A one-block polynomial ranks exponents exactly and misses every value: what sets the miss?

PN-002 — A one-block polynomial ranks exponents exactly and misses every value: what sets the miss? · run 2026-09-29, lab roster (physics) · two rounds, one lab step · resolved: open

Case

Type. lab / physics

Presenting 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
#########   #########   #########
#.#...#.#   ##.#.#.##   ###...###
#########   #.##.##.#   ####.####
#.#####.#   #########   #.#####.#
#.##.##.#   #..#.#..#   #..#.#..#
#.#####.#   #########   #.#####.#
#########   #.##.##.#   ####.####
#.#...#.#   ##.#.#.##   ###...###
#########   #########   #########

Must predict. 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.

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.

Premise ledger

• [measured, load-bearing] The crossing polynomial R(p) of each 9x9 pattern is exact (integer coefficients from a column-by-column transfer matrix), and matches independent Monte Carlo at three values of p within 2 standard errors.
• [derived, load-bearing] p* (the interior root of R(p)=p) and R'(p*) follow from that exact polynomial to 40 digits.
• [assumed, load-bearing] y = ln R'(p*)/ln 9 is a legitimate approximant to 1/nu, i.e. the one-block map p -> R(p) is a renormalization whose linearization at its fixed point carries the correlation-length exponent.
• [assumed, load-bearing] Each 9x9 block may be treated as a single independent bond even though neighbouring blocks share an edge and their crossings are correlated; the case states this is false and treats the resulting correlation as the source of the miss.
• [assumed, load-bearing] The Monte Carlo 1/nu at L = 81, 729, 6561 with 8,000 samples is the true carpet exponent up to the quoted +/-0.004-0.006, i.e. the estimator's systematic low bias of ~0.01-0.02 seen on the square lattice does not shift the carpet values differently from one another.
• [assumed, load-bearing] The square-lattice calibration (estimator reads 0.741 where the exact value is 0.75) transfers to the carpets as a uniform, pattern-independent offset.
• [measured, load-bearing] All five known patterns share the same minimum left-right cut (4 bonds at step 1, 4^g at step g) and the same shortest crossing path (8 bonds), so their misses are comparable to one another.
• [derived] The lowest-order term of 1 - R in q = 1 - p is exactly (number of minimum cuts)*q^4 on all five, so the minimum-cut count is read off the exact polynomial.
• [assumed] The three candidate accounts (finite-cell error, threshold-exponent identity, zero locations) exhaust the natural explanations, so their failure licenses seeking a fourth mechanism.
• [assumed] The cut entropy s computed exactly at step g = 4 is close enough to its asymptotic value to serve as a pattern-level structural label.
• [assumed, load-bearing] The observed spread of misses (0.009 to 0.051) is larger than the combined statistical and systematic uncertainty, so the ordering of misses is a real signal rather than estimator noise.
• [assumed, load-bearing] X, Y and Z are structurally of the same class as the five known patterns (same 64/81 carpet construction, same crossing transition), so any mechanism fitted on the five extrapolates to them.
• [measured, load-bearing] At the one-block p* the 81x81 (two-step) lattice already crosses with probability 0.89-0.98, i.e. the two-step system is past its own threshold at p*.
• [assumed, load-bearing] That y orders the five measured exponents exactly is not a coincidence of five points, so the same ranking rule holds for the three unknowns.

Number kinds

• [count] 9x9 grid, 64 cells kept of 81, 17 removed — integers fixed by the generator's structure
• [count] the eight symmetries of the square
• [dimensional exponent] carpet dimension ln 64 / ln 9 = 1.893 — fixed by the self-similar scaling relation, mass 64 per linear factor 9
• [count] minimum left-right cut of 4 bonds at step 1 (4^g at step g) — exact count, zero spread, same on all five
• [count] shortest crossing path of 8 bonds on every pattern
• [count] minimum-cut counts 16, 48, 64, 256, 100 — exact counts confirmed by brute force at step 1
• [unclassifiable] cut entropy s (step 4): 0.693, 1.058, 1.153, 1.680, 0.870, and 0.923 / 1.058 / 1.220 for X, Y, Z — exactly computed as ln(count)/cut size, but the case names no scaling relation fixing its value and it is not set by measurement
• [unclassifiable] one-block fixed points p* (0.7183-0.7934) — exact roots of a pattern-specific polynomial; no dimensional or counting route sets their value
• [unclassifiable] R'(p*) values 3.454, 3.326, 3.122, 2.874, 3.175 (and 3.373, 3.345, 3.301) — exact derivatives of a pattern-specific polynomial
• [dimensional exponent] y = ln R'(p*)/ln 9 (0.4805-0.5642, and 0.5533/0.5495/0.5435) — fixed by the rescaling relation with linear factor 9 applied to the linearized map
• [computed exponent] measured 1/nu: 0.547, 0.524, 0.477, 0.4294, 0.5170 — Monte Carlo slope fit with bootstrap errors of +/-0.004-0.006 on both sides
• [computed exponent, target] y - 1/nu: +0.017, +0.023, +0.041, +0.051, +0.009 (to be predicted for X, Y, Z) — difference of an exact quantity and a measured one; an uncorrected estimator bias of about 0.01-0.02 sits inside it, so the absolute size is partly artefact
• [dimensional exponent] square-lattice exact 1/nu = 0.75 — exact 2D percolation value nu = 4/3; quoted as universal and it is, but the calibration around it is measured
• [free] estimator reading 0.741 on the square lattice and its implied bias of about -0.01 (0.73 on larger sizes, bias 0.01-0.02) — set purely by the estimator's finite-size behaviour; also a difference divided against an absolute exponent and never measured on the carpets
• [free] run-to-run spread 0.004 and bootstrap error 0.004-0.006 — set by sample count, not by structure
• [free] two-step crossing probability at p*: 0.89-0.98 — measured, with standard error 0.0002-0.0003
• [free] largest threshold gap 0.060 paired with smallest exponent miss 0.009 — difference of a one-block fixed point and an extrapolated measured threshold

Target form. number — For X, Y and Z, give the sign and size of y - 1/nu and at least their rank order, from a pattern-computable quantity, plus the mechanism that reproduces the five known misses including why G5's is smallest.. Bound: not fired.

Bound note. Bound not added — target has a known route (computed exponent).

Round 1

Ledger (Ledger)
TAKE
What is exactly true by identity: y is defined as ln R′(p*)/ln 9, computed from an exact polynomial to 40 digits — it is not an estimate of anything, it is what it is. 1/ν as reported is a Monte Carlo slope with a quoted statistical error of ±0.004–0.006 and an admitted, unmeasured-on-these-patterns systematic bias of ~0.01–0.02 from the square-lattice calibration. So y − 1/ν is a subtraction of an exact number from a number that is itself (bias + noise + true value), and no part of the case computes the true carpet 1/ν a second, independent way to check it. There is no closed identity here — R(p) is not checked against 1/ν by any algebraic relation, only against Monte Carlo crossing probabilities at three p values, which is a different quantity (R itself, not its threshold derivative). The one two-ways-computed quantity in the whole case is R(p): exact polynomial vs. 10⁶-sample Monte Carlo, and it closes to within 2 standard errors. Nothing analogous closes 1/ν. Given that, the "miss" y − 1/ν cannot be partitioned into signal and estimator bias from the case as given — the ledger cannot certify that the ordering of misses reflects the pattern rather than the uncorrected bias interacting differently with each pattern's finite-size approach to threshold (note the two-step lattice is already past threshold at p*, meaning finite-size corrections are large and pattern-dependent, not uniform).

PREDICTION
(none — my axis has no computation that produces a signed prediction for X, Y, Z; it can only flag that the bias term is uncontrolled)

ASSUMES
none

CHANGES MY MIND
A second independent computation of the true carpet 1/ν (e.g., an exact transfer-matrix or exact-enumeration bound at g=2 or g=3, not Monte Carlo) that lets y − 1/ν be checked as a closed identity the way R(p) was checked against Monte Carlo.

Framing (Framing)
TAKE
The named object is "1/ν of the carpet," but that name is doing double duty for two different objects. y = ln R′(p*)/ln 9 is a property of a single fixed 9×9 block under an approximation (block-independence) that the case itself states is false. "Measured 1/ν" is a property of the full infinite-genus lattice, extracted from finite-size scaling of an estimator with a known, uncalibrated bias on exactly this kind of geometry. These are not two measurements of one quantity converging with a correction term — they are two different objects that happen to share a name and a unit. The "miss" y − 1/ν is therefore not an error signal about the renormalization scheme; it is a category mismatch dressed as a residual. Chasing "what sets the miss" as if it's a single correctable quantity (finite-cell error, threshold identity, zero-location) will keep failing because there is no single mechanism to find — one side is exact algebra on a decimated block, the other is a biased slope-estimator on a lattice that (per the case's own admission) is already past threshold at two steps. The object this case should be about is: the estimator-bias function b(pattern) itself, specifically whether the square-lattice bias (0.01–0.02) is pattern-independent or scales with something like cut entropy/connectivity of the carpet. Until that bias is measured per-pattern, "y − 1/ν" is not a percolation quantity at all — it's an artifact of subtracting an exact number from an uncharacterized-bias number, and ranking or fitting it is fitting noise-plus-name-collision.

PREDICTION
The claimed exact ordering (y ranks 1/ν exactly across X,Y,Z as it did for G1–G5) will fail: at least one pair among X, Y, Z will invert the y-order relative to measured 1/ν once bias-corrected values are used.

ASSUMES
That the square-lattice bias is not in fact pattern-independent (contra the ledger's assumed premise), i.e. that carpets with higher cut entropy/connectivity show larger finite-size bias in this percentile-width estimator.

CHANGES MY MIND
If the bias, directly measured per-pattern (e.g. via an independent exact or exact-enough 1/ν check, or via extending L further), turns out uniform to within 0.002 across all eight patterns — then "miss" is a single well-defined quantity after all, and my category objection dissolves.

Hypothesis (Hypothesis)
TOOL
One-block real-space renormalization (Migdal–Kadanoff-style block transformation) applied to bond percolation on a self-similar carpet: p_{n+1} = R(p_n), with fixed point p* and derivative R'(p*) identified with the correlation-length exponent via y = ln R'(p*)/ln b (b = linear rescaling factor, here 9).

REQUIRES
1. The map p → R(p) must be a faithful coarse-graining: the crossing probability of a block, treated as replacing the block by a single renormalized bond, must correctly generate the crossing probability of the next generation lattice — i.e., blocks recombine independently once replaced by an effective bond.
2. The linearization of R at its fixed point must actually control the divergence of the correlation length under iteration — this requires that iterating the single-block map is asymptotically equivalent to the true many-body RG flow near p_c.
3. Finite-size/finite-order corrections (from truncating the RG at one block, ignoring correlations between adjacent blocks) must vanish or be sub-dominant in the regime where p*, R'(p*) are evaluated.
4. The lattice must be self-similar enough over the levels compared (step 1 block vs. asymptotic L→∞ behavior) that a single level's derivative already approximates the asymptotic multiplier.

SATISFIED
1. No — the case states this explicitly: "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." This is a stated violation, not a hedge.
2. Not checkable from the case as a clean yes/no — no independent check (e.g., a second-generation R₂(p) built from two RG steps, whose fixed-point derivative should converge toward the same y if the flow is truly RG-like) is reported. The one datum bearing on it — that the 81×81 lattice at one-block p* is already at crossing probability 0.89–0.98 — is evidence against requirement 2: the two-step system is already far past its own threshold at the one-step fixed point, meaning the "fixed point" is not stable/self-consistent across levels. This is a red flag, functionally a "no."
3. No — this is the same violation as (1), restated: the correlation between adjacent blocks is exactly the truncation error being ignored, and it is stated as present, not estimated as small.
4. Not checkable — no second RG level's y is computed and compared to the one-block y for any pattern, so self-similarity of the flow (as opposed to the geometric self-similarity of the carpet) is unverified.

TOOL
Finite-size Monte Carlo exponent estimator (percentile-width scaling, slope of log-width vs log-L) as a stand-in for the true 1/ν.

REQUIRES
1. The estimator must be unbiased, or have a bias that is at least uniform/pattern-independent, so that differences (y − 1/ν) across patterns reflect the RG error rather than the estimator's own systematic drift.
2. Sizes L = 81, 729, 6561 must be in the asymptotic scaling regime for all five (and three new) patterns alike.

SATISFIED
1. Not checkable from the case, and explicitly flagged as assumed, not verified: the ledger states the square-lattice bias (~0.01–0.02, itself size-dependent: 0.741 at moderate L, 0.73 at larger L) "has not been measured on these patterns," and its transfer to the carpets as a uniform offset is listed as an unverified LOAD-BEARING assumption. Given the bias itself changes with fitting range on the square lattice, there's a live possibility it depends on the local geometry (cut multiplicity, block topology) that differs across G1–G5, in which case it is not uniform and directly contaminates the y − 1/ν ranking the case treats as signal.
2. Not checkable — no convergence check (e.g., 1/ν fit restricted to the two largest sizes only, compared to the three-size fit) is reported per pattern, though the case notes one instance where adding two more sizes shifted a single-pattern fit by 0.004 — comparable to the smallest quoted miss (G5: 0.009). That alone means precondition 2 is marginal at best for the tightest cases.

TAKE
The whole case stands on y = ln R'(p*)/ln 9 being a legitimate approximation to 1/ν, and its own stated friction (blocks are not independent bonds) is precisely the unmet precondition of the RG tool being used — not a subtle caveat but an admitted violation. Worse, the check that would tell us whether the one-block linearization is even self-consistent — comparing y from step 1 to y from step 2 — is not done; instead we get indirect evidence (81×81 already past threshold at p*) that the one-step fixed point is not stable across levels. On the measurement side, the 1/ν estimator's bias is calibrated only on the square lattice, found to itself depend on fitting range, and its transfer to the carpets is asserted as uniform without measurement — yet the smallest miss (0.009) is inside the same magnitude as a single-pattern refit shift (0.004) reported elsewhere. So both halves of "y − 1/ν" carry unquantified, uncontrolled systematic error, and no tool in the case establishes that the residual pattern (G5 smallest, G4 largest) is RG structure rather than a mix of one-block truncation error and estimator bias riding on different local geometries. I cannot certify the ranking as real from the precondition record given.

PREDICTION

ASSUMES
none beyond what's in the case.

CHANGES MY MIND
A reported two-step RG derivative y₂ (from R₂(p), the true two-block-composed map) for at least G1–G5 that converges toward the one-block y within the quoted misses — that would certify requirement 2/3 above and let the miss be attributed to something other than an unchecked truncation error.

Legitimacy (Constraint)
TAKE
The target quantity y − 1/ν is not a clean measurement of a single effect; it is the difference between an exact number and a Monte Carlo slope that carries an unmeasured, pattern-dependent bias. The case admits the square-lattice estimator reads 0.741 against a true 0.75 — a −0.012 bias — but assumes without evidence that this offset is pattern-independent when transferred to five different carpets with different p* (0.718–0.793), different finite-size crossover behavior, and different two-step crossing probabilities already at 0.89–0.98 (i.e., already saturated past threshold, which will distort a 20th–80th percentile width estimator differently depending on how saturated it is). The misses span 0.009–0.051, only 2–8× the quoted bootstrap error of 0.004–0.006, and the systematic bias is explicitly flagged as unmeasured on the carpets at 0.01–0.02 — the same order as several of the misses themselves. G5's "smallest miss" (0.009) is within 1.5–2 bootstrap sigma of being indistinguishable from a pure bias artifact. I cannot certify that the ordering of misses is a structural signal rather than a pattern-dependent restatement of estimator bias correlating with how saturated each carpet is at p* two steps in. The three "failed" mechanistic accounts were tested against numbers that may not be real differences at all.

PREDICTION
When 1/ν for X, Y, Z is remeasured with bias-corrected estimator (calibrated per-pattern via the two-step saturation probability at p*, not a uniform offset), at least one of the three y − 1/ν values will shift by more than 0.012 (i.e., more than the assumed uniform bias) relative to its naively-computed value — falsifying the uniform-offset premise.

ASSUMES
That a per-pattern bias correction is measurable via the two-step saturation probability at p* (the case gives this number but never uses it for calibration) — none of this is stated as a method in the case, only as raw data.

CHANGES MY MIND
A direct measurement of the estimator's bias on at least two of the existing carpets (not just the square lattice), showing it is uniform to within 0.003 regardless of p* or two-step saturation probability, would remove my objection and let the five misses stand as real.

Lateral (Lateral)
TAKE

This is Migdal–Kadanoff bond-moving renormalization dressed as a "carpet CLG" (corner-linked-generator) calculation — the same object used for hierarchical lattices since the 1970s, and the shared theorem is exactly the one the case's stated premise flags as false: the single-bond RG identity R(p) treats a block as one independent bond, so the linearization at the fixed point carries the eigenvalue of a hierarchical (tree-like, no loop feedback between blocks) renormalization group, not the true one of a lattice with shared edges. Migdal-Kadanoff is well known to be exact on a hierarchical lattice and systematically biased on the "real" lattice it's meant to approximate, always in the same direction relative to the true exponent, because bond-moving discards exactly the connectivity correlations between neighbor blocks that lower 1/ν. That's a structural, named result, not a resemblance: it's the standard MK inequality/bias direction. The magnitude of the bias for a given pattern is controlled by how much inter-block sharing the true lattice has relative to what the one-block map assumes — and cut entropy s is a decent proxy for how much shared boundary structure (hence correlation) a pattern's minimal cuts have, since higher s means more distinct minimal paths crowding the same 4-bond cut, meaning more geometric entanglement between adjacent blocks the single-bond treatment ignores.

PREDICTION

Ordering by cut entropy s (X=0.923, Y=1.058, Z=1.220) predicts miss order X < Y < Z, with y − 1/ν(Z) > 0.041 (i.e., Z's miss exceeds G3's, since s(Z)=1.220 > s(G3)=1.153), and y − 1/ν(X) < 0.023 (since s(X) < s(G2)).

ASSUMES

Cut entropy s is monotonic proxy for inter-block correlation strength (case doesn't state this as established, only lists s alongside the data).

CHANGES MY MIND

If Z's measured miss comes out smaller than X's despite s(Z) > s(X), the MK-bias/cut-entropy mechanism is wrong and I'd need a different shared invariant.

House 1

VERDICT

The case's own framing is the first casualty: "the miss follows no obvious feature of the pattern" is false. Rank the five by cut entropy s — G1 0.693, G5 0.870, G2 1.058, G3 1.153, G4 1.680 — and the misses run 0.017, 0.009, 0.023, 0.041, 0.051: monotone except one 0.008 inversion at G1/G5, which is inside the quoted error. Lateral's Migdal–Kadanoff reading is the only seat that both names a mechanism and produces falsifiable numbers, and it is right in kind: bond-moving discards exactly the shared-edge correlations, and s counts how many distinct minimal cuts crowd the same 4-bond boundary. But Legitimacy and Framing outrank it on ordering: the estimator's ~0.012 low bias, measured only on the square lattice and never on a carpet, is a precondition of the measurement tool that carries the disputed claim, and it is of the same size as G5's entire miss. Mechanism is probably settled; the number it must fit is not yet certified.

DISAGREEMENT

Not "which mechanism." Lateral says inter-block correlation, quantified by s; Legitimacy and Framing say the quantity being explained (y − 1/ν) is an exact number minus a biased number, with the bias plausibly pattern-dependent through each carpet's saturation at p*. Those are compatible: a bias floor of ~0.012 plus a structural term rising with s reproduces the table, and makes G5's 0.009 mostly floor — which is why it is smallest. The genuine conflict is with the case's framing. The case presents the miss as a single physical residual with no structural correlate, and offers block-independence as the source. Block-independence being violated is uncontested — the case states it, Hypothesis confirms it is an unmet precondition. So it cannot be the disagreement. The disagreement is whether the ordering of misses is signal. Hypothesis's other point — that the one-block fixed point is not self-consistent across levels (81×81 crosses at 0.89–0.98 at p*) — is the sharpest unmet precondition on the table, and it contaminates both halves at once: it breaks the RG flow and it puts the L=81 point outside the estimator's scaling regime.

FORCED

Forced: y itself (an exact derivative of an exact polynomial, 40 digits — not an estimate, per Ledger); the q⁴ leading term of 1−R equalling the minimum-cut count (dual-cut identity); the common minimum cut 4^g and 8-bond shortest path for G1–G5, which is what makes their misses comparable at all; the sign of the estimator bias (low, hence inflating every miss by the same construction).

Fitted or inherited: the 0.01–0.02 bias magnitude — inherited from one square-lattice calibration that itself moves with fitting range (0.741→0.73), then assumed uniform across carpets; the choice of s as the controlling variable — fitted on five points with one inversion; the claim that y's exact ranking of five exponents is not chance (1/120 under the null, suggestive, not established).

Model artefact: p* is not a threshold — the gap to the measured threshold reaches 0.060, and the two-step lattice crosses at 0.89–0.98 there. So account (2) in the FRICTION was never a candidate; p* and the measured threshold are different objects, and their difference carries no information about the exponent. Account (3)'s zero-sum cancellation is likewise an artefact of writing an exact derivative as a sum over roots; nothing forces those terms to be separately meaningful.

TESTS

1. [lab] Read the q⁴ coefficient and shortest-path order off R(p) for X, Y, Z — predicts min-cut 4, path 8, i.e. same class as G1–G5 if the ledger's extrapolation premise holds; any pattern with cut ≠ 4 or path ≠ 8 is off-scale and its miss is not comparable. Nearly free; do it first.

1. [lab] Y is G2 with rows 2/3 (and 7/8) transposed: identical s = 1.058, p* 0.7815 vs 0.7803, R′ 3.345 vs 3.326 — predicts y − 1/ν(Y) = 0.023 ± 0.006 if s controls the miss; a deviation > 0.012 kills s as the controlling variable and points to a p*-or-geometry-specific term.

1. [lab] Re-fit 1/ν for G4 and G5 on the two largest sizes only (729, 6561) and compare to the three-size fit — predicts shifts equal to within 0.003 if the bias is uniform (ledger premise holds, misses stand); shifts differing by > 0.012 and correlated with the two-step crossing probability at p* confirm Legitimacy's pattern-dependent bias and require re-deriving the whole miss table.

1. [lab] Run the percentile-width estimator on one carpet at L = 81, 729, 6561, 59049 (reduced samples) and on the square lattice at matched sizes — predicts the carpet bias tracks the square-lattice bias within 0.003 if the calibration transfers; a larger carpet-specific bias confirms that L = 81 sits past threshold and outside the scaling regime, as Hypothesis's precondition check implies.

1. [lab] Compute s at g = 5 and 6 by the same dual shortest-path counting — predicts s stable to ~0.01 if the g=4 value is a legitimate asymptotic label; drift that re-orders G1 and G5 would resolve the one inversion in Lateral's fit rather than refute it.

1. [lab] Build the composed two-step map R₂(p) (exact transfer matrix if tractable, else 10⁶-sample MC crossing curve on 81×81), take y₂ = ln R₂′(p₂*)/ln 81 — predicts y₂ − 1/ν substantially smaller than y − 1/ν, with ordering preserved, if the miss is one-block truncation (MK); predicts no shrinkage if the miss is estimator bias.

1. [seat] Task Lateral with the G1/G5 inversion: produce a boundary-weighted entropy (minimal cuts weighted by shared-edge incidence, not raw count) — predicts G5 below G1 if inter-block sharing, not cut multiplicity, is the true variable; failure to separate them leaves s a proxy fitted on five points.

1. [seat] Task Ledger with a second, non-Monte-Carlo route to 1/ν at g = 2–3 (exact width from enumerated crossing distribution) — predicts closure of y − 1/ν as a checked identity, the way R(p) closed against MC; if no such route exists, the target quantity stays half-certified and every prediction below is conditional.

1. [lab] With tests 3–5 in hand, fit miss = b + c·s and predict: X ≈ 0.015–0.022, Y ≈ 0.023, Z ≈ 0.040–0.048, all positive, order X < Y < Z — predicts this if the MK-plus-bias-floor decomposition holds; an inversion (Z below X) falsifies it, as Lateral concedes.

RESIDUE

Not closed. The sharper case drops the Monte Carlo entirely: take two patterns with identical s and identical minimum-cut count but different p* (Y/G2 is the nearest available pair; construct an exact-degenerate pair if one exists), and ask whether their misses differ. That isolates the structural term from the p*-dependent finite-size term without ever needing the estimator's bias to be uniform. Second residue: compute y₂ and y₃ for G4 (largest miss) and G5 (smallest) and see whether the sequence y_g converges to a common limit at all — if it does not, then Framing is right that y and 1/ν are two objects sharing a name, the "miss" is a category residual, and no mechanism fitted to it will extrapolate to X, Y, Z no matter how well it ranks the five.

Lab report

LAB REPORT (runs 061, 062; measured after round 1)

Three new patterns, same class as G1–G5 (shortest path 8, minimum cut 4 at step 1, cut size 4^g):
y measured 1/ν y − 1/ν
X 0.5533 0.5522 ± 0.0037 +0.0011
Y 0.5495 0.5134 ± 0.0036 +0.0361
Z 0.5435 0.5281 ± 0.0039 +0.0154
Pair slopes (81→729, 729→6561): X 0.5525, 0.5519; Y 0.5260, 0.5008; Z 0.5254, 0.5308.

Minimum-cut counts obey N(g+1) = c·N(g)^4 exactly (checked g = 1→4):
G1 c = 1, G2 c = 3, G3 c = 4, G4 c = 36, G5 c = 81/2500, X c = 2, Y c = 3, Z c = 9.
Y and G2 have identical cut counts at every generation (N₁ = 48, c = 3); their y − 1/ν are 0.036 and 0.023.
On 5 of 19 other patterns in the same family, c is not constant across generations.

Two-step exponent y₂ = ln P₂′(p₂*)/ln 81, at the 81×81 lattice's own fixed point:
y₂ y (one block) measured 1/ν
square 0.7385 0.7213 0.7411 (exact 0.75)
G1 0.5811 0.5642 0.547
G2 0.5657 0.5470 0.524
G3 0.5428 0.5181 0.477
G4 0.4965 0.4805 0.4294
G5 0.5523 0.5258 0.5170
X 0.5725, Y 0.5659, Z 0.5533.
y₂ > y on every lattice; y₂ orders the measured exponents on all 28 pairs of the original eight.

Round 2

Ledger
TAKE
The new run gives an exact second rung: y₂ = ln R₂′(p₂*)/ln 81 is computed from a composed exact object exactly as y₁ was, no Monte Carlo involved. y₂ > y₁ on every one of eight lattices — that ordering is exact and unconditional, not a fit. This is the closure I asked for in round 1, but it does not close the way the square-lattice calibration suggested it would. On the square lattice, y₁=0.7213, y₂=0.7385, true=0.75: monotone approach, shrinking gap, consistent with a single convergent sequence y_g → 1/ν. On Y, the same construction gives y₁=0.5495, y₂=0.5659 — moving away from the measured 1/ν=0.5134, gap widening from 0.036 to 0.053. That is not an estimator artifact on my side of the ledger: y₁, y₂ are both exact. So either the measured 1/ν(Y) is the unreliable number here (its own pair-slopes, 0.5260 then 0.5008, are still falling with L, unconverged), or y_g does not converge monotonically to 1/ν on every pattern the way it does on the square lattice. Either way, no single extrapolation rule fitted to the square lattice transfers uniformly. Separately: Y and G2 share identical minimum-cut counts at every generation (N₁=48, c=3, exactly) yet differ in miss by 0.013 — an exact tie in cut structure with an unequal outcome, which is a clean ledger fact against any account resting on cut multiplicity alone.

PREDICTION
y₃(Y), computed exactly from the composed three-step map at its own fixed point, will exceed y₂(Y)=0.5659 (continuing y₁<y₂<y₃), while the Monte Carlo pair-slope for Y at the next size step (6561→59049) will fall further, below 0.50.

ASSUMES
none beyond treating y₃ as computable by the same exact construction used for y₁, y₂.

CHANGES MY MIND
y₃(Y) computed exactly and found to reverse — i.e. y₃ < y₂ — closing the gap toward the measured 1/ν(Y) rather than widening it; that would restore a single convergent-sequence account and let me retract the claim that no uniform extrapolation exists.

Hypothesis
TOOL
One-block RG map p→R(p), y = ln R′(p*)/ln 9, now tested against its own g=2 extension y₂ = ln R₂′(p₂*)/ln 81.

REQUIRES
1. (as before) the linearization at one block should already approximate the linearization of the true infinite-g flow, i.e. y should be close to y_∞, not just correctly ordered.
2. If (1) fails, the sequence y_g must at least converge monotonically toward the true exponent as g grows, so the one-block value is a controlled, not accidental, underestimate.

SATISFIED
1. No, and now measurably no: y₂ ≠ y on every lattice (square: 0.7213→0.7385; G1–G5, X,Y,Z all shift by 0.013–0.019). One block is not close to the asymptotic derivative.
2. Yes, now checkable and confirmed on the one calibration case with a known exact answer: square lattice y=0.7213, y₂=0.7385, exact=0.75 — monotone approach from below. Same direction (y₂>y) on all eight carpets. This is real evidence the flow is a legitimate RG in the sense that matters (precondition 2 of my round-1 Tool block) — I was wrong to call it merely "not checkable." It is checkable, and it passes, at least in direction.

TOOL
Minimum-cut recursion N(g+1) = c·N(g)^4 as grounds for treating cut entropy s as a stable structural label (and as the controlling variable for the miss).

REQUIRES
1. c must be constant across generations (else "s at g=4" is not close to its asymptotic value, per the round-1 assumption ledger flagged this explicitly).
2. Patterns with matching cut structure should show matching miss, if s (or equivalently minimum-cut multiplicity) is what's doing the explanatory work.

SATISFIED
1. Yes for G1–G5, X, Y, Z (checked g=1→4, constant c) — but no on 5 of 19 other patterns in the family, so this precondition is pattern-dependent, not generic, and the case's own extrapolation premise ("X,Y,Z are structurally of the same class") is only accidentally safe here — it could easily not have been.
2. No — decisively falsified: Y and G2 have identical cut counts at every generation checked, yet y−1/ν is 0.036 vs 0.023, a gap (0.013) bigger than several of the misses being explained. Cut structure alone does not fix the miss. My round-1 test-9 fit (miss = b + c·s) is falsified in kind, even before checking X's near-zero miss.

TOOL
Finite-size MC estimator, precondition that L=81,729,6561 lie in the asymptotic scaling regime uniformly across patterns.

REQUIRES
Pair-slopes (81→729 vs 729→6561) should agree within bootstrap error for a given pattern if the fit is in its asymptotic regime.

SATISFIED
No for Y specifically: slopes 0.5260 and 0.5008, a 0.025 spread — six times the quoted ±0.004 bootstrap error, and larger than Y's own quoted miss of 0.036. X and Z pair-slopes agree within ~0.001–0.005 (fine). This is exactly the failure I flagged in round 1 as "not checkable" — it is now checkable, and it fails for the one pattern (Y) whose reported miss is by far the largest of the eight. Y's "measured 1/ν = 0.5134" is not a stable number; its true value is unresolved by the given data.

TAKE
Round 1's central objection stood on two unmet preconditions. One of them is now partly answered in the tool's favor: y₂>y, moving toward the exact value on the square-lattice calibration and preserving order on all 28 pairs, is real evidence that the one-block map sits on a legitimate, slowly-converging RG trajectory rather than being numerology — I update toward crediting the RG mechanism. But the other precondition — that the MC estimator is in its asymptotic regime, uniformly across patterns — is now directly falsified for Y, whose pair-slopes disagree by six sigma of the quoted error, more than Y's own reported miss. That means Y's "measured 1/ν" is not a fact, and the miss table's biggest entry rests on an unconverged fit. Separately, the attempt to explain the miss structurally (cut entropy/minimum-cut multiplicity) is now falsified outright: Y and G2 share identical cut recursion at every generation but differ in miss by 0.013 — more than the gap the mechanism was invoked to explain. So the case has traded one open precondition question for two sharper diagnoses: the RG side is more legitimate than I credited, and the measurement side is worse — non-uniformly so, concentrated in Y.

PREDICTION
Extending the L=81,729,6561 fit to L=59049 for pattern Y will raise its measured 1/ν by more than 0.015 (i.e., above 0.529), shrinking y−1/ν(Y) to below 0.020, while X and Z shift by less than 0.005 in the same extension.

ASSUMES
None beyond the case; the prediction follows from the observed non-convergent pair-slopes for Y versus the stable ones for X, Z.

CHANGES MY MIND
A refit of Y at larger L that reproduces the same slope (~0.50–0.51) as the 729→6561 pair — confirming Y really has converged to a lower exponent — would mean the large miss is real structure, not an artifact of an unconverged fit, and would force me to drop this objection for Y specifically.

Legitimacy
TAKE
The new numbers sharpen my axis rather than closing it. Two results matter. First, X's pair-slopes (0.5525 → 0.5519, a 0.0006 shift) are stable well within bootstrap error, and its y − 1/ν is 0.0011 — statistically zero. That is a clean, legitimate measurement, and it directly falsifies my own proposed uniform +0.01–0.02 bias floor: if the estimator carried a pattern-independent low bias of that size, X could not read out at 0.0011. So "uniform bias" as I framed it is dead. But Y and Z tell the opposite story: Y's pair-slopes move from 0.5260 to 0.5008, a 0.026 drift — roughly seven times the quoted bootstrap error of 0.0036 — and Z's move from 0.5254 to 0.5308, a 0.0054 drift, also outside error. These are not stable numbers converging to a limit; they are still moving between the two largest sizes probed. A single reported "measured 1/ν" for Y or Z is not a legitimate summary of the data — it's a snapshot of an unconverged sequence, and nothing in the case bounds where it settles. The Y/G2 identical-cut-count pair now diverging in miss (0.036 vs 0.023) is consistent with this: the difference could be geometry, or it could just be that Y hasn't reached its asymptotic slope yet while G2's fit happened to land closer to it.

PREDICTION
At L = 59049 (one more decade), Y's pair-slope will continue to drop, landing below 0.49 — i.e., further from its currently reported 0.5134 by more than 0.02 — while Z's slope will land within 0.5250–0.5320 (i.e., essentially unchanged from its current 729→6561 value).

ASSUMES
That the drift seen in the 81→729→6561 pair-slopes for Y and Z is monotonic in this next decade rather than oscillating; the case gives no autocorrelation or higher-size data to rule out oscillation. None of this is stated as tested in the case.

CHANGES MY MIND
If a fourth size point for Y shows the pair-slope stabilizing to within 0.006 of 0.5134 (i.e., the drift stops rather than continues), I'll accept that Y's reported 1/ν is converged and that its large miss (0.036) is a real structural signal, not an artifact of an unfinished fit.

Lateral
TAKE

The s/c-based Migdal–Kadanoff reading is dead as a quantitative mechanism. The lab report contains its own falsification: Y and G2 have identical minimum-cut counts at every generation (c = 3, same s = 1.058) yet their misses are 0.036 and 0.023 — an 0.013 gap between two patterns my proxy called indistinguishable. My round-1 ordering prediction (miss rising with s: X<Y<Z) is also wrong; measured order is X<Z<Y. So: no boundary-counting invariant (cut multiplicity, cut entropy) sets the miss. That part of my take is retracted.

What survives, and what I now name explicitly, is the RG identification itself, not the proxy: y₂ > y on every one of eight lattices, and y₂ orders the measured exponents correctly on all 28 pairs while y did not order perfectly. That is the signature of a convergent iterated renormalization sequence approaching a fixed-point eigenvalue — the theorem is Wegner's correction-to-scaling structure: a one-step (or n-step) RG estimate of a critical exponent differs from the true value by a term controlled by the irrelevant eigenvalue of the linearized map, not by any static geometric count of the generating block. The miss is a leading correction-to-scaling amplitude, and its size is pattern-specific because each block's transfer matrix has its own subleading eigenvalue — it is not read off cuts, entropy, or c.

PREDICTION

For X, Y, Z, compute y₃ (three-step, 729×729 fixed point). Predict the ratio (y₃−y₂)/(y₂−y) is roughly constant across X, Y, Z, landing in 0.3–0.6 for all three (single dominant irrelevant eigenvalue), rather than scattering by more than a factor of 2 between any pair.

ASSUMES

That the g→∞ sequence y_g is dominated by one irrelevant eigenvalue near the relevant fixed point (standard RG correction-to-scaling premise); not stated or verified in the case.

CHANGES MY MIND

If (y₃−y₂)/(y₂−y) differs by more than a factor of ~3 across X, Y, Z, single-irrelevant-eigenvalue control is wrong and the miss is genuinely pattern-idiosyncratic with no shared theorem left to name.

House final

VERDICT

The weight has moved off mechanism entirely. The square lattice is the one place truth is known, and there y − 1/ν is negative (0.7213 vs MC 0.7411, exact 0.75), while y_g rises with g toward the exact value. On the carpets y_g rises identically, yet y sits above the measured exponent. Both cannot be a single RG truncation story: if y_g approaches 1/ν from below on carpets as on the square, then the true carpet exponents lie above y, and every "miss" is a deficit in the Monte Carlo, not an excess in the renormalization. Two independent lab facts point the same way: X, whose pair-slopes are stable to 0.0006, has a miss of 0.0011 (statistically zero); Y, whose pair-slopes drift 0.026 — seven bootstrap sigma — carries the largest miss and is the single pattern that inverts the y and y₂ orderings. The ranking rule fails exactly where the fit is unconverged. That is the tell.

DISAGREEMENT

The genuine conflict is between Hypothesis/Legitimacy (the MC estimator's asymptotic-regime precondition is falsified, non-uniformly, concentrated in Y and Z) and Lateral/Ledger's round-2 reading (the miss is a Wegner correction-to-scaling amplitude set by each block's irrelevant eigenvalue). Hypothesis named an unmet precondition of the measurement tool, and that tool supplies one of the two numbers in the disputed quantity — so it outranks the mechanism dispute. You cannot fit a correction amplitude to a number that is still moving between the two largest sizes probed.

But the deeper conflict is with the case's own framing. The case says the miss is an RG error caused by block correlation. Block non-independence is uncontested — the case states it, Hypothesis confirms it, nobody argues — so it was never the disagreement, and I rank it below. Worse, the case's framing has the sign backwards relative to its own calibration lattice. And the lab's claim that "y₂ orders the measured exponents on all 28 pairs of the original eight" cannot include X, Y, Z: y₂ gives Y > Z, measurement gives Z > Y. Either "the original eight" excludes the new patterns — in which case nothing has been extended — or the claim is false as written.

FORCED

Forced by structure: y and y₂ are exact derivatives of exact composed polynomials — not estimates (Ledger, round 1, still standing). y₂ > y on all eight lattices, direction unconditional. The square-lattice sequence 0.7213 → 0.7385 → 0.75 fixes the sign of the RG truncation: y underestimates the true exponent. The q⁴ = min-cut identity, the common cut 4^g and 8-bond path for all eight, and the exact recursion N(g+1) = c·N(g)⁴ are counting identities.

Fitted or inherited: every reported "measured 1/ν" — a slope from three points, two of which are outside the scaling regime for Y. The +0.01–0.02 uniform bias floor: dead, killed by X's 0.0011 (Legitimacy retracted it correctly). Cut entropy s as controlling variable: dead, killed by G2/Y sharing c = 3 and N₁ = 48 at every generation with misses 0.023 vs 0.036, and by the measured order X < Z < Y against the predicted X < Y < Z.

Model artefact: the miss table's sign convention. y − 1/ν is positive on carpets and negative on the square only because the carpet denominators are low-biased; the quantity is a difference of one exact number and one unconverged one, and the case reifies it as a physical residual. p*'s 0.060 threshold gap and the zeros-of-R cancellation remain artefacts of round 1, unchanged.

TESTS

1. [lab] Audit the "y₂ orders all 28 pairs" claim: list the eight lattices explicitly and recheck against the table — predicts the set excludes X, Y, Z (claim intact but unextended) if bookkeeping is right; predicts the claim is false as written if it includes them, since y₂(Y) > y₂(Z) while 1/ν(Y) < 1/ν(Z). Free; decisive for whether any ranking rule survives.

1. [lab] Recompute pair-slopes for G1–G5 and square from stored samples and regress miss on slope drift Δ = |slope₁ − slope₂| — predicts miss ≈ a·Δ with X (Δ=0.0006, miss 0.0011) and Y (Δ=0.026, miss 0.036) collinear if the miss is an unconverged-fit deficit; predicts scatter at fixed Δ if it is a correction-to-scaling amplitude. [tests the "spread is real signal" premise and the "MC is the true exponent" premise]

1. [lab] Extend Y to L = 59049, reduced samples — predicts 1/ν(Y) rises above 0.525 and its miss falls below 0.020, restoring both the y and y₂ orderings, if Hypothesis is right; predicts the slope holds at ≤ 0.51 if Y's exponent is real and the ranking rule is simply broken.

1. [lab] Compute y₃ exactly for the square lattice, where 1/ν = 0.75 is known — predicts y₃ ≈ 0.744 and a three-point extrapolated y_∞ = 0.750 ± 0.003 if y_g is a convergent RG sequence (settles the "y is a legitimate approximant" premise once, globally); an extrapolated limit off by > 0.01 kills that premise for every pattern at once.

1. [lab] Apply the same three-point extrapolation per carpet to get y_∞ — predicts y_∞ > y > measured 1/ν on all eight, i.e. the carpet MC is low by 0.03–0.08 and the miss's sign is a measurement artefact, if convergence is from below as on the square; predicts y_∞ ≈ measured if block-correlation truncation really is the source. [tests the block-independence premise the case leans on]

1. [lab] Measure the two-step crossing probability at p* for X, Y, Z and correlate the full eight against miss — predicts X near the 0.89 end, Y and G4 near 0.98, if saturation at L = 81 drives the deficit; no correlation kills the saturation mechanism. [tests the load-bearing 0.89–0.98 datum, currently reported and never used]

1. [seat] Lateral: compute (y₃ − y₂)/(y₂ − y) for all eight — constant within a factor ~2 supports a single irrelevant eigenvalue and a Wegner amplitude; scatter beyond factor 3 leaves no shared theorem, as Lateral concedes.

1. [seat] Ledger: produce the second non-Monte-Carlo route to 1/ν (exact enumerated percentile width at g = 2–3) for G5 and Y — closes y − 1/ν as a checked identity the way R(p) closed, or confirms that it cannot be closed and every miss stays half-certified.

1. [seat] Hypothesis: check whether the 5-of-19 patterns with non-constant c also show unstable pair-slopes — predicts the "same class" extrapolation premise is safe only for constant-c patterns, which would make X, Y, Z's membership lucky rather than structural.

RESIDUE

Not closed. The sharper case eliminates the Monte Carlo from the target entirely: compute y₁, y₂, y₃ for the square lattice, the triangular lattice and one carpet, and ask whether the exactly-computed sequence extrapolates to the exactly-known answer in the two cases where the answer is known. If it does, the carpet y_∞ becomes a prediction and the MC becomes the thing being tested, which is the correct direction of inference and inverts the case. Second residue: the G2/Y degenerate pair — identical cut recursion at every generation, different p* (0.7803 vs 0.7815) and different R′. If their misses still differ after test 3 resolves Y, then something not visible in any cut count sets the amplitude, and the object to name is the subleading transfer-matrix eigenvalue, not the geometry.

BACKPROP

open. Write back: (a) an edge — "miss = y − 1/ν" is not a percolation quantity until the MC side is in its scaling regime; the estimator's precondition failure on Y outranks the mechanism debate, and the ranking rule's only failure coincides with it; (b) a correction — the case's sign is inverted by its own calibration lattice: on the square, y undershoots the exact exponent, so on carpets the deficit is in the measurement, not the renormalization; (c) a probe — per-pattern estimator bias as a function of the two-step saturation at p*, a number already in hand and never used. Retract cut entropy as the controlling variable (dead by G2/Y) and retract the uniform bias floor (dead by X). Next case: the exact y_g sequence on lattices with known exponents, run as a test of the estimator rather than of the RG.