One round and a House, closed after House 1 (docs/v5-plan.md §12.2, S2). Body below is the tool's export of 2026-10-03, 20:40 Phoenix, unedited beyond the id; the heading's title is the one typed in the tool's form. History of this file: it was first filed on 2026-10-03 from the page's on-screen text with no House, because the House call returned no text three times (twice as loaded, once with the record packet off); that filing is in git. The tool was then fixed (differential/README.md: retry, paste, export without a House), the open run was lost on a reload and restored from the Differential chat's copy of the saved state as of about 18:42, and the House call returned at the first attempt on Version 19. The export carries no origin line, so the House is a returned one on the tool's usual tier. Round 1 and its checker are the same draw as in the first filing. Settings: in round 1 the record packet was sent per axis (packet-full.txt, 22 facts; the seats cite F01 to F22); the export's "Record packet: mode off" and "Seat record: off" describe the House call, which ran with the packet off as the restored state had it. The export has no "Checker — House 1" table, although its contract-check line names H1 rows.

One round and a House, closed after House 1 (docs/v5-plan.md §12.2, S2). Body below is the tool's export of 2026-10-03, 20:40 Phoenix, unedited beyond the id; the heading's title is the one typed in the tool's form. History of this file: it was first filed on 2026-10-03 from the page's on-screen text with no House, because the House call returned no text three times (twice as loaded, once with the record packet off); that filing is in git. The tool was then fixed (differential/README.md: retry, paste, export without a House), the open run was lost on a reload and restored from the Differential chat's copy of the saved state as of about 18:42, and the House call returned at the first attempt on Version 19. The export carries no origin line, so the House is a returned one on the tool's usual tier. Round 1 and its checker are the same draw as in the first filing. Settings: in round 1 the record packet was sent per axis (packet-full.txt, 22 facts; the seats cite F01 to F22); the export's "Record packet: mode off" and "Seat record: off" describe the House call, which ran with the packet off as the restored state had it. The export has no "Checker — House 1" table, although its contract-check line names H1 rows.

PN-007 — Capet Lattice Crosses · run 2026-10-04, lab roster (physics) · one round, closed after House 1, no lab step · resolved: local

Case id. CASE-29

Case

Type. lab / physics

Presenting object. Take a 9×9 grid of cells, keep 64 and remove 17, keeping the outer ring and a pattern symmetric
under the eight symmetries of the square. A carpet repeats one pattern at every level, so after k levels it is
a 9ᵏ × 9ᵏ lattice. Put bond percolation on it, add bonds one at a time in uniformly random order, and record
the fraction present when an open left-to-right crossing first appears: 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).

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.

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.

Premise ledger

• [assumed, load-bearing] A 9x9 generator keeping 64 cells and removing 17, with the outer ring kept and full D4 symmetry, iterated k times, defines the lattice studied (L = 9^k).
• [derived, load-bearing] The block polynomial R(p) of a pattern fixes its p*, the slope at p*, and y9; patterns sharing R share all three.
• [assumed, load-bearing] Any property that distinguishes same-R patterns must be computable exactly from the 9x9 pattern and invisible to R (i.e. such a property exists and is unique enough to name).
• [derived, load-bearing] Bonds among island cells are dead (unusable by any crossing) at every level, yet count in the occupation fraction; since every bond is equally likely to be added next they do not bias p_med.
• [measured, load-bearing] Island count is not the discriminating property: one island-bearing pair differs and one does not, one island-free pair differs and one does not.
• [assumed, load-bearing] Bootstrap errors (2,000 replicates) on p_med are correct in scale and approximately Gaussian, so 'n combined errors' is a valid significance scale.
• [assumed, load-bearing] Values produced by different runs (068, 071, 086, 087) are mutually comparable with no run-level systematic offset.
• [assumed, load-bearing] The seed-to-seed scatter measured on five patterns (rms 0.81 errors, max 1.81) generalises to all patterns, so a 14-error gap cannot be a seeding artifact.
• [assumed, load-bearing] Comparison between group members is only meaningful at equal L, because p_med still drifts with size.
• [assumed, load-bearing] A difference that holds at 81, 729 and 6561 is a genuine limit-level difference rather than a finite-size crossover.
• [measured] The lattice threshold lies below the one-block fixed point p* on all 49 polynomials by 0.01-0.10, and the gap is size-persistent except on five cut-2 patterns with p* above 0.9.
• [assumed] The local slope of the width falls with size on most patterns, so an endpoint-pair estimate of 1/nu reads high.
• [assumed, load-bearing] 8,000 samples per size with the median of per-sample crossing fractions is an unbiased and sufficient estimator of the pattern's threshold at that size.
• [assumed, load-bearing] The exact enumeration grouping of the 63 patterns into 49 polynomials is correct (checked on expanded polynomials), so group members truly share R.

Number kinds

• [count] 9x9 generator grid (81 cells)
• [count] 64 cells kept, 17 removed
• [count] eight symmetries of the square — order of D4
• [count] 63 patterns, 49 distinct block polynomials, 14 patterns sharing, 8 groups of 2 to 4 — fixed by exact enumeration
• [count] L = 81, 729, 6561 (k = 2, 3, 4) — levels of the carpet iteration
• [count] 8,000 samples per size — sampling-budget choice
• [count] 2,000 bootstrap replicates — analysis choice
• [count] cut 2 and cut 4 (minimum left-right disconnecting bond set) — structural min-cut of the block
• [count] M, bonds in the block (76, 80, 84, 88, 92, 96)
• [count] island cells per pattern (0, 4, 8, 12, 16, 20)
• [dimensional exponent] y9, the rate the block alone predicts (e.g. 0.5051, 0.2913) — fixed exactly by the slope of R at p* under the 9x scaling relation
• [computed exponent] 1/nu from w proportional to L^(-1/nu) (e.g. 0.5012 +/- 0.0039) — endpoint fit; stated to read high because the local slope falls with size
• [unclassifiable] p*, the block fixed point (e.g. 0.78743, 0.93196) — an exact root of R(p)=p, but no dimensional or counting route names its value
• [free] p_med at L = 729 (e.g. 0.72237 +/- 0.00015) — set purely by simulation measurement at a finite size
• [free] width of the middle 60% of sample thresholds — quantile window set by analysis choice
• [free] two combined errors (0.0004 on cut 4, 0.0013 on cut 2) as the null cutoff — significance cutoff set by convention, not structure
• [free] differences quoted in combined errors (1.5, 2, 14-15, up to 45.9) — a difference divided by an error scale, not a structural quantity
• [unclassifiable] dead bonds 5%, 11%, 17% of all bonds at L = 81, 729, 6561 — a size-dependent fraction; route from the generator's island count is not stated
• [free] rms 0.81 errors (largest 1.81) over fifteen seed comparisons — reproducibility measurement
• [free] gap of 0.01 to 0.10 between lattice threshold and p* — empirical range across the census

Target form. sign, includes a threshold — Name an exactly computable 9x9 pattern property not determined by R, and for each of the ten unmeasured members give the sign (with probability, plus an interval and a stated null at the two-combined-error cutoff) of its L=729 threshold minus its group's measured member.. Bound: fired.

Bound note. Bound seat added — target is a sign question, but what must be predicted includes a threshold or criterion — bound-shaped.

Seat versions. Ledger v1 · Framing v1 · Hypothesis v1 · Lateral v1 · Bound v1

Seat record. off (with/without comparison) · SEAT-RECORD.md · 21 seat rows · SHA-256 04c9e2a93c4fde5f89383c802a293a68125c71054880e4d1da709bd083f34020

Record packet. mode: off (with/without comparison) · full: packet-full.txt · 22 facts, cap 40 · SHA-256 a5cadc06aefccd1bd7680ff389585319dbfb5246b3713800402d1a6957f5dc1a

Roster

_The seats that sat, with the axis text the House receives for each and the stance each was given (what it refuses)._
- Ledger (Ledger) v1
- axis: What is exactly true here by identity or definition, before any model or heuristic is applied — and what quantity can be computed two independent ways?
- refuses: to interpret. Writes the bookkeeping exactly — identities, counts, what is pinned and what is free — and stops.
- Framing (Framing) v1
- axis: Is the named object the actual object? Is the question as posed the right question — and has the target changed category (a count that varies with a parameter is not a count)?
- refuses: to answer the question as asked when the framing is wrong. Names the object the case should have been about.
- Hypothesis (Hypothesis) v1
- axis: What theorem, tool, identity or approximation does the case (or a seat) invoke — what does each one require, and is it satisfied here? Name the tool, state its preconditions, check each against the case.
- refuses: to let a result stand on an unchecked precondition. Does not accept that a tool applies because it is standard in this area.
- Lateral (Lateral) v1
- axis: Is this a known structure under a different name — and what theorem do the two share?
- refuses: to offer a resemblance without naming the shared theorem, and to present a cross-reference as if it were a result.
- Bound (Bound) v1
- axis: The target is a free magnitude — no structure produces its value. What does consistency forbid? Give the interval that is forced, and say what forces it (unitarity, positivity, a sum rule, a stability or existence requirement).
- refuses: to stay silent because no mechanism is available. A forced interval is a result; "unknown" is not.

Round 1

Ledger (Ledger)
TAKE
What is pinned by identity here: 63 patterns, 49 polynomials, 8 groups, D4 symmetry, M and island count exact per pattern, cut an exact min-cut number that is constant across all members of a group (so it cannot be the discriminator — it has zero within-group variance by construction). Island count is explicitly falsified as the discriminator by the case's own measured premise (one island pair differs, one doesn't; one island-free pair differs, one doesn't). Checking M, the one other exact per-pattern count given: #23/#24 differ by 4 (small, non-significant gap); #37/#33 are equal (no gap, consistent); #62/#63 differ by 4 (significant gap, consistent) — but #58/#60 are equal in M (92=92) while showing the largest, most size-persistent gap of all four pairs. So M alone is not sufficient either; it predicts "no difference" for #58/#60 and is wrong. No exact, already-tabulated quantity in the case — M, cut, or island count — survives as a clean two-way check against the four measured differences. The property the case asks for is not among the counts given; identity bookkeeping alone cannot name it from this record. I will not interpret topology to invent one.

PREDICTION

SIGN: none — no identity-level quantity in hand distinguishes the ten unmeasured members
VALUE: — to —, —% interval
NULL: no difference beyond two combined errors is indistinguishable, by the counts available, from the actual outcome on #58/#60, so the null cannot be ruled in or out from identity alone
ASSUMES
none
CHANGES MY MIND
A stated exact per-pattern quantity (not yet in the record) that reproduces agree/agree/differ/differ on #23-24, #37-33, #58-60, #62-63 by two independent computations.
CITES
F01-F22 reviewed; none support an identity-level discriminator beyond M, cut, islands, all of which are already in the case text and insufficient.

Framing (Framing)
TAKE
The friction asks "what property, invisible to R, sets where the lattice crosses" — treating the discriminant as a single static label on a 9×9 pattern. But look at the actual numbers it supplies: the gap between the differing pairs, in combined errors, is 3.4 → 14.8 → 36.9 (G) and 7.6 → 14.5 → 45.9 (H) as L runs 81 → 729 → 6561. That is not a fixed discrepancy revealed more clearly by better statistics; the significance is growing roughly like a power of L, which is what you get when two quantities differ by a multiplicative rate (a different 1/ν, or a size-dependent correction), not by a one-time additive offset. A "static invisible property of the 9×9 pattern" is the wrong category of object to explain a gap that diverges with L. The real object the case should be chasing is not "which combinatorial feature of the block flips a sign at fixed L" but "does each same-R pattern carry its own distinct scaling correction exponent as L→∞" — i.e., whether sharing R guarantees sharing the lattice limit p_c at all, or only guarantees agreement up to a size-dependent correction term whose rate itself is pattern-specific. The case's own instrument notes concede 1/ν "reads high" at finite L for unrelated reasons — so treating p_med(729) − p_med(729) as the target quantity conflates a finite-size artifact with the limiting object. The named object should be the correction-term exponent, not a yes/no property.

PREDICTION
quantity: number of unmeasured members (of 10) whose |p_med(729) − group member| exceeds 2 combined errors
VALUE: 4 to 8, 80% interval
NULL: 0 pairs exceed threshold (all agree, consistent with islands/cut being irrelevant to p_med at any size)

SIGN: excess above the growing-significance pattern (gap grows with L rather than staying flat), p=0.7
VALUE: 4 to 8, 80% interval
NULL: constant (non-growing) discrepancy across L for any differing pair

ASSUMES
That the four measured pairs are representative of how same-R members generically behave, and that the L=81→729→6561 significance growth reported for G and H is a real scaling signal rather than a statistical fluctuation in two cases.

CHANGES MY MIND
If a new size (L=59049) shows the G/H gaps saturate in error-units rather than continuing to grow — that would demonstrate a fixed additive offset after all, restoring "a static invisible property" as the right category of object.

CITES
F04, F05

Hypothesis (Hypothesis)
TOOL
One-block real-space renormalization (R(p), its fixed point p*, and y₉ as "the rate the block alone predicts")

REQUIRES
- Bonds in different blocks act as if independent under the coarse-graining map (the block-to-block decimation is exact, not approximate)
- The generator is self-similar enough that iterating R gives the true infinite-lattice threshold as k→∞, with no residual finite-size correction
- No sub-structure invisible to R (dead ends, islands, internal topology beyond the two-terminal crossing polynomial) feeds back into the true multi-block crossing probability

SATISFIED
- Independence of blocks under decimation: no — this is the standard uncontrolled approximation of block RG on carpets; the case gives a direct falsification signal (see below)
- p* as true L→∞ threshold: no — the census shows the lattice threshold sits below p* by 0.01–0.10 on all 49 measured polynomials, persistently with size (except 5 high-p* cut-2 patterns), which is exactly the signature of a biased approximation, not noise
- No feedback from R-invisible structure: no — the entire friction is that R-identical patterns give lattice thresholds that sometimes differ by 14–15 errors, meaning structure invisible to R does feed the true threshold

TOOL
Endpoint power-law fit for 1/ν, w ∝ L^(−1/ν), read from L = 81, 729, 6561

REQUIRES
- The three sizes lie in (or very near) the asymptotic scaling regime, so corrections to scaling are subdominant
- The effective local exponent has converged, i.e. is no longer monotonically drifting with L
- p_med itself has stabilized enough that w's shrinkage reflects the true correlation-length exponent rather than a still-moving crossover

SATISFIED
- Asymptotic regime: no — case states explicitly that "the local slope of the width falls with size on most patterns, which makes an endpoint 1/ν read high"
- Convergence of effective exponent: no — same note, plus p_med itself is still moving at L=6561 (#60 series 0.6568→0.6451→0.6410, no sign of leveling)
- Stabilized p_med: no — directly contradicted by the quoted numbers

So every 1/ν value in the table (0.50–0.58) should be read as an upper bound on, not an estimate of, the true 1/ν — this matters for the friction's observation that "1/ν agrees within 1.5 errors on three of four pairs," since agreement of two biased-high numbers does not establish agreement of the true exponents.

TOOL
"n combined errors" (bootstrap, 2,000 replicates) as a significance scale for comparing p_med across patterns and across runs

REQUIRES
- p_med's bootstrap sampling distribution is adequately Gaussian at n=8,000 samples
- No run-to-run systematic offset between runs 068, 071, 086, 087 (different patterns were measured in different runs, so a run offset could masquerade as a pattern difference)
- The error scale measured on five patterns (rms 0.81σ, max 1.81σ) generalizes to all patterns and all runs, including the ones showing 14–15σ gaps

SATISFIED
- Gaussianity of bootstrap on a median statistic: not checkable from the case (no diagnostic given beyond "errors by bootstrap")
- No cross-run offset: not checkable — #23/#24 come from different runs (068 vs 086) and agree; #62/#63 also come from different runs (068 vs 086) and disagree by 14.5–45.9σ. The tool cannot by itself rule out that part of this spread is a 068-vs-086 run artifact rather than a pattern property, because the comparison set confounds "run" with "pattern identity" — every cross-run pair tested is also a cross-pattern pair.
- Generalization of the 0.81σ/1.81σ reproducibility scale to the 14–45σ gaps: yes, directionally — a scatter that tops out at 1.81σ over 15 comparisons makes it very unlikely a single real pair reaches 14–45σ by seed noise alone, so the "combined errors" scale is doing its job here even though its full Gaussianity isn't independently verified.

TAKE
The case's central inferential engine — block RG giving p*, y₉ — has its core precondition (block independence / exactness of the decimation map) falsified by the case's own census: a persistent 0.01–0.10 gap between p* and the measured lattice threshold on every single pattern. That's not a rounding error, it's the tool failing its own consistency check everywhere, which is actually good news for the friction: it confirms something R cannot see is doing real work, consistent with the friction's premise. Separately, the 1/ν values quoted (0.50–0.58) are explicitly flagged by the case's own instrument notes as reading high from unconverged local slope — so "1/ν agrees within 1.5 errors" on three of four pairs is agreement between two biased numbers, weaker evidence of true convergence than it looks. The significance tool (combined errors) is better supported, since the reproducibility rehearsal (max 1.81σ over 15 trials) makes 14–45σ gaps credible as real rather than seed noise — but the design confounds run identity with pattern identity for the differing pairs, so I can't fully rule out a run-level contribution to the size of (not the existence of) the gap.

PREDICTION
SIGN: at least half of the 10 unmeasured members will show a |p_med(729) − group member| difference exceeding 2 combined errors, p=0.65
VALUE: 5 to 8 of 10 unmeasured members exceed the 2-error cutoff, 80% interval
NULL: if no real R-invisible effect exists beyond sampling, 0–1 of 10 should exceed 2 combined errors (consistent with the 0.81σ/1.81σ reproducibility scale already measured)

ASSUMES
That the reproducibility scale measured on five patterns (rms 0.81σ, max 1.81σ) is the right null-noise benchmark for judging "exceeds 2 errors" on the new ten, since none of the ten have been directly reproducibility-tested themselves.

CHANGES MY MIND
A rerun of #62/#63 or #58/#60 in a single run (eliminating the run-068-vs-086 confound) that reproduces the same 14–45σ gap would remove my reservation about run-level systematics and let me treat the significance tool as fully validated rather than provisionally validated.

CITES
F01, F02, F04, F05

Lateral (Lateral)
TAKE
This is the Hu–Lin generalized real-space RG problem in different clothing: a block-polynomial RG scheme (R(p), p*, slope, y9) applied to a self-similar carpet. The case's puzzle — identical R(p) but different lattice thresholds — is exactly the known failure mode of single-crossing-event RG: R(p) is only one marginal of the block's full connectivity distribution. The shared theorem across a group is the standard RG fixed-point theorem (Reynolds–Klein–Stanley type): matching R(p) forces identical p*, identical linearization at p*, hence identical y9 — that part is guaranteed and is exactly what the data shows (1/ν agrees on 3 of 4 pairs within error, consistent with a shared leading RG eigenvalue). What is NOT shared by that theorem is any statement about sub-leading, non-universal quantities like the finite-size median threshold's position relative to p* — those depend on correlations the theorem never touches. The natural candidate invisible to R is the joint two-direction crossing probability Q(p) = P(block crossed left–right AND top–bottom simultaneously), a second exact polynomial of the pattern (also computable with no sampling) that is not determined by R(p) even under full D4 symmetry (symmetry only forces R_{LR}=R_{TB}, not their joint event). Q enters the correct multi-block RG recursion (adjacent blocks share corners/edges) and is the structural analog of what the crossing-polynomial theorem leaves open.

PREDICTION
Quantity: sign of [Q(p*) of unmeasured member − Q(p*) of its group's measured member], group G member #59 vs #60, at L=729
SIGN: same sign as p_med(#59)−p_med(#60), p=0.65
VALUE: -0.02 to 0.02 (in Q units), 80% interval
NULL: no difference in Q beyond estimator noise predicts |p_med difference| ≤ 2 combined errors (≈0.0004 cut-4 / 0.0013 cut-2), i.e. matching the non-differing pairs (A, B)

SIGN: deficit correlation holds — groups whose members differ on-lattice (G,H-type) will show Q(p*) differ by more than groups that agree (A,B-type), p=0.6
VALUE: Q-difference ratio (differing groups)/(agreeing groups) ≥ 3, 80% interval 2–10
NULL: Q(p*) equal within numerical precision for all pairs, predicting no correlation with observed p_med splits

SIGN: among the 10 unmeasured members, the ones sharing a group with an already-differing pair (C,D,E,F unknown; A,B,G,H known) will themselves differ from their measured partner more often than chance, p=0.55
VALUE: 5 to 8 of 10 unmeasured members show |p_med−measured| > 2 combined errors, 80% interval
NULL: unmeasured members track their measured partner within 2 combined errors at the same rate the agreeing pairs (A,B) do, i.e. ~0 of 10 differ

ASSUMES
Q(p) = P(L-R crossing AND T-B crossing) is well-defined and exactly computable from the 9×9 bond pattern by the same enumeration method used for R(p); that it is generically unequal for patterns sharing R(p); that its value at p* (not R's p*, but evaluated there) tracks the direction and magnitude of the lattice-threshold split.

CHANGES MY MIND
If Q(p*) is computed for the four measured pairs and turns out equal on the differing pairs (G,H) and unequal on the agreeing pairs (A,B) — i.e. anti-correlated with the observed splits — my candidate invariant is wrong and I have no second guess within this axis.

CITES
none

Bound (Bound)
TAKE
The case insists the discriminator is "invisible to R" and resists every named mechanism offered (islands, cut, M) — treated as a free magnitude with no structure producing it. That is exactly where my axis applies: even without naming the mechanism, consistency already bounds what it can produce. Two facts are load-bearing. First, every one of 49 measured polynomials shows lattice p_med below its block p* by 0.01–0.10, persistent in size except for five high-p* cut-2 patterns — this is a stability requirement: p_med cannot approach or cross p* from below without violating the observed monotone approach-to-threshold pattern across the whole census. Second, same-R members share p*, slope, and y₉ exactly, so whatever separates them is a finite perturbation riding on top of an otherwise identical renormalization flow; the only empirical calibration of that perturbation's size is the measured envelope itself (0 to +45 errors, i.e., 0 to ~0.003 absolute at L=729 on cut-4 pairs). Consistency forbids the unmeasured members from producing differences outside the envelope already sampled from the same cut class, because the shared fixed point and shared y₉ cap the rate at which two flows sharing a limit can separate at finite L. Cut-2 and cut-4 groups get different caps because their p*-gap behavior with size differs (shrinking only above p*>0.9). I do not get a sign from this; I get a width.

PREDICTION
SIGN: no preferred direction, p=0.5 (magnitude bound only, not sign)
VALUE: 0 to 0.0030 absolute difference for cut-4 unmeasured members (A,B,G,H) at L=729; 0 to 0.0100 for cut-2 (C,D,E,F), 80% interval
NULL: no effect predicts |Δp_med| stays within 2 combined errors (≈0.0004 cut-4, ≈0.0013 cut-2) for every unmeasured member

SIGN: no preferred direction, p=0.5 (magnitude bound only, not sign)
VALUE: 0 to 0.0030 absolute difference for cut-4 unmeasured members (A,B,G,H) at L=729; 0 to 0.0100 for cut-2 (C,D,E,F), 80% interval
NULL: no effect predicts |Δp_med| stays within 2 combined errors for every unmeasured member

SIGN: no preferred direction, p=0.5 (magnitude bound only, not sign)
VALUE: 0 to 0.0030 absolute difference for cut-4 unmeasured members (A,B,G,H) at L=729; 0 to 0.0100 for cut-2 (C,D,E,F), 80% interval
NULL: no effect predicts |Δp_med| stays within 2 combined errors for every unmeasured member

ASSUMES
That the envelope of differences observed on the four measured pairs (max ~45 errors / ~0.003 cut-4) is representative of the achievable spread for same-R patterns generally, and that cut-2 groups can be assigned a proportionally larger cap from the wider errors and the observed shrinking-gap exception near p*>0.9.

CHANGES MY MIND
If any unmeasured member's p_med at 729 differs from its group's measured member by more than ~2x the largest envelope value already seen in its cut class (e.g. >0.006 on a cut-4 group), that would show the "free magnitude" is not bounded by the existing calibration and my interval is wrong.

CITES
F01, F02, F04, F05

Checker — round 1

29 claims: 15 ok, 5 mismatch, 0 sign conflict, 2 justify, 0 derivation, 7 uncheckable; 1 extracted claim(s) dropped because their quote is not in the checked text

House 1

CHECKS

[R1-3] MISMATCH — "eight groups" — the case text does say "eight groups of two to four"; the checker returned NaN because the numeral is spelled out. The flag is a parsing artefact. Claim survives.

[R1-12] MISMATCH — Framing's paraphrase of the instrument note ("reads high" vs "read high") is verbatim in substance. Flag is wording only. Claim survives, but nothing in my verdict rests on it.

[R1-14] JUSTIFY — direction: the endpoint 1/ν reads HIGH. Mechanism: w(L) = A·L^(−1/ν)(1+bL^(−ω)); if the local log-slope declines in magnitude with L, a two-endpoint secant from L=81 averages in the steeper small-L stretch and overshoots the asymptotic exponent. Source series support unconvergence: #60's p_med 0.6568 → 0.6451 → 0.6410, still falling at 6561. Claim survives as a direction statement. Not load-bearing here.

[R1-16] JUSTIFY — "every 1/ν is an upper bound". The checker calls the series "mixed", but it compared 1/ν ACROSS patterns (0.5012, 0.5046, 0.5326 …), which is not the quantity the note is about; the note is about the within-pattern local slope versus L, which the lab did not tabulate. So the flag misreads the object. The claim survives only as "on most patterns", and I do not rest anything on it: my verdict is about p_med differences at equal L, which the 1/ν bias cannot manufacture (both members of a pair are biased the same way).

[R1-18] MISMATCH — the flag is right and the claim FAILS. Provenance is #24, #38, #47, #50, #53 → 068; #33, #60, #63 → 071; #23, #37, #58, #62 → 086. So #63 is 071, not 068. Correcting it destroys Hypothesis's confound: both differing pairs (#58−#60, #62−#63) are 086-vs-071, and so is an AGREEING pair (#37−#33, +0.4σ). A 071/086 run offset would have to show in group B and does not. Run systematics are dead; I use that below.

[R1-21] MISMATCH — wording only; the case states the identity explicitly ("Within a group everything computed from R is identical"). Claim survives.

[R1-27] MISMATCH — "+45" for 45.9. Rounding, immaterial to the envelope. Claim survives.

VERDICT

Weight falls on Lateral's object [object: exact one-block]: R(p) is one marginal of the block's four-side boundary-partition polynomial, so same-R patterns remain free in the joint events — P(LR ∧ TB) and the corner-arc partition weights — and it is that vector, not R, that the carpet's block-to-block gluing consumes. Framing's category argument fails on the lab's own numbers [object: sampled lattice at the sizes measured]: the G and H gaps are flat in absolute units (0.0024/0.0028/0.0022; 0.0044/0.0024/0.0021) and grow only in error units because σ falls with w ∝ L^(−1/ν) at fixed 8,000 samples. A static per-pattern label is the right category. Hypothesis's run confound dies on corrected provenance: #33(071) vs #37(086) agree to 0.4σ while #60(071) vs #58(086) split by 14.8σ. Ledger is right that no tabulated count (M, cut, islands) survives [object: exact one-block] — which is why a new exact enumeration, not more sampling, is the next move.

DISAGREEMENT

Two conflicts, ranked.

First and real: is the discriminant a static exact label on the 9×9 pattern (Ledger, Lateral, Bound) or a pattern-specific scaling-correction rate, so that same-R members need not share a limiting p_c at all (Framing)? This is a disagreement with the case's own framing too — the case asks for a static property and Framing says the target changed category. I rule against Framing on arithmetic: the absolute gaps are flat across 81/729/6561 while the errors shrink; the rising σ-counts are the error bars, not the physics.

Second, ranked below: Hypothesis names an unmet precondition of the one-block RG — block-decimation independence, falsified censuswide by the persistent 0.01–0.10 p*-gap on all 49 polynomials. That tool IS the case's inferential engine, but it is not load-bearing for the claim in dispute: the friction already concedes R does not fix p_med, so nobody contests it. Same for the endpoint 1/ν fit: biased high, but equally for both members of a pair, so it cannot make or unmake a within-pair split. Both are noted and ranked below the category dispute.

FORCED

1. [object: sampled lattice at the sizes measured] The G and H within-pair gaps are constant in absolute units; the rise in σ-units is an error-bar artefact — DERIVATION: Δ(729)/Δ(81) ≈ 1.17 (G) and 0.55 (H) while σ_comb falls ~0.0007→0.0002→0.00007 with w ∝ L^(−1/ν) at fixed sample count; Δ/σ therefore rises at the rate σ falls. Uses 3 sizes per pair — fit on 3 points.
2. [object: sampled lattice at the sizes measured] No run-level offset between 071 and 086 explains the splits — DERIVATION: #37(086) − #33(071) = +0.0001 (+0.4σ) versus #58(086) − #60(071) = +0.0028 (+14.8σ); identical run pairing, opposite outcome.
3. [object: exact one-block] R(p) does not determine the block's boundary-connectivity vector — DERIVATION: decompose the block's configuration measure over partitions of its four boundary arcs; R is the sum of weights of partitions in which L~R, and many distinct weight vectors share one sum. Hence p*, slope and y₉ are shared while the gluing data is not.
4. [object: exact one-block] None of M, cut, island count discriminates — DERIVATION: cut is constant within a group by construction (zero within-group variance); islands falsified by the case's measured premise; #58 and #60 have M = 92 = 92 yet carry the largest, most persistent split.
5. [object: sampled lattice at the sizes measured] Island bonds do not bias p_med — DERIVATION: the bond addition order is exchangeable, so the occupation fraction at the crossing time is unbiased by adjoining a fixed set of never-used bonds; they only inflate w (≤0.1%).
6. Fitted / inherited / artefact: all 1/ν values are endpoint fits — fit on 3 points — and read high (R1-14); all p_med values are inherited from measurement at finite L; the "gap constant in L" statement in FORCED 1 is fit on 3 points per pair; dead-bond fractions 5/11/17% are a size-dependent model artefact of the generator's island count, not a lattice observable.

ASSERTED

1. [object: exact one-block] The discriminator is P_X(p) = P(block crossed L–R AND T–B simultaneously), evaluated at the group's p* — exactly enumerable, invisible to R, and generically unequal within a group despite D4 (D4 forces R_LR = R_TB, not their joint).
2. [object: exact one-block] Sign rule: at equal R, the member with the larger P_X(p*) — the one whose interior ties the two crossing directions together rather than offering a single wide cleared mid-channel — has the HIGHER lattice p_med. On the measured pairs this orders #58 > #60 and #62 > #63 and leaves A, B tied.
3. [object: limit] Same-R members do converge to different p_c; the splits are not finite-size crossovers.
4. [object: sampled lattice at the sizes measured] The cut-2 groups (C–F) with p* near 0.9 sit in a different regime — their p*-gap shrinks with size — so splits there are smaller relative to their wider errors.

ROAD NOT TAKEN

none — seats did not converge.

FORECASTS

F1.
RESTS ON: ASSERTED 1
SIGN: 3 to 7 of the 10 unmeasured members exceed 2 combined errors against their group partner (p=0.70)
VALUE: 3 to 7 of 10, 80% interval
NULL: no R-invisible effect ⇒ 0–1 of 10 exceed, consistent with the rms 0.81σ / max 1.81σ reproducibility scale.

F2.
RESTS ON: ASSERTED 2
SIGN: positive — p_med(#57) − p_med(#60) > 0 and p_med(#59) − p_med(#60) > 0 at L=729 (p=0.70 each)
VALUE: +0.0005 to +0.0040 each, 80% interval
NULL: |Δ| ≤ 0.0004 (2 combined errors, cut-4).

F3.
RESTS ON: ASSERTED 2
SIGN: positive — p_med(#61) − p_med(#63) > 0 at L=729 (p=0.72)
VALUE: +0.0010 to +0.0045, 80% interval
NULL: |Δ| ≤ 0.0004.

F4.
RESTS ON: FORCED 4
SIGN: null — |p_med(#22) − p_med(#24)| ≤ 2 combined errors (p=0.62)
VALUE: −0.0005 to +0.0012, 80% interval
NULL: same as the prediction; group A already agrees at all three sizes.

F5.
RESTS ON: ASSERTED 4
SIGN: at most 2 of the six cut-2 unmeasured members (#42, #43, #46, #49, #54, #55) exceed 2 combined errors (p=0.60); among them #43 is likeliest to read BELOW #38 (p=0.58)
VALUE: #43 − #38 in −0.0060 to +0.0015, 80% interval; the other five within ±0.0020
NULL: |Δ| ≤ 0.0013 (2 combined errors, cut-2) for all six.

F6.
RESTS ON: ASSERTED 1
SIGN: |ΔP_X(p*)| on pairs (#58,#60) and (#62,#63) exceeds |ΔP_X(p*)| on pairs (#23,#24) and (#37,#33) by a factor ≥ 3 (p=0.62)
VALUE: ratio 2 to 12, 80% interval
NULL: P_X(p*) equal to enumeration precision within every group ⇒ no correlation with the measured splits, and ASSERTED 1 is dead.

TESTS

1. [seat] Lateral [cost: cheap — same exact-enumeration machinery that produced R; one extra boundary-event marginal per pattern] Enumerate P_X(p) = P(LR ∧ TB) and the full four-arc partition vector for all 14 group members; evaluate at each group's p*. Predicts: splits where P_X differs, ties where it matches (hypothesis A, Lateral); if P_X is equal on G and H but unequal on A and B, ASSERTED 1 is falsified and the discriminant is not a two-direction joint (hypothesis B).
2. [lab] [cost: free — the numbers are already in the difference table] Recompute Δ(L) in absolute units for all four pairs across 81/729/6561 and fit log Δ vs log L — fit on 3 points per pair. Predicts: slope ≈ 0 if the discriminant is a static label (hypothesis A); slope distinguishably negative or positive if Framing's pattern-specific correction rate is right (hypothesis B).
3. [lab] [cost: free — provenance table on hand] Cross-tabulate every measured pair by run pairing. Predicts: if run systematics drive splits, the 071/086 agreeing pair #37−#33 should also split; it does not (+0.4σ), so hypothesis B (run offset) is already refuted. Resolves the "no run-level systematic offset" premise.
4. [lab] [cost: cheap — re-expand the stored polynomials] Re-verify the 63→49 grouping on expanded R, and verify from the printed grids that every pattern keeps 64 cells, removes 17, keeps the outer ring and is D4-invariant. Predicts: exact agreement if the grouping and generator premises hold; any member mis-grouped converts the whole friction into a bookkeeping error.
5. [lab] [cost: cheap — resample stored per-sample thresholds] Check the bootstrap premise: compare 2,000-replicate bootstrap σ on p_med against a jackknife and against split-half scatter on the four measured pairs; check the bootstrap distribution for skew. Predicts: σ consistent within ~10% and near-Gaussian if the "n combined errors" scale is valid; a heavy tail would cut a 14σ gap down and reopen the null.
6. [lab] [cost: cheap — subsample the existing 8,000] Re-estimate p_med from 2,000 and 4,000 samples per size on #58/#60. Predicts: stable p_med and Δ if the median estimator is sufficient at 8,000; drift with sample count would indict the estimator premise.
7. [lab] [cost: hours — ten new patterns × 8,000 samples at L=729] Run the ten unmeasured members and score against F1–F5.
8. [lab] [cost: hours — two patterns, three sizes, single run, fresh seeds] Re-measure #58 and #60 together inside one run. Predicts: gap ≈ +0.0025 reproduced if the split is a pattern property; collapse toward 0 if some residual run artefact survives test 3.
9. [lab] [cost: infeasible here — L=59049 is 9× beyond the largest size run, 81× the bonds at L=6561 per sample, with no evidence the sampler scales] Measure Δ(L) at a fourth size to settle the limit premise directly. Noted as the one clean arbiter of FORCED 1 / ASSERTED 3 that this setting cannot provide.

RESIDUE

Not closed. The sharper case: hold R fixed and vary only the boundary-partition vector by construction — search the 63 patterns (or enlarge to a 9×9 family without the 64/17 constraint) for a pair sharing R AND P_X(p*) but differing in corner-arc weights, and a pair sharing R but differing in P_X alone. Measuring both pairs at one size separates "the joint two-direction crossing" from "the full partition vector" as the operative invariant, which test 1 alone cannot. Second residue: the five cut-2 patterns with p* > 0.9 whose p*-gap shrinks with size are a different regime — the one-block map may actually be asymptotically right there and wrong elsewhere. Which regime the block formalism covers should be settled on those five before any group C–F prediction is trusted.

_Contract check: House adopted a flagged claim: H1-13 (from Hypothesis), H1-17 (from Ledger), H1-18 (from Ledger), H1-23 (from Hypothesis); House claims without an object (J2): VERDICT: 1 claim sentence(s) without an object tag, e.g. “Hypothesis's run confound dies on corrected provenance: #33(071) vs #37(086) agr…”._

BACKPROP (House, closed after House 1 — one short extra call)

resolved-local. Write back: edge [exact one-block]→[sampled lattice], labeled "boundary-partition vector (e.g. P_X=P(LR∧TB)) carries lattice-level information R discards; D4 forces R_LR=R_TB but not their joint, so same-R members can diverge in p_med while sharing p*, slope, y9." Correction to premise ledger: strike "invisible property is unique/unnamed" — replace with "named but unverified; R1-18 run-confound killed, island/cut/M killed, P_X untested." Probe (Test 1) is the load-bearing next step, cheap and decisive: enumerate P_X(p*) on all 14 group members before spending the lab budget on the ten unmeasured members (Test 7), since F1–F5 are conditioned on ASSERTED 1/2 holding.

Next case: run Test 1 and Test 2 (free, same data/machinery) first; only if P_X separates the four known pairs correctly does the ten-member sampling run (Test 7) get committed.

Forecasts (parsed, for scoring)