   #########   #.#...#.#   #.#.#.#.#   ####.####
#########   ##.#.#.##   #########   #########   #.##.##.#
#.##.##.#   #.##.##.#   ###.#.###   ###...###   ##.###.##
#########   #########   #########   #########   #########

Measured: the polynomials R(p) are exact (integer coefficients, from a transfer matrix; checked against 10⁶-sample Monte Carlo at three values of p, all within 2 standard errors). The lowest-order term of 1 − R in q = 1 − p is exactly (number of minimum cuts)·q⁴ on all five. The shortest crossing path is 8 bonds on every pattern. The cut entropy s is ln(number of minimum cuts at step g) ÷ (cut size at step g), computed exactly to g = 4. The measured 1/ν comes from Monte Carlo on the full lattice at L = 81, 729, 6561 (8,000 samples each, per-sample threshold, width between the 20th and 80th percentiles, 1/ν from the slope of log width against log L). That estimator's run-to-run spread is 0.004, and on the square lattice (exact 1/ν = 0.75) it reads 0.741, so it carries a bias of about −0.01 that has not been measured on these patterns. On the 81×81 lattice (two steps) the crossing probability at the one-block p* is 0.89–0.98 on these patterns: the two-step lattice is already past its own threshold at p*.

Stated: the renormalization treats each block as a single independent bond, which a carpet is not — neighbouring blocks share an edge, so their crossings are correlated.

The three patterns to predict (their exponents are being measured now):

X           Y           Z
#########   #########   #########
#.#...#.#   ##.#.#.##   ###...###
#########   #.##.##.#   ####.####
#.#####.#   #########   #.#####.#
#.##.##.#   #..#.#..#   #..#.#..#
#.#####.#   #########   #.#####.#
#########   #.##.##.#   ####.####
#.#...#.#   ##.#.#.##   ###...###
#########   #########   #########

provenance. Exact quantities: the crossing polynomial R(p) of each 9×9 pattern was computed exactly with integer coefficients by a column-by-column transfer matrix over connectivity states; p* (the interior root of R(p) = p) and R′(p*) are from that exact polynomial to 40 digits. As a check, R(p) matched an independent Monte Carlo of 10⁶ samples per point at p = 0.5, 0.7, 0.8 on every pattern, all within 2 standard errors. The minimum-cut counts and cut entropy are exact counts (shortest-path counting on the planar dual with zero-cost edges contracted), confirmed at step 1 by brute force over all bond subsets of the minimum size. Measured quantities: 1/ν is from Monte Carlo on the full carpet at L = 81, 729 and 6561, with 8,000 samples per size. Each sample gives its own crossing threshold; the width is the gap between the 20th and 80th percentiles of those thresholds, and 1/ν is minus the slope of log(width) against log(L), with a 1,000-resample bootstrap. The bootstrap error is about ±0.004–0.006 per value. Eight independent repeats on the square lattice gave a run-to-run spread of 0.004, matching the bootstrap, so the error bars are honest. But on the square lattice (exact 1/ν = 0.75) the same estimator reads 0.741 on average, and 0.73 when refitted on larger sizes, so it carries a systematic low bias of about 0.01–0.02 that has not been measured on the carpets. Two single-size fits on one carpet differed by 0.004 when two more sizes were added. The "measured threshold" is a finite-size extrapolation from the same samples. The two-step (81×81) crossing probability at p* is Monte Carlo with 2×10⁶ samples, standard error about 0.0002–0.0003. The unknown exponents of X, Y and Z are being measured with the same estimator, sizes and sample counts.

Must predict. For X, Y and Z: the sign and size of y − 1/ν for each, and at least their order, from a quantity computable from the pattern (or from the one- and two-step lattices) before the exponents are measured — together with the mechanism that also accounts for the five known misses, including why G5's is smallest.

Not part of the case — the codifier's sealed prediction of the panel (2026-09-29, before it runs)

The codifier's own lead, withheld from the case text so the panel cannot be steered by it: the slope of the two-step defect at p*, P₂′(p*) − R′(p*)², ranked the known misses (post-hoc, ρ = 0.88 over eight lattices including the square and two other cut exponents) and orders X > Y > Z in the miss (values −1.76, −5.77, −6.32); run 061 tests that. Predictions of what the panel will say: 1. At least two of the four seats name correlations between neighbouring blocks — the shared edge — as the source of the miss; at most one proposes a quantity computed from the two-step lattice's slope. 2. At least one seat attributes part of the miss to the measurement's own finite-size bias rather than to the renormalization. 3. The House will not commit to a numeric prediction for X, Y, Z, and will rank G5's small miss as the case's key anomaly. Scored against the panel record once it is filed; the panel's X, Y, Z predictions are scored against run 061.

Scored 2026-09-29, after the panel ran (PN-002)

The codifier's sealed panel predictions. (1) Held: in round 1 three seats (Lateral, Legitimacy, Framing's category point) named correlation between neighbouring blocks or its consequences, and none proposed a two-step slope quantity of its own. (2) Held: Legitimacy and Framing attributed part of the miss to the estimator's bias. (3) Falsified on both halves: the first House gave numbers for X, Y, Z (0.015–0.022, 0.023, 0.040–0.048) and called G5's small miss "mostly floor", not the key anomaly.

The panel's numbers against run 061. All three House predictions miss (X 0.001, Y 0.036, Z 0.015), and the order X < Y < Z fails on Y–Z. Y's deviation (0.013) exceeds the House's own kill threshold for s (0.012). The codifier's D₂′ lead (X > Y > Z) also failed (run 061, clause B).

The final House, checked against the record it did not have. Its verdict inverts the case: on the square y undershoots the exact exponent and y₂ climbs toward it, so on carpets, it argues, the true exponent lies above y and the "miss" is a Monte Carlo deficit. The record argues against that inversion. On every original carpet the pair slope falls with L (runs 051–056: H 0.560 → 0.534, U 0.539 → 0.509, T, V, A, B, U′ likewise), which puts the asymptotic exponent below the three-size fit, not above it — probe P-048's Deduction 3, filed before run 061 landed. Its test 3 also adopts Hypothesis's prediction that Y's 1/ν rises at larger L, against Y's own falling slopes. The House's test 2 (miss collinear with slope drift) can be read off the record now: Spearman(|drift|, miss) over the ten carpets is 0.49 — B drifts 0.046 with a 0.009 miss, A drifts 0.013 with a 0.051 miss — so drift is not the whole miss. Its test 1 misreads the lab report: "the original eight" excludes X, Y, Z, and y₂'s order does break on Y–Z, as the report's own table shows. Still standing and worth running: test 4 (y₃ on the square, where the answer is known) and the residue's reframing — compute y_g on lattices with known exponents and use the exact sequence to test the estimator, not the other way round.

Run status: the panel leaves the case open. Runs 061–063 bear on it; none settles it.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

• Run 058 — confirmed (2026-09-29, measured). lab/runs/058-reliability-polynomial-rg.md: CONFIRMED 2026-09-29.
• Run 059 — falsified (2026-09-29, measured). lab/runs/059-commutator-and-bias.md: FALSIFIED 2026-09-29, on clause A3.
• Run 060 — falsified (2026-09-29, measured). lab/runs/060-crossing-polynomial-zeros.md: FALSIFIED 2026-09-29, on clause 2.