Carpets that share a block polynomial share its rate but not always their threshold — what in the pattern, invisible to the polynomial, sets the threshold?

---

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. A carpet repeats one pattern at every level, so after k levels it is a 9ᵏ × 9ᵏ lattice. Put bond percolation on it, add bonds one at a time in uniformly random order, and record the fraction present when an open left-to-right crossing first appears: the sample's threshold. Over 8,000 samples, p_med is the median threshold and w the width of the middle 60%; 1/ν is the rate at which w shrinks, w ∝ L^(−1/ν), read from L = 81, 729 and 6561.

Each pattern has a block polynomial R(p): the probability that one 9×9 block is crossed left to right when each bond is open with probability p. There are 63 such patterns and only 49 distinct polynomials. Fourteen patterns share theirs with another, in eight groups of two to four. Within a group everything computed from R is identical: its fixed point p*, its slope there, and the rate y₉ that the block alone predicts.

friction. A pair from one group has been measured on the lattice in four groups. On three of the four the rate 1/ν agrees within 1.5 errors. The threshold does not follow: at L = 729 it agrees within 2 errors on two pairs and differs by 14 to 15 errors on the other two, and the difference holds at every size. The pairs that differ are not picked out by the obvious feature. Some members have kept cells that are cut off from the ring inside the block (islands, which no crossing can use at any level); one pair with an island member differs and one does not, and one pair with no islands differs and one does not. What property of a 9×9 pattern, invisible to R, sets where the lattice crosses?

background.

The eight groups (rows 0 to 8, # kept, . removed). M is the number of bonds in the block; "island cells" are kept cells not connected to the ring.

group A  (cut 4, p* 0.78743, y9 0.5051)
#22          #23          #24
#########    #########    #########
#.#####.#    #.#####.#    #.#####.#
###...###    ###.#.###    ###.#.###
##.###.##    ##..#..##    ##.#.#.##
##.#.#.##    ####.####    ###...###
##.###.##    ##..#..##    ##.#.#.##
###...###    ###.#.###    ###.#.###
#.#####.#    #.#####.#    #.#####.#
#########    #########    #########
M 84, 8 isl.    M 84, 0 isl.    M 80, 4 isl.

group B  (cut 4, p* 0.76095, y9 0.5349)
#33          #37
#########    #########
##.###.##    ##.###.##
#..###..#    #.#####.#
####.####    ###...###
###...###    ###...###
####.####    ###...###
#..###..#    #.#####.#
##.###.##    ##.###.##
#########    #########
M 88, 0 isl.    M 88, 0 isl.

group C  (cut 2, p* 0.93196, y9 0.2913)
#38          #42          #43
#########    #########    #########
###...###    ###.#.###    ###.#.###
##.###.##    ##.#.#.##    ##.###.##
#.#####.#    #.#####.#    #.#.#.#.#
#.##.##.#    ##.#.#.##    ####.####
#.#####.#    #.#####.#    #.#.#.#.#
##.###.##    ##.#.#.##    ##.###.##
###...###    ###.#.###    ###.#.###
#########    #########    #########
M 84, 20 isl.    M 76, 16 isl.    M 76, 0 isl.

group D  (cut 2, p* 0.92797, y9 0.2817)
#46          #47
#########    #########
###.#.###    ###.#.###
####.####    ####.####
#.#.#.#.#    #.##.##.#
##.#.#.##    ##.....##
#.#.#.#.#    #.##.##.#
####.####    ####.####
###.#.###    ###.#.###
#########    #########
M 76, 4 isl.    M 84, 0 isl.

group E  (cut 2, p* 0.84847, y9 0.3658)
#49          #50
#########    #########
####.####    ####.####
##..#..##    ##.#.#.##
##.###.##    ###.#.###
#.##.##.#    #..#.#..#
##.###.##    ###.#.###
##..#..##    ##.#.#.##
####.####    ####.####
#########    #########
M 84, 12 isl.    M 80, 4 isl.

group F  (cut 2, p* 0.82860, y9 0.3601)
#53          #54          #55
#########    #########    #########
####.####    ####.####    ####.####
###...###    ###.#.###    ###.#.###
##.###.##    ##..#..##    ##.#.#.##
#..#.#..#    #.##.##.#    #.#...#.#
##.###.##    ##..#..##    ##.#.#.##
###...###    ###.#.###    ###.#.###
####.####    ####.####    ####.####
#########    #########    #########
M 88, 8 isl.    M 84, 8 isl.    M 80, 8 isl.

group G  (cut 4, p* 0.71783, y9 0.5179)
#57          #58          #59          #60
#########    #########    #########    #########
#########    #########    #########    #########
##.....##    ##..#..##    ##..#..##    ##.#.#.##
##.###.##    ##..#..##    ##.#.#.##    ###...###
##.#.#.##    ####.####    ###...###    ##.....##
##.###.##    ##..#..##    ##.#.#.##    ###...###
##.....##    ##..#..##    ##..#..##    ##.#.#.##
#########    #########    #########    #########
#########    #########    #########    #########
M 92, 8 isl.    M 92, 0 isl.    M 88, 4 isl.    M 92, 0 isl.

group H  (cut 4, p* 0.68353, y9 0.5391)
#61          #62          #63
#########    #########    #########
#########    #########    #########
###...###    ###...###    ###.#.###
##..#..##    ##.#.#.##    ##.....##
##.#.#.##    ##.....##    ###...###
##..#..##    ##.#.#.##    ##.....##
###...###    ###...###    ###.#.###
#########    #########    #########
#########    #########    #########
M 92, 4 isl.    M 92, 4 isl.    M 96, 0 isl.

"Cut" is the minimum number of bonds whose removal disconnects the block's left side from its right.

Measured members, lattice, 8,000 samples per size:

Differences within the four measured pairs, in combined errors:

• A fresh draw of #60 (new seeds, 8,000 per size) gave p_med 0.64543 ± 0.00011 at 729 against the stored 0.64510 ± 0.00014, 1.8 errors apart; against it #58 − #60 is +0.0025.
• The census as a whole: the lattice threshold lies below the block's fixed point p* on every one of the 49 measured polynomials, by 0.01 to 0.10, and that gap does not shrink with size except on five cut-2 patterns whose p* is above 0.9.
• Unmeasured members: #22 (A), #42 and #43 (C), #46 (D), #49 (E), #54 and #55 (F), #57 and #59 (G), #61 (H).

Must predict. A run will sample the ten unmeasured members at L = 729, 8,000 samples each, and compare each p_med with the measured member of its group (#24, #38, #47, #50, #53, #60, #63). 1. Which property? Name one quantity that can be computed exactly from a 9×9 pattern, with no sampling, and that R does not determine. Say what it gives on the four measured pairs: it should agree where the thresholds agree and differ where they differ. 2. What it predicts. For each of the ten unmeasured members, the sign of its p_med at 729 minus its group's measured member, with a probability; a value interval with stated coverage; and what the null (no difference beyond two combined errors) predicts.

Provenance. Block polynomials and their groups from the exact enumeration of runs 068 and 075, groups checked on the expanded polynomials. Lattice values for #24, #38, #47, #50, #53 from run 068; #33, #60, #63 from run 071; #23, #37, #58, #62 from run 086; the fresh draw of #60 from run 087. p_med is the median of the per-sample thresholds; errors by bootstrap, 2,000 replicates.

Instrument notes. - Objects. p* and y₉ are exact one-block values. p_med and 1/ν are sampled-lattice values at the sizes measured. None is a limit. - Error scale. p_med at 729: about ±0.00012 to ±0.00015 on the cut-4 groups, ±0.0004 to ±0.0005 on the cut-2 groups. Two combined errors is about 0.0004 on cut 4 and 0.0013 on cut 2. - Size. The threshold still moves with L (#60: 0.6568, 0.6451, 0.6410), so members are compared at equal size. The local slope of the width falls with size on most patterns, which makes an endpoint 1/ν read high. - Dead bonds. A threshold is the fraction of all the lattice's bonds present, including bonds among island cells that no crossing can use. Every bond is equally likely to be added next, so this does not shift the threshold as an occupation fraction; it widens w by at most 0.1%. On #24 and #62 dead bonds are 5%, 11% and 17% of all bonds at L = 81, 729, 6561; members with more island cells have more. - Same samples. p_med, w and 1/ν of a pattern come from the same samples. - Reproducibility. On five patterns drawn twice with different seeds, the two p_med values at each of three sizes differ with rms 0.81 errors (largest 1.81) over fifteen comparisons.

---

Not part of the case — the codifier's sealed predictions (2026-10-03, before any panel runs)

The codifier's own lead, held with conviction: R records one event, left side joined to right. A block on a lattice is also entered from one side and left by an adjacent one. Two patterns can agree on the first and differ on the second. So the quantity is the block's corner polynomial: the probability that the middle cell of the left side is joined to the middle cell of the top side, each bond open with probability p. The lead is that it differs between #58 and #60 and between #62 and #63, and agrees (or differs by far less) between #33 and #37 and between #23 and #24; and that among the ten, the member with the larger corner polynomial near its group's threshold crosses earlier. Nothing has been computed for this. Where it breaks: if all four measured pairs differ in the corner polynomial by similar amounts, the corner is not what the threshold reads, and the next candidate is a two-block quantity that no single block's polynomial holds.

Predictions of what the panel will do: 1. At least two seats name a connectivity of the block other than left to right (a turn, a top-to-bottom crossing conditioned on a left-to-right one, or a connection between boundary cells). 2. At least one seat attributes the differences to the islands or to the dead-bond normalisation, against the instrument note. 3. The House's quantity is a single-block exact quantity, not a two-block one.

Scored against the panel record once it is filed. The panel's forecasts for the ten members are scored against the run this case shapes.

Settings proposed (v5-plan §12.2): S1, five seats in round 1 (Ledger, Framing, Hypothesis, Lateral, Bound; the target is numeric). S2, one round and a House, since no lab step lands between rounds. S3, the record not loaded (PN-005 was loaded; alternate). S4, one draw.

Scored 2026-10-03, after the panel ran (PN-007; one round, no House)

The codifier's sealed panel predictions — 0 of 2 held; the third is unscorable. 1. Failed. One seat, not two, named a connectivity of the block other than left to right: Lateral's Q(p), the probability that the block is crossed left to right and top to bottom at once. Ledger checked the counts in the case (M, cut, islands) against the four measured pairs and found none sufficient; Framing asked for a correction exponent; Hypothesis audited the tools; Bound gave a width. 2. Failed. No seat attributed the differences to the islands or the dead bonds; Ledger cited the case's own measured premise against them. 3. Unscorable. No House: its call returned no text three times.

A note on the record (codifier). Framing read the gap's growth in error units (3.4 → 14.8 → 36.9; 7.6 → 14.5 → 45.9) as two rates diverging. The case's table gives the gap itself for #62 − #63 as +0.0044, +0.0024, +0.0021: it shrinks and levels while the errors shrink eightfold. The checker did not flag it and there was no House to. Hypothesis's objection that every differing pair is also a cross-run pair is a design point the run can close at no cost: redraw the four measured members of groups G and H beside the ten. Lateral's Q is exact and, by planarity, is the probability that one cluster touches all four sides; it and the codifier's corner polynomial are the two level-1 entries of P-053's table that R does not fix.

The run this case shapes (to be sealed): the ten unmeasured members at L = 729, 8,000 samples, with #58, #60, #62, #63 redrawn in the same run; and, exactly, R, the corner polynomial and Q on all fourteen group members (exact_events.py's method). It scores the panel's forecasts (panels/SEAT-RECORD.md) and the codifier's lead above.

Scored 2026-10-03, night — House 1 (PN-007, closed after House 1)

The codifier's third sealed prediction held; the tally is 1 of 3. The House's quantity is a single-block exact one: P_X(p), the probability that the block is crossed left to right and top to bottom at once, evaluated at the group's p*. It is Lateral's Q under another letter.

The codifier's lead beside the House's. Both are exact one-block quantities that R does not fix. The lead: the corner polynomial, and the member with the larger corner crosses earlier (lower p_med). The House: P_X, and the member with the larger P_X crosses later (higher p_med). The House asserts, without computing it, that this orders #58 above #60 and #62 above #63 and leaves groups A and B tied. Nothing has been computed for either. The exact run decides both, and the House's residue names how to tell them apart: a pair sharing R and P_X that differs in corner weights, and a pair sharing R that differs in P_X alone.

Notes on the record (codifier). - The House corrected the provenance error the checker flagged (R1-18): #63 is run 071's. Both differing pairs and the agreeing pair #37, #33 are then run 086 against run 071, so a run offset cannot explain the splits. That removes the reason for redrawing #58, #60, #62, #63 in the same run; the redraw stays in the sampled seal only as a cheap control. - The House's FORCED 1 says the G and H gaps are constant in absolute units. On G they are (0.0024, 0.0028, 0.0022). On H the gap halves between 81 and 729 (0.0044 to 0.0024, about three combined errors at 81) and is flat only from 729 on. Its conclusion against Framing stands, since nothing grows, but "constant" is true of three sizes on G and of two on H. - ASSERTED 3, that members of a group converge to different limits, rests on three sizes, and the House tags it as a limit claim itself. - The House's first test asks for the block's full four-side partition vector. That is the one-block form of the table of side pairs the lab's recorder measures on the lattice (codifier's note; the panel was not shown it).

The runs this case shapes, in order. 1. Exact, no sampling: R, P_X, the corner polynomial and the four-side partition vector on all twenty-two members of the eight groups, at each group's p*. It scores the House's F6, Lateral's ratio forecast and the codifier's lead. (A count to keep straight: the eight groups hold twenty-two patterns, twelve measured and ten not. The case's "fourteen" is 63 − 49, the patterns beyond one per polynomial; the note above of 2026-10-03 and the House's test 1 both say fourteen members and mean these twenty-two.) 2. Sampled, sealed after (1) is scored so its sign predictions can come from exact numbers: the ten unmeasured members at L = 729, 8,000 samples each. It scores the House's F1 to F5 and the seats' forecasts.

Reading 2026-10-03, late — every group shares one core (R27_tree_groups/carpet_perc/READING-TWINS.md)

Codifier's note; not part of the case and not shown to a panel.

The codifier's lead is falsified, and so is the House's quantity, by derivation. In every one of the eight groups the members have the same core: the 2-connected part of the pattern that holds the ring. They differ by islands, by pendant parts hanging on one core cell, and in group B by the position of four cells. No path between two ring cells uses a pendant part, so every one-block quantity between ring cells is identical within a group. The corner polynomial and the chance of both crossings at once both have zero difference on every pair. The exact run named above is dropped: its outcome is foretold.

What separates the measured pairs. #58 has four pendant parts and #60 eight; #62 has none and #63 four. In both pairs the member with more parts crosses earlier, by 0.0028 and 0.0024. #23 and #24 have four parts each on the same cells, differing only in depth, and agree. #33 and #37 have none and agree.

New lead, held with conviction. A pendant part is a block entered and left by the same side. One level up it gives the core block it hangs on a way round its own interior, so the carpet crosses earlier. More pendant parts, earlier crossing; islands, no change; depth of a part, little change. Where it breaks: if #22 agrees with #24, or #57 with #59.

The run this case now shapes. One sampled run: the ten unmeasured members at L = 729, 8,000 samples each. Sign predictions from the pendant counts, the islands-only pairs (#54 and #55 against #53; #61 against #62) as controls. Draft at _incoming_twins/, to a sighted pass before sealing.

Sealed 2026-10-03, late: run 098 (lab/runs/098-pendant-parts-size.md). The sighted pass found that containment fixes the sign of nearly every difference, so the run's clauses are sizes: four one-cell parts on cores A and G against the 0.0024 on H, the cost of a second cell on a part, and whether the cut-2 groups have the same scale. All nineteen lattices are drawn in the run. The draft at _incoming_twins/ is superseded by the run file.

Scored 2026-10-03, late — run 098, confirmed (lab/runs/098-pendant-parts-size.md)

The ten unmeasured members are drawn, with nine redraws beside them. Seven of the ten differ from the measured member this case named for their group, beyond the panel's bars: #22, #42, #43, #49, #57, #59, #61. The three that do not are #46 (against #47) and #54 and #55 (against #53, islands only). The property the case asked for is not a one-block quantity at all: within a group the threshold moves with the pendant parts, by about 0.002 to 0.003 for four of them on the cut-4 cores and by 0.027 on group C, where they hang on the bottleneck. The panel's forecasts are scored in panels/SEAT-RECORD.md: the House's F1 and F3 hold, F2 holds in sign, F4 (no difference for #22) and F5 (at most two of six on cut 2) fail; three seats' counts hold; Bound's cap fails on cut 2.