Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why?

PN-004 — Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why? · run 2026-09-30, lab roster (physics) + Reductio (one-off, round 2) · two rounds, one lab step · resolved: open

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: replace each kept cell
by a copy of the pattern, and again. A stacked carpet alternates two patterns A and B: A at the coarsest
level, B inside every kept cell of A, A inside those, and so on; after 2k levels it is an 81^k × 81^k lattice.
Put bond percolation on it and ask for an open left-to-right crossing. The correlation-length exponent 1/ν is
measured from how the width of the crossing transition shrinks with lattice size (L = 81, 729, 6561; 8,000
samples per size). For a pattern, C₁ is the minimum number of bonds whose removal disconnects the left side
of the 9×9 block from the right; minimum cuts multiply across levels, so a stacked carpet's cut exponent
ln(C_A·C_B)/ln 81 is exactly the average of its parents' ln C/ln 9, as is its fractal dimension.

Friction. Two stacks were measured. With A and B of the same cut class (C₁ = 4 for both), the stacked
exponent equals the average of the parents' exponents within noise. With A of cut 4 and B of cut 2, it falls
below the average by 3.5 standard errors. Dimension and cut exponent average exactly in both cases, and
building the stack in either order gives the same exponent, as it must. What makes the exponent add over
levels in one case and not in the other?

Background. | pattern | C₁ (cut class) | measured 1/ν |
|---|---|---|
| X | 4 | 0.5508 |
| Y | 4 | 0.5154 |
| T | 3 | 0.3743 |
| U′ | 2 | 0.2755 |

(All ±0.004 per measurement. Across other patterns of cut 4, 1/ν ranges from 0.43 to 0.55.)

The one-block estimate (the slope of the exact 2-cycle of the two patterns' crossing polynomials, per factor
81) gives 0.5503 for X with Y and 0.4485 for X with U′; the parents' one-block values average to 0.5514 and
0.4370. The run-to-run spread of the estimator is about 0.0045 per measurement.

Must predict. Two more stacked pairs are being measured the same way: X with T (cut 4 with cut 3) and
U′ with T (cut 2 with cut 3). For each: is the stacked 1/ν at, below, or above the mean of its parents, and
by how much? Give it from a mechanism that also says why cut-matched parents average.

Provenance. Stacks built level by level from the patterns' 9×9 generators (the builder reproduces a plain
two-level carpet when both levels are the same pattern). 1/ν is the endpoint slope of log width from L = 81 to
6561, which for three sizes equally spaced in log L equals the least-squares slope; bootstrap errors from
1,000 resamples. Parents' values are their recorded measurements by the same estimator.

---

Premise ledger

• [derived, load-bearing] Minimum cuts multiply across levels, so a stacked carpet's cut number is C_A·C_B and its cut exponent ln(C_A·C_B)/ln 81 is exactly the average of the parents' ln C/ln 9.
• [derived, load-bearing] The fractal dimension of a stacked carpet is exactly the average of the two parents' dimensions.
• [assumed, load-bearing] 1/ν extracted as the endpoint slope of log crossing-transition width over L = 81 to 6561 is the asymptotic correlation-length exponent, i.e. three sizes suffice and finite-size corrections are negligible at this range.
• [derived, load-bearing] The two build orders (A over B, B over A) must give the same exponent, so order is not a mechanism for the deficit.
• [assumed, load-bearing] Cut class C₁ is the structural label that controls whether exponents average, i.e. patterns sharing C₁ belong to one class for this purpose.
• [assumed, load-bearing] The X–Y agreement (−0.26 SE) is consistent with exact averaging rather than a small deficit hidden by noise.
• [assumed, load-bearing] The quoted ±0.004 per measurement and the bootstrap errors capture the full uncertainty, so a 3.5 SE gap is a real effect and not estimator bias.
• [assumed, load-bearing] The parents' recorded 1/ν values are directly comparable to the stacks' because the same estimator, sizes and sample counts were used.
• [assumed, load-bearing] Averaging of exponents under stacking is the null hypothesis; deviation from the parental mean is the signal to be explained.
• [assumed, load-bearing] The one-block (crossing-polynomial 2-cycle) estimator tracks the same quantity as the Monte Carlo estimator and its deficit/surplus pattern is informative about the mechanism.
• [assumed, load-bearing] Local slopes drifting downward between the two size windows (e.g. 0.4112 → 0.3846) reflect a genuine crossover in the mixed-cut stack, not a common systematic of the estimator.
• [assumed] When parents have unequal cut classes, the narrower cut (smaller C₁) dominates the crossing bottleneck at alternating levels, so the effective exponent is pulled below the arithmetic mean.
• [measured, load-bearing] Measured 1/ν for X, Y, T, U′ are directly obtained from the case's own simulations.
• [derived, load-bearing] The mean of both build orders is the correct estimate of the stacked exponent for each pair.

Number kinds

• [count] 9×9 grid of cells
• [count] 64 cells kept, 17 removed
• [count] eight symmetries of the square
• [count] 81^k × 81^k after 2k levels (linear factor 81 per two levels)
• [count] C₁ = 4 (cut class of X and Y) — flagged: patterns sharing this count have 1/ν spread 0.43–0.55, so the label does not fix the exponent
• [count] C₁ = 3 (cut class of T)
• [count] C₁ = 2 (cut class of U′)
• [dimensional exponent] cut exponent ln C/ln 9 (and ln(C_A·C_B)/ln 81 for stacks) — fixed by the scaling relation that minimum cuts multiply across levels
• [computed exponent] 1/ν = 0.5508 (X), 0.5154 (Y), 0.3743 (T), 0.2755 (U′), all ±0.004
• [computed exponent] stacked 1/ν = 0.5298 ± 0.0039, 0.5343 ± 0.0037, 0.3979 ± 0.0037, 0.3990 ± 0.0041
• [computed exponent] local slopes 0.5433, 0.5163, 0.5558, 0.5129, 0.4112, 0.3846, 0.4144, 0.3836
• [computed exponent] difference from parental mean: −0.0010 and −0.0147 — flagged: a difference of two same-type estimates, each carrying the full estimator uncertainty
• [free] −0.26 and −3.5 standard errors — flagged: a difference divided by an estimated error, so it inherits any misstatement of the error bar
• [computed exponent] one-block estimates 0.5503 and 0.4485; parental one-block averages 0.5514 and 0.4370
• [free] run-to-run spread of the estimator ≈ 0.0045 per measurement — set by sampling effort, not by structure
• [count] L = 81, 729, 6561
• [free] 8,000 samples per size; 1,000 bootstrap resamples — parameter choices of the run
• [free] 1/ν range 0.43 to 0.55 across other cut-4 patterns — flagged: a spread attached to a quantity treated as class-determined
• [unclassifiable, target] stacked 1/ν for X with T, and its offset from the parental mean 0.4626 — the deficit's size has no named route in the case; only its sign is argued
• [unclassifiable, target] stacked 1/ν for U′ with T, and its offset from the parental mean 0.3249 — the deficit's size has no named route in the case; only its sign is argued

Target form. number — For each of the two new stacks, give the stacked 1/ν relative to the parental mean — at, below or above, and by how much — from a mechanism that also explains why cut-matched parents average exactly.. Bound: fired.

Bound note. Bound seat added — no known route for the target (unclassifiable).

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

Round 1

Ledger (Ledger)
TAKE
Two things are exact by construction and must be kept separate from what is measured. (1) Cut number multiplies across levels, so ln(C_A·C_B)/ln81 = average of ln C_A/ln9 and ln C_B/ln9, identically, for every pair, matched or not. (2) Fractal dimension averages exactly by the same counting argument. Neither identity references 1/ν at all — 1/ν is obtained from an independent computation (Monte Carlo width-scaling), and nothing in the case states or derives an identity forcing it to average. So "exponents add over levels" for X,Y is an empirical near-equality (−0.26 SE), not a proven identity, and its failure for X,U′ is not a violation of anything pinned.

The two independently computed values that are available for the same object are the multi-size MC slope and the one-block 2-cycle slope. For X,Y both sit near the parental average (MC 0.5321 vs one-block 0.5503, means 0.5331/0.5514 — same sign of near-agreement). For X,U′ they disagree in sign of offset: MC is below the parental mean (0.3984 vs 0.4131) while one-block is above it (0.4485 vs 0.4370). That is a pinned bookkeeping fact, not an interpretation: the two routes to "the stacked exponent" do not agree for the mismatched pair, meaning the object does not yet have a scale-independent exponent at these sizes — local slopes are still descending (0.4112→0.3846, 0.4144→0.3836) toward the smaller-C₁ parent.

PREDICTION
Using ΔlnC as the only pinned scaling variable and the single available (ΔlnC, deficit) point (ΔlnC=ln4−ln2=0.693 → −0.0147), linear extrapolation gives: X,T deficit ≈ −0.0061 (stacked 1/ν ≈ 0.4565, below mean 0.4626); U′,T deficit ≈ −0.0086 (stacked 1/ν ≈ 0.3163, below mean 0.3249). Both below parental mean, X,T deficit smaller in magnitude than U′,T.

ASSUMES
Linear scaling of the deviation with ΔlnC between cut classes is not stated in the case; it is introduced here only as an extrapolation from the one mismatched data point available, offered as bookkeeping projection, not as a mechanism.

CHANGES MY MIND
If the X,T or U′,T deficits do not scale monotonically with ΔlnC (e.g., X,T deficit larger in magnitude than U′,T despite smaller ΔlnC), the linear-in-ΔlnC bookkeeping fails and the pinned-vs-free split needs a different variable.

Framing (Framing)
TAKE
The case treats "1/ν" as one fixed number that either averages or doesn't. It isn't one number here — it's an endpoint slope of log-width over three sizes, and the case's own local-slope columns show it still moving: 0.5433→0.5163, 0.5558→0.5129 (matched-cut, converging toward each other) versus 0.4112→0.3846, 0.4144→0.3836 (mixed-cut, still sliding the same direction, not stabilizing). A two-point slope taken from a drifting sequence is a parameter-dependent count, not a converged exponent — you cannot ask "why does the converged exponent average" about a quantity you haven't shown has converged. The object the question should be about is the L-dependence of the local slope itself (the flow/crossover behavior), not a single scalar 1/ν. Separately, the case already contains proof that "cut class" doesn't fix 1/ν — cut-4 patterns range 0.43–0.55 — so matching C₁ is being used as an explanatory label for something (near-equal 1/ν) that in the X,Y case is really just X and Y happening to sit close in that 0.43–0.55 spread, not a class effect. The real object of study should be named "asymptotic 1/ν(L→∞) versus local slope at finite L," and the deficit question is really a question about crossover length, wrongly posed as a question about exponent addition.

PREDICTION
Adding L=59049 to the X–U′ stack will show the local slope continue falling below 0.38 (not plateau near 0.398), i.e. the reported 0.3979±0.0037 is not the asymptotic value but an intermediate point on a still-descending crossover.

ASSUMES
That the builder can feasibly run one more size (59049) with comparable sample count to resolve whether the local slope has plateaued.

CHANGES MY MIND
If the local slopes for both X–Y and X–U′ stacks are shown to be flat (within error) across three independent size windows (not just two), demonstrating 1/ν has actually converged at L=81 already — then the "count varies with parameter" objection dissolves and the averaging-vs-deficit question is legitimately about a fixed number.

Hypothesis (Hypothesis)
TOOL
Finite-size scaling extraction of 1/ν as the endpoint slope of log(width) vs log L over three sizes (L = 81, 729, 6561)

REQUIRES
- A single dominant scaling length governs the crossing-width transition (width ~ L^{-1/ν}) over the measured range
- Corrections to scaling are subdominant at L = 81, i.e., the local slope has converged to its asymptotic value by the smallest size used
- No second, competing divergent length scale (e.g., from a structurally distinct sublattice) that would make the "width" a mixture of two power laws rather than one

SATISFIED
- Single dominant length: not checkable from the case as stated — the case simply asserts "1/ν is the endpoint slope," treating this as settled
- Corrections subdominant: no, for the mixed pair. The local slopes 0.4112 → 0.3846 (X–U′) and 0.4144 → 0.3836 (U′–X) show a real, order-of-magnitude-larger-than-noise drift across the two size windows (drop of ≈0.027, versus a stated per-measurement spread of 0.0045). The matched pair drifts too (0.5433→0.5163) but lands back on the parental mean; the mixed pair drifts and lands off it. A drifting local slope is the signature of an unconverged, non-single-power-law regime — exactly the precondition this tool needs and does not have at these three sizes for the mismatched stack
- No competing length scale: no — a stack of two patterns with different C₁ has, by construction, two different natural cut geometries alternating level by level, which is precisely a candidate second length scale. The tool assumes this away by fiat, it does not verify it

TOOL
Multiplicative cut number / additive-in-log cut exponent and fractal dimension across levels

REQUIRES
- The minimum cut of a two-level nested block equals the product of the level-wise minimum cuts (no cheaper cut exists that exploits both levels jointly)
- Self-similarity is exact so the same relation holds at every level of the recursion

SATISFIED
- Product-of-cuts: yes — this is a combinatorial identity of the construction (a cut must sever every A-block, and inside each severed region a B-cut is separately required), and the case reports it holding exactly for both stacks
- Exact self-similarity: yes, by the stated construction (replace each kept cell by a copy, recursively)
This tool is sound and fully discharged; it is not where the deficit can be hiding.

TOOL
(Implicit) linear averaging of 1/ν under stacking, argued by analogy to the cut-exponent/dimension averaging
— no named theorem in the case; this is the actual load-bearing inference being tested

REQUIRES
- The renormalization map for the stacked lattice's crossing probability must factorize, near the relevant fixed point, into a product of the two parents' individual RG derivatives evaluated at (approximately) the same fixed point
- Equivalently: both patterns' single-block RG flows must be close to each other (nearly commuting, nearly co-located fixed points) for the composite log-derivative to split into an arithmetic mean of the parents' own log-derivatives
- "Same cut class C₁" is being used in the case as a stand-in for this requirement

SATISFIED
- Shared/nearby fixed point: no, not guaranteed by C₁ alone — the case's own background table shows cut-4 patterns spanning 1/ν = 0.43–0.55, so "same C₁" does not imply "same or nearby RG derivative." The X–Y match (0.5508 vs 0.5154, both cut 4) is close only because those two exponents happen to be close, not because C₁ matches
- The real discriminator is exponent proximity, not cut-class identity: X–Y differ by 0.035 and average cleanly (−0.26 SE); X–U′ differ by 0.275 and fall short (−3.5 SE). This is consistent with a precondition of "closeness of parent exponents" being satisfied for one pair and violated for the other, while the case's own stated precondition ("cut classes match") is a different, unverified claim that happens to coincide with exponent-closeness in these two examples only

TAKE
The case names one tool that is genuinely a theorem here — multiplicative cuts imply additive-in-log cut exponent and dimension — and that tool is fully satisfied and fully discharged; it explains the two topological invariants averaging exactly, nothing more. It is then silently extended, by analogy and via the proxy "same C₁," to a second, unstated tool: that 1/ν itself averages when RG derivatives combine linearly. That tool has real preconditions (shared or nearby fixed point / linearizable, near-commuting maps) which the case never checks and which its own background data (cut-4 spread 0.43–0.55) shows C₁ does not guarantee. The local-slope drift in the mismatched stack (0.411→0.385) is independent evidence that the finite-size-scaling tool's own precondition — single dominant length scale, converged by L=81 — is violated exactly where averaging fails and roughly satisfied where it holds. So the deficit is not a mystery attached to "cut class mismatch"; it is what happens when you apply a linear-averaging tool whose precondition is closeness of the parents' RG data, mislabeled by the case as "cut class matching."

PREDICTION
U′–T (parents 0.2755, 0.3743, exponent gap 0.0988, mean 0.3249) will show a deficit smaller than 0.006 (stacked 1/ν > 0.319), while X–T (parents 0.5508, 0.3743, exponent gap 0.1765, mean 0.4626) will show a deficit larger than 0.008 (stacked 1/ν < 0.4546) — even though both pairs span the same one-step cut-class gap (2→3 and 4→3), because deficit size tracks parent-exponent distance, not cut-class distance.

ASSUMES
That deficit magnitude is monotonic in |1/ν_A − 1/ν_B| rather than in |C₁,A − C₁,B|; the case gives only two data points and does not state which variable controls magnitude, so this ordering is an extrapolation from those two points, not a given fact.

CHANGES MY MIND
If U′–T's deficit turns out to be equal to or larger than X–T's deficit despite U′–T's much smaller exponent gap, that would kill the exponent-proximity mechanism and support the case's own cut-class-based story instead.

Legitimacy (Constraint)
TAKE
Before any mechanism is entertained, check what the two data points can actually support. The X–U′ deficit rests on treating ±0.004 as the full uncertainty on each parent and propagating it linearly to get "3.5 SE." But 1/ν is extracted from only three system sizes with 8,000 samples each, via an endpoint slope — a two-point-equivalent estimator with no independent handle on finite-size corrections. The local slopes for X-with-U′ drop from 0.411 to 0.385 between the two windows; for X-with-Y they drop from 0.543 to 0.516/0.556. Both pairs show the same-sized drift (~0.03), which is the signature of a common finite-size correction to the estimator, not evidence that one pair "averages" and the other doesn't. The X–Y case looks clean only because its parents are close in value, so a shared systematic partially cancels in the difference-from-mean; the X–U′ case has parents far apart, so the same systematic shows up as a large apparent deficit. Until someone extends to a fourth size and confirms the slope estimate is converged rather than still drifting, I can't certify that −0.0147 is a structural effect rather than an artifact of a short extrapolation applied asymmetrically to unequal parents. The one-block numbers (0.5503 vs 0.4485, exact by construction) disagreeing with Monte Carlo by comparable margins to the "deficit" itself is a second warning sign, not a confirmation.

PREDICTION
A fourth size point (L=59049) will move X-with-U′'s 1/ν up by more than 0.008, cutting the reported 3.5 SE deficit to under 2 SE, while X-with-Y shifts by less than 0.003.

ASSUMES
That the endpoint-slope estimator's finite-size correction scales with the spread between parent exponents (i.e., is worse for less-similar stacks) — not stated in the case.

CHANGES MY MIND
A four-size (or more) fit showing the X-with-U′ local slope has already converged (flat across at least two consecutive windows) rather than still monotonically drifting — that would certify the deficit as real rather than an artifact of estimator range.

Lateral (Lateral)
TAKE
Strip this down to what's actually invariant vs. what isn't. C₁ multiplies across levels exactly because minimum cut is a max-flow/min-cut quantity — it's linear under series composition by construction, so ln C and the fractal dimension MUST average. No claim needed there beyond the theorem that already names itself in the case.

1/ν is a different animal: it's the local Lyapunov exponent of an RG recursion on crossing-probability polynomials (Migdal-Kadanoff-type hierarchical RG). Composing two different generators A then B is composing two different maps, and the exponent of that composition is a Lyapunov exponent of a matrix/map cocycle — this is the same object Furstenberg's theorem on products of non-commuting maps governs. Its headline result: the Lyapunov exponent of a product of two different operators equals the average of their individual exponents only when they share an invariant structure (here: the same crossing-probability fixed point, i.e. same universality class); otherwise subadditivity bites and the composite exponent sits below the average. X and Y share C₁=4 and (apparently) the same fixed point regime, so they average. X and U′ have different C₁, hence different fixed points — the composite map's derivative is evaluated off both individual fixed points, and Furstenberg subadditivity pulls it down. This is a resemblance to a named theorem, not just a label: cut class is a proxy for "shares a fixed point," not itself the mechanism.

PREDICTION
X+T (ΔC₁=1) falls below its parental mean 0.4626 by less than X+U′ did in absolute terms: stacked 1/ν = 0.452–0.458 (deficit 0.005–0.011, roughly 1.3–3 SE, strictly less deficit-SE than X+U′'s 3.5). U′+T (ΔC₁=1) similarly deficits by 0.003–0.009 from 0.3249, landing at 0.316–0.322.

ASSUMES
That deficit magnitude scales monotonically with |ΔC₁| (distance between fixed points) rather than being a step function that fires whenever C₁ differs at all — the case gives only two data points (match/no-match), not a gradient, so this is an extrapolation.

CHANGES MY MIND
If X+T's deficit (ΔC₁=1) turns out equal to or larger than X+U′'s (ΔC₁=2) in SE terms, the mechanism isn't fixed-point distance/Furstenberg subadditivity — it would mean the deficit is triggered by presence of any lower class, not distance, and I'd retract the "shares a theorem with Furstenberg" claim in favor of a threshold story.

Bound (Bound)
TAKE
This is a hierarchical lattice, so the crossing probability obeys an exact real-space RG recursion p' = f_A(p) for a pure A-carpet, with 1/ν = ln f_A'(p*_A)/ln 9 at the pattern's own attractive fixed point p*_A ∈ (0,1) — existence of that fixed point in the open interval, with f_A'(p*_A) > 1, is required for a sharp transition to exist at all (stability). For the stack, the composed map is f_A∘f_B, and by the chain rule its derivative at the stack's own fixed point p* is f_A'(p*)·f_B'(p*) — evaluated at one shared point p*, not at each parent's own p*_A, p*_B separately. Only when p*_A = p*_B does this reduce, by continuity/symmetry, to the parents' own derivatives and hence exact averaging of ln-derivatives. When p*_A ≠ p*_B, monotonicity and the concavity/convexity of increasing crossing-polynomial maps forces f_A' evaluated off p*_A to differ from f_A'(p*_A) in a fixed direction (away from a stable fixed point the derivative of such maps is bounded above by its value at the fixed point on the relevant side), which forces the composed derivative — hence 1/ν_stack — to lie at or below the parental average, never above. That is the forced interval: 1/ν_stack ∈ (0, (1/ν_A+1/ν_B)/2], equality iff p*_A=p*_B. This is a stability/existence constraint on the RG map, not a fitted mechanism.

PREDICTION
Both remaining stacks show deficits, not surpluses: 1/ν(X,T) falls strictly below 0.4626, landing near 0.450 ± 0.006, and 1/ν(U′,T) falls strictly below 0.3249, landing near 0.312 ± 0.006 — with the X,T deficit smaller in magnitude than the U′,T deficit (since T's cut class, 3, is closer to X's 4 than to U′'s 2 is far from T's 3... actually closer to U′), i.e. |deficit(U′,T)| > |deficit(X,T)|.

ASSUMES
That distinct cut classes generically imply distinct self-dual crossing fixed points p*_A ≠ p*_B for these patterns (not independently verified, only inferred from the X–Y vs X–U′ contrast).

CHANGES MY MIND
A measured stacked exponent for any mismatched-cut pair that exceeds the parental average by more than 1 SE would falsify the forced-inequality direction and overturn this take.

Checker — round 1

31 claims: 24 ok, 1 mismatch, 0 sign conflict, 2 justify, 4 uncheckable

House 1

CHECKS

[R1-29] MISMATCH — Lateral, "X and Y share C₁=4 … X and U′ have different C₁." The flag is an extraction artefact: the recomputation read U′ into the cut-4 set, which Lateral never claimed. The arithmetic survives. The inference does not: the case's own background (cut-4 patterns spread 0.43–0.55) shows C₁=4 does not imply a shared fixed point, so Lateral's "same C₁ ⇒ same invariant structure" is unsupported independently of the flag. I do not rest anything on Lateral's Furstenberg mapping.

[R1-28] JUSTIFY — Legitimacy, "endpoint-slope reads low when parents are far apart in 1/ν." Mechanism: with two alternating generators the width is a mixture of two power laws; an endpoint slope over a window where the local slope is still descending reads the lower asymptote-ward value. Direction in the source series: supported — X–U′ falls 0.4112→0.3846, X–T-style mixed pairs would do the same. But the differential part of the claim is not supported: the matched pair drifts by the same ~0.027 (0.5433→0.5163) and still lands on the mean. So "reads low" survives; "reads low in proportion to parent spread" is untested and is exactly what Test 2 settles.

[R1-31] JUSTIFY — Bound, "composed derivative lies at or below the parental average, never above." Mechanism: chain rule at the stack's single fixed point p*, with each parent's log-derivative evaluated off its own p*. Direction in the MC series: supported (−0.0147, local slopes falling). Direction in the exact series: contradicted. The one-block 2-cycle is precisely the chain-rule object Bound invokes, and for X,U′ it reads 0.4485 against a parental one-block mean of 0.4370 — a surplus, opposite sign ([R1-5] vs [R1-6]). Bound's interval is therefore not forced; it is an empirical read of one drifting estimator. I do not adopt it as a bound.

VERDICT

The weight falls on Hypothesis and Framing, not on the case's own story. Two estimators of the same object disagree in sign for the mismatched pair: Monte Carlo reads 0.0147 below the parental mean, the exact one-block 2-cycle reads 0.0115 above it. A sign conflict between an exact chain-rule computation and a three-point endpoint slope is not a mechanism waiting to be named; it is a warning that the three-point slope has not converged. The mixed stack's local slopes are still descending monotonically toward the weaker parent at the largest size measured. So the "deficit" is, on present evidence, a crossover length showing up as an exponent. And the label the case uses to sort the cases — cut class — is disqualified by the case's own background: cut-4 patterns span 0.43–0.55, wider than the entire effect. X and Y average because their exponents are 0.035 apart, not because C₁ matches.

DISAGREEMENT

The case's framing is the thing in conflict. It asks "why does the exponent add when cut classes match?" — presupposing (i) that 1/ν is a converged exponent at L ≤ 6561, and (ii) that C₁ is the sorting variable. Both presuppositions are attacked from inside the case's own data. The panel's visible fight — Lateral/Bound (structural subadditivity of composed RG maps, cut class as proxy for fixed point) versus Hypothesis/Framing (crossover; parent-exponent distance is the real variable) — is downstream of that. Hypothesis named an unmet precondition of the tool the whole dispute rests on: single dominant length scale, corrections subdominant by L=81. That tool is load-bearing for the claim actually in dispute (is there a deficit at all?), so it outranks the mechanism quarrel. Legitimacy's error-bar complaint is a weaker version of the same point and is ranked under it; Ledger's pinned/free split is uncontested bookkeeping and ranks below both.

FORCED

Forced: C_A·C_B and hence ln(C_AC_B)/ln81 = mean of parents' cut exponents; identically, for every pair, matched or not. Fractal dimension likewise. Order-independence of the stacked exponent (same 2-cycle up to conjugation) — and the data confirm it, 0.5298/0.5343 and 0.3979/0.3990. That the composed map's derivative is f_A′(p*)f_B′(p*) at one shared p* — chain rule, exact.

Not forced: that this product equals the parents' average — that needs p*_A = p*_B, which nobody has checked. Bound's inequality direction is a model artefact of assumed concavity; the exact one-block number for X,U′ violates it.

Inherited from measurement: all four parent 1/ν, both stack values, the 0.0045 spread.

Fitted: every predicted magnitude on the table. Ledger's −0.0061/−0.0086 and my own numbers below are linear-in-one-variable extrapolations, fit on 1 point. Hypothesis's and Lateral's orderings are fit on 2 points (match/mismatch).

My stance, stated as such: both new stacks read below their parental means at these sizes, by an amount tracking parent-exponent gap, not ΔC₁. X,T ≈ 0.453, deficit ≈ −0.0094 (fit on 1 point), ~2.5 SE. U′,T ≈ 0.320, deficit ≈ −0.0053 (fit on 1 point), ~1.4 SE, i.e. not significant. Both deficits shrink toward zero if the window is extended — which is the point.

TESTS

1. [lab] [cost: cheap — the crossing polynomials are already in hand] Compute the one-block 2-cycle slope for X–T and U′–T and compare to the parents' one-block averages (0.4626-analogue, 0.3249-analogue) — predicts a surplus for both (as X,U′ gave +0.0115) if the MC deficit is a crossover artefact; predicts a deficit matching the MC sign if the composed-map subadditivity of Bound/Lateral is real. This is the sign conflict that decides the case.

1. [lab] [cost: free — local slopes for parents and stacks are already recorded] Window-matched deficit: form the deficit using only the 729→6561 local slopes (stack vs parents' same-window slopes) and separately the 81→729 window. Predicts the deficit shrinks with window if crossover ([R1-28] direction); predicts it is flat or grows if structural. Also discharges "parents comparable to stacks" and "drift is genuine, not a common systematic" — if the matched pair's drift equals the mixed pair's, the drift is estimator-common.

1. [seat] Hypothesis [cost: cheap] Solve f_X, f_Y, f_T, f_U′ for their fixed points p*_A and tabulate against C₁. Predicts p*_X ≈ p*_Y with p*_U′ far off if fixed-point proximity is the variable; predicts p* clustering by C₁ (T apart from both X and U′) if the case's cut-class label is the variable. This is the direct test of the ASSUMED premise "C₁ is the structural label," which no seat resolved.

1. [lab] [cost: cheap] Fit width = aL^{−1/ν}(1 + bL^{−ω}) with ω shared across all four stacks (3 points per stack, but ω pooled, so identified). Predicts asymptotic 1/ν for X,U′ recovering ≥0.408 (deficit under 1 SE) if crossover; predicts 1/ν stable near 0.398 if real.

1. [lab] [cost: hours] Independent-seed replicate of X–U′ at all three sizes, 8,000 samples, to compare replicate spread against the 0.0037 bootstrap error. Predicts spread ≈ 0.0045 (bootstrap understates by ~20%, softening 3.5 SE to ~2.9) if the quoted errors are roughly right; predicts a larger spread if not. Discharges the "error bars capture full uncertainty" premise.

1. [lab] [cost: infeasible here — L=59049 is 3.5×10⁹ sites per sample at 8,000 samples, far beyond the largest run so far] Framing's and Legitimacy's decisive test: a fourth size. Predicts local slope continues below 0.38 and the fitted 1/ν rises toward 0.413 if crossover; predicts a plateau at ~0.398 if structural. Named because it is the clean test, ranked last because it cannot be run.

1. [seat] Ledger [cost: free] Power analysis on the X–Y null: given σ=0.0039, what deficit would have been invisible at −0.26 SE? Predicts anything up to ~0.006 is undetectable — which, if true, means "X,Y averages exactly" is an upper bound, not a measurement, and the matched/mismatched dichotomy is a detection-threshold effect.

RESIDUE

If Test 1 returns one-block surpluses for both new pairs while MC returns deficits, the case is not about stacking at all — it is about what an endpoint slope does to a two-scale crossing distribution, and the sharper case is: for a stack of two patterns with fixed points p*_A, p*_B, derive the crossover length L× at which the local slope leaves the p*_A basin, and show the measured deficit is a single function of L×/L rather than of ΔC₁ or Δ(1/ν). If instead Test 3 finds p* clustering by C₁ and Test 2 finds a flat deficit, the case survives as posed and the sharper version is a third cut class: build a stack of two different cut-4 patterns from opposite ends of the 0.43–0.55 spread. Same C₁, large exponent gap — that single measurement separates the case's label from its rival outright.

_Contract check: House adopted a flagged claim: H1-26 (from Legitimacy); House TESTS without a cost tag or reason: 3 (cost tag has no reason), 4 (cost tag has no reason), 5 (cost tag has no reason), 7 (cost tag has no reason); House fitted numbers without a "fit on n points" label: 2 line(s)._

Checker — House 1

30 claims: 16 ok, 1 mismatch, 0 sign conflict, 2 justify, 11 uncheckable

Lab report

LAB REPORT — run 070 (cross-class stacks), for round 2

Machine control: the pattern T alone, measured on the new machine with the same
estimator: 1/ν = 0.3745 ± 0.0037 (recorded 0.3743) — holds.

One-block values, computed before any sample (exact 2-cycle of the two patterns'
crossing polynomials, per factor 81; both orders give the same value):
| pair | stacked y | parents' one-block y | their mean |
| X with T | 0.4984 | X 0.5533, T 0.4392 | 0.4962 |
| U′ with T | 0.3851 | U′ 0.3206, T 0.4392 | 0.3799 |
One-block fixed points p* of the parents: X 0.740, Y 0.782, T 0.832, U′ 0.943.

Stacks (L = 81, 729, 6561; 8,000 samples per size; 1/ν = endpoint slope 81→6561):
| stack | order | 1/ν | local slopes (81→729, 729→6561) |
| X with T | X over T | 0.4706 ± 0.0040 | 0.4938, 0.4474 |
| | T over X | 0.4775 ± 0.0040 | 0.4718, 0.4832 |
| U′ with T | U′ over T | 0.3211 ± 0.0039 | 0.3547, 0.2874 |
| | T over U′ | 0.3438 ± 0.0037 | 0.3621, 0.3255 |
(local slopes ± about 0.008)

| pair | mean of both orders | mean of parents | difference | in standard errors | gap between orders |
| X, T | 0.4740 | 0.4626 | +0.0115 | +2.7 | 0.0069 |
| U′, T | 0.3324 | 0.3249 | +0.0075 | +1.7 | 0.0227 |
The two orders of a pair are the same infinite object. The U′,T orders differ by
4.1 standard errors at L = 6561.

Parents' last local slopes (729→6561): X 0.5520, T 0.351, U′ 0.249.
On last local slopes: X,T stacks 0.4653 against parents 0.4515 (+0.014);
U′,T stacks 0.3065 against parents 0.300 (+0.007).

Earlier pairs, for comparison (same estimator): X with Y −0.0010 (−0.26 s.e.);
X with U′ −0.0147 (−3.5 s.e.).

Estimator note: a local slope still falling with L means the endpoint slope reads
ABOVE the limiting exponent, not below it.

Round 2

Ledger
TAKE
Round 1's linear extrapolation is falsified: both new pairs came in ABOVE the parental mean (+2.7 SE, +1.7 SE), not below. The pinned facts from this report are these. First, the one-block value is exact and order-independent by construction (both orders give one number: 0.4984 for X,T; 0.3851 for U′,T) — this is the identity, not a measurement. Its deviation from the parental one-block mean is small and same-signed as MC for the new pairs (+0.0022, +0.0052 vs MC's +0.0115, +0.0075), unlike X,U′ where one-block and MC disagreed in sign. Second, and this is the load-bearing bookkeeping fact of this round: the two build orders of U′,T disagree by 4.1 SE at L=6561 (X,T orders by ~1.7 SE). The forced identity says the infinite stacked object is order-independent — the one-block computation demonstrates this exactly. So a 4.1 SE order-split in the finite-L MC estimate is not a violation of that identity; it is direct, pinned evidence that L=6561 has not reached the regime where order-independence (and hence any stable exponent) holds. Third, the case's own estimator note is now on record: a still-falling local slope means the endpoint slope reads ABOVE the limit. Local slopes are falling in 3 of 4 order-runs here. Two independent quantities — one-block exact value, and order-agreement of the MC estimate — both say: not yet converged.

PREDICTION
At the next size (L=59049), the order-gap for U′,T will shrink to under 2 standard errors (i.e., |U′-over-T minus T-over-U′| < 0.0104 at that L's bootstrap error scale), while for X,T it will shrink to under 1 SE.

ASSUMES
That order-gap shrinkage with L is a valid proxy for approach to the order-independent asymptote; the case does not state this, it follows only from treating order-independence as the target identity.

CHANGES MY MIND
If the order-gap does not shrink (or grows) at the next size, order-independence is not being approached by L=6561, and the whole endpoint-slope-as-asymptotic-exponent premise fails outright rather than merely converging slowly.

Framing
TAKE
The new data delivers the smoking gun for the framing objection, and it comes from the panel's own "forced" column. Order-independence of the stacked exponent is derived as exact — A-over-B and B-over-A are the same infinite object. At U′,T the two orders disagree by 4.1 standard errors at L=6561. That is not noise around a converged number; it is direct proof that "1/ν" as read off three finite sizes is not yet the invariant the case's algebra guarantees exists. A quantity that must be order-independent in the limit but is order-dependent at the largest size measured is, by definition, a parameter-dependent count, not the converged exponent — exactly my axis's complaint. The estimator note the lab attaches (a still-falling local slope reads ABOVE the limit, not below) also overturns the naive reading Bound and Lateral were using: X,T and U′,T show positive deficits now, X,U′ showed negative, X,Y near zero — the sign flips pair to pair with no stable rule, which is what you'd expect from a slope still in transit, not from a structural law keyed to C₁. The object worth naming is still the crossover length L×(A,B) set by how far p*_A and p*_B sit apart (0.740 vs 0.943 for U′; 0.740 vs 0.832 for T); the reported "deficits/surpluses" are L×/L artifacts, and the U′,T order-split is the cleanest evidence yet that L=6561 is inside that crossover, not past it.

PREDICTION
At L=59049, the two orders of U′,T will converge to within 1 standard error of each other, and their common value will sit below 0.3324 (the current two-order mean) by at least 0.010 — i.e. the "surplus" shrinks by at least two-thirds as the order-gap closes.

ASSUMES
That order-convergence and value-convergence track the same crossover length (so closing the 4.1-SE order gap coincides with the endpoint slope moving toward its asymptote), and that a run at L=59049 with comparable sample count is feasible.

CHANGES MY MIND
If a run at the next size shows the U′,T order-gap holding steady or widening (not closing) while the mean value stays pinned near +0.0075 above parents, that would mean the discrepancy is a real, order-sensitive structural effect rather than an unconverged crossover — and I'd concede the object is well-posed as asked.

Hypothesis
TOOL
Order-independence of the stacked exponent (A-over-B = B-over-A), justified in the case as forced by construction
REQUIRES
- The self-similar recursive identity for an infinite (or sufficiently converged) lattice
- At the finite sizes actually sampled, any order-dependence observed must be smaller than the estimator's own noise, or the "forced" identity is not what's being measured

SATISFIED
- Exact identity for the infinite object: yes, by construction (uncontested)
- Order-independence at L = 6561, the size actually used to report 1/ν: no. U′-over-T gives 0.3211 ± 0.0039, T-over-U′ gives 0.3438 ± 0.0037 — a 4.1 SE gap between two quantities that must be identical. X-T's gap (0.0069) is smaller but still a nontrivial fraction of σ. This is direct, quantitative proof that the finite-size estimator has not converged for these pairs; it is no longer an inference from drifting local slopes, it's a violation of a "must-hold" identity.

TOOL
Finite-size scaling endpoint slope (L = 81 → 6561) as estimator of asymptotic 1/ν — carried over from round 1, now sharpened by the case's own estimator note
REQUIRES
- Local slope must have stopped drifting by the window used, or the known bias direction must be accounted for
- The case's own stated rule: a local slope still falling with L reads ABOVE the true asymptotic value

SATISFIED
- Converged by L = 81: no, for three of four new series. U′-over-T: 0.3547 → 0.2874 (falling). T-over-U′: 0.3621 → 0.3255 (falling). Both falling means, by the case's own stated rule, the reported mean 0.3324 overestimates the true asymptotic value — i.e. the measured "surplus" over the parental mean (0.3249) is itself inflated and could shrink or invert with more levels.
- X-T is mixed: one order falls (0.4938→0.4474), the other rises (0.4718→0.4832). Averaging a falling and a rising series and calling the mean converged is not licensed by the tool; it's exactly the order-dependence problem flagged above wearing a different hat.
- This tool's precondition is violated in a directionally informative way this round, not merely "unchecked."

TOOL
One-block exact 2-cycle value as ground truth for the asymptotic exponent
REQUIRES
- The one-block quantity equal the true L→∞ limit of the finite-size estimator (no further multi-level renormalization correction beyond the first block)

SATISFIED
- Not checkable from the case — no derivation is given that the two coincide.
- But the pattern across pairs is striking and was absent from the MC numbers: one-block deviation (X,U′ +0.0115 > U′,T +0.0052 > X,T +0.0022 > X,Y −0.0011) tracks the one-block fixed-point gap |Δp*| (0.203 > 0.111 > 0.092 > 0.042) almost linearly. No such clean ordering exists in the MC deviations (which flip sign between X,U′ and the other two mismatched pairs). This is evidence the one-block/fixed-point object is closer to the real invariant than the raw endpoint slope is, even without a proof of exact equality.

TAKE
The sign flip kills my round-1 mechanism outright: X,T and U′,T both show surplus, not deficit, so "deficit tracks |Δ(1/ν)|" is falsified (X,U′ has the largest exponent gap, 0.275, and the only deficit; X,T's gap of 0.177 gives a surplus). What survives is the tool-precondition argument, now confirmed rather than merely suspected: the order-independence identity — which must hold exactly — measurably fails at L=6561 for U′,T (4.1 SE) and marginally for X,T, which is direct proof the endpoint-slope estimator has not converged for mismatched stacks. The case's own estimator note (falling local slope ⇒ reads above asymptote) then predicts the U′,T surplus is partly or wholly an artifact that should shrink with L. Meanwhile the one-block exact numbers — immune to finite-size convergence issues — show a consistent surplus pattern tracking the fixed-point separation |Δp*| (X,U′ > U′,T > X,T > X,Y), not cut-class distance and not exponent distance. So the real structural variable is plausibly |Δp*|, and the MC "deficit" for X,U′ (opposite sign from its own one-block value!) is the clearest single piece of evidence in the whole case that the endpoint-slope tool is unreliable exactly where the case's central claim depends on it.

PREDICTION
A correction-to-scaling fit (or a direct L=59049 measurement) of U′,T will pull the fitted asymptotic 1/ν down from the current 0.3324 to below 0.320 — i.e. the apparent surplus over the parental mean (0.3249) will shrink to under half its current size — while the corresponding fit for X,T will stay at or above 0.469, moving less than 0.005 from its current value of 0.4740, because only U′,T's two orders both show falling local slopes.

ASSUMES
That the bias direction stated in the estimator note ("falling local slope reads above asymptote") applies uniformly to both orders of a stack and not just to single-pattern carpets — the case states the rule in general terms but only demonstrates it for unstacked patterns.

CHANGES MY MIND
If the U′,T order-gap (currently 0.0227, 4.1 SE) does NOT shrink at L=59049 or under a correction-to-scaling fit, that would mean the order-dependence is not a finite-size artifact at all, and I'd have to abandon the "unconverged estimator" reading entirely in favor of some genuine order-sensitive structural effect the case's own "must be order-independent" argument failed to anticipate.

Legitimacy
TAKE
The new data both vindicates and corrects my axis-level objection. The "forced" fact is that the two build orders are the same infinite object — the case states this outright, and my Round-1 stance rested on trusting that the error bars and the endpoint-slope estimator were adequate to see through to that infinite object at L≤6561. They are not: U′-over-T and T-over-U′ disagree by 4.1 standard errors, and X-over-T's local slopes fall (0.494→0.447) while T-over-X's rise (0.472→0.483) — opposite trends for a quantity proven order-independent. That is not a small residual; it's direct, structural proof that at these sizes the endpoint slope has not converged to the invariant it's supposed to measure. My specific directional claim was wrong, though: the lab's estimator note says a falling local slope reads ABOVE the asymptotic exponent, not below — so my predicted "X-with-U′ rises toward the parental mean at L=59049" has the sign backwards; if anything the true asymptotic value sits further below 0.3979, making that deficit larger, not smaller, while U′,T's currently-measured surplus (+1.7 SE, also built on a falling slope) should shrink or invert. I retract the magnitude/sign table I gave for X,T and U′,T — both were fit on one point and are now falsified in sign for at least one pair. What survives is the diagnosis, sharpened: order-disagreement this large at L=6561 means none of these "differences from parental mean," matched or mismatched, are yet legitimate estimates of a limit.

PREDICTION
At L = 59049, the two build orders of U′-with-T will converge to within 1 SE of each other, and their common mean 1/ν will be ≤ 0.315 (i.e. below the current 0.3324, moving toward or past the parental mean of 0.3249) — a downward correction of at least 0.010 from the current reading.

ASSUMES
That order-convergence and value-convergence track together, i.e. the same finite-size correction responsible for the order gap also biases the order-averaged value — not separately established in the case.

CHANGES MY MIND
If a fourth size shows the two orders of U′-with-T already agreeing to <1 SE while the order-averaged value stays flat at 0.332–0.334, that would certify the current reading as converged and retire my legitimacy objection for this pair.

Lateral
TAKE
Retracting the sign of my round-1 mechanism; keeping — and sharpening — the class of theorem. Furstenberg-style subadditivity predicted a monotone deficit growing with ΔC₁. The data kill that outright: X,T and U′,T (both ΔC₁=1) come in with surpluses (+2.7, +1.7 SE in MC; +0.0022, +0.0052 in the exact one-block), while X,U′ (ΔC₁=2) gave a deficit in MC but a surplus in its own exact one-block value (+0.0115). Sign isn't a function of cut-class distance at all. But look at the exact (non-extrapolated) one-block surpluses against the one-block fixed points p*: X 0.740, T 0.832, U′ 0.943. Ordering |Δp*|: X–T=0.092→0.0022, U′–T=0.111→0.0052, X–U′=0.203→0.0115 — monotone, and roughly quadratic (ratios ≈4.9 vs predicted (0.203/0.092)²≈4.9). That's not Furstenberg; that's the standard second-order correction to a composed-map exponent from a convex renormalization map evaluated off a common fixed point — the same structure as Jensen's inequality applied to the crossing-polynomial recursion. C₁ never enters except as an accidental proxy for p*-separation. The MC "deficit" for X,U′ is a finite-size read of a slowly-converging composite, not the asymptotic sign.

PREDICTION
Stack two cut-4 patterns from opposite ends of the quoted 0.43–0.55 spread (Δ1/ν ≈ 0.10–0.12, comparable to X–U′'s Δp*). One-block surplus over the parental mean will be positive and in the range +0.008 to +0.016 — same order as X,U′'s +0.0115 — despite identical C₁.

ASSUMES
That one-block p* values scale monotonically with 1/ν across the untested cut-4 patterns (not directly given), and that the observed quadratic fit is not itself an artifact of having only three (p*,surplus) points.

CHANGES MY MIND
If that same-C₁, wide-spread pair's one-block value lands within noise of the exact parental average (surplus ≲0.003) despite Δp*≈0.1–0.15, the curvature/fixed-point-separation mechanism is dead and cut-class reclaims sole explanatory status.

Bound
TAKE
Round 1 wrong on sign. I argued the shared-fixed-point chain rule forces 1/ν_stack ≤ parental mean, "never above." The exact one-block 2-cycle — the correct infinite-volume object, not the finite-L endpoint slope — now shows surplus in all three mismatched pairs measured: X,U′ +0.0115; X,T +0.0022; U′,T +0.0052. That's not noise, it's the exact computation, and it's the same direction three times. My concavity assumption about f_A′ off its own fixed point was backwards.

What's still forced, and now correctly signed: the stacked map's own fixed point p* is pinned by f_A(f_B(p*))=p*, strictly between p*_A and p*_B when they differ (intermediate value, existence forced) — here p*_X=0.740 < p*_T=0.832 < p*_U′=0.943, all distinct, so every mismatched stack has p*_A ≠ p*_B and the degeneracy that gives exact averaging (p*_A=p*_B) never fires. The forced interval is now [mean, mean + something set by |p*_A−p*_B|·f″], not (0, mean]. I retract the upper cap and keep only the floor: 1/ν_stack ≥ mean of parents whenever p*_A≠p*_B, by the exact one-block object; equality only at p*_A=p*_B.

The MC endpoint slope reading a deficit is a finite-size artifact — the note confirms falling local slopes mean the true limit is displaced further from the naive reading, not that the sign flips, but the two estimators disagree and only the exact one is load-bearing.

PREDICTION
The exact one-block y for a third mismatched cut-4/cut-4 pair (two patterns both C₁=4 but with p* differing by more than 0.03, e.g. a wide-spread pair from the 0.43–0.55 range) will also show a positive one-block surplus over the parental mean, of size roughly proportional to (p*_A−p*_B)², i.e. greater than +0.003.

ASSUMES
That the one-block 2-cycle slope is a monotonic, convex function of p near each parent's own fixed point (the specific curvature sign that turns "p* between p*_A,p*_B" into "surplus" rather than "deficit") — inferred from the three data points, not derived independently.

CHANGES MY MIND
A measured or exact one-block value for any future mismatched pair that comes in below the parental mean by more than its quoted precision would kill the floor claim entirely, and I'd have no forced direction left — only the bare fact that p*_A≠p*_B breaks exact averaging, with sign undetermined.

Reductio
TAKE
Taking VERDICT exactly true (mixed-cut deficits are a crossover artifact; endpoint slope reads low while local slopes still descend toward the weaker parent), six pushes:

Limits/Neighbours: applied to the two new pairs, VERDICT predicts both read BELOW their parental mean, by ~0.0094 (X,T) and ~0.0053 (U′,T) — stated explicitly in the VERDICT text. The lab report shows both read ABOVE the mean: X,T +0.0115 (+2.7 SE), U′,T +0.0075 (+1.7 SE). Wrong sign, both cases.

Joint consistency: VERDICT's mechanism ("descending local slope ⇒ endpoint reads low") is directly negated by the lab's own estimator note: "a local slope still falling with L means the endpoint slope reads ABOVE the limiting exponent, not below it." The report's own local slopes descend (e.g. 0.4938→0.4474) and the endpoint reads above the mean, exactly as the note — not VERDICT — predicts.

Invariance/Forced: order-independence ("same infinite object") is contradicted at U′,T — the two orders differ by 4.1 SE at L=6561, and X,T orders differ by 0.0069, both larger than the matched-pair order gaps (~0.001–0.005), with no account offered.

Transforms (negation/Test 2): the "shrinks with window" signature is absent — last-window deficits (+0.014, +0.007) track the overall ones, not decaying toward zero.

PREDICTION
Most exposed: VERDICT's sign claim for X,T and U′,T ("both read below parental mean"). Already falsified by the lab table (+0.0115, +0.0075); no further test needed — record it as contradicted rather than untested.

ASSUMES
None beyond what VERDICT already asserts; I did not introduce new premises, only applied VERDICT's stated mechanism to the new pairs and to the report's own estimator note.

CHANGES MY MIND
If a fourth, larger lattice size (L=59049) showed the X,T and U′,T local slopes continuing to fall and the endpoint slope crossing over from surplus to deficit — i.e. the sign flips toward the mean from above as predicted by a genuine crossover approaching from the opposite side — that would restore VERDICT's picture. Short of that size, VERDICT's own explicit sign prediction for these two pairs stands contradicted by the data in hand.

Checker — round 2

33 claims: 25 ok, 2 mismatch, 0 sign conflict, 1 justify, 5 uncheckable

House final

CHECKS

[R2-7] MISMATCH — Ledger calls the "falling local slope ⇒ endpoint reads ABOVE the limit" rule "the case's own estimator note." It is the lab report's note, not the case's. Attribution error only; the quoted content and direction are exact, and Ledger uses it correctly. Claim survives.

[R2-14] JUSTIFY — "the reported mean 0.3324 overestimates the true asymptotic value." Mechanism: the endpoint slope over 81→6561 is the average of the two window slopes; when the local slope is still descending, that average sits above the limiting value. Source series: U′-over-T 0.3547→0.2874 falls, T-over-U′ 0.3621→0.3255 falls — both orders descending, so the estimator reads high. Direction supported. Claim survives; the MC +0.0075 surplus for U′,T is inflated and may be entirely estimator bias.

[H1-26] JUSTIFY — my round-1 adoption of Legitimacy's "endpoint slope reads LOW when parents are far apart." The checker confirms the source series falls (0.4112→0.3846), and under the lab's now-stated rule a falling local slope reads high, not low. My direction was backwards. This claim does not survive, and my round-1 sign prediction for X,T and U′,T rested on it — Reductio is right, it is contradicted by data in hand, not merely untested. I retract it and rest nothing on it.

[H1-27] JUSTIFY — Bound's round-1 cap, "composed derivative at or below the parental average, never above." The source series rises (0.4370→0.4485). Now confirmed three times in the exact object (+0.0115, +0.0022, +0.0052). The flag is right; the cap is dead, and Bound has retracted it.

[H1-16] MISMATCH — my round-1 remark that the recomputation "read U′ into the cut-4 set." The re-flag parses my own quoting of the earlier flag. Bookkeeping artefact; nothing in this round rests on it, and Lateral has withdrawn the Furstenberg mapping independently.

VERDICT

The weight has moved to the exact one-block object, and it points at a variable the case never names. Across all four pairs the one-block deviation from the parental mean is monotone in the fixed-point separation |Δp*|: X,Y 0.042→−0.0011; X,T 0.092→+0.0022; U′,T 0.111→+0.0052; X,U′ 0.203→+0.0115. Cut class does not order this; exponent gap does not order it (X,T has the second-largest Δ1/ν and the smallest deviation). Averaging is exact when p*_A = p*_B and fails at second order in Δp* otherwise — C₁ matched for X,Y only because their p* happen to be close. The Monte Carlo endpoint slope is not a competing witness: U′,T's two build orders, which must be the same object, differ by 4.1 SE at L = 6561. A quantity that violates a forced identity at the largest size measured has not converged, and its sign — deficit for X,U′, surplus for X,T and U′,T — is noise on a crossover, not a law.

DISAGREEMENT

The case's framing is the thing in conflict, twice over. It sorts by cut class, and its own background (cut-4 patterns spanning 0.43–0.55) plus the p* table disqualify that label. It treats "1/ν" as a converged number, and the order-split falsifies that at the size it reports. The panel's residual fight is now narrower and sharper than either: does the one-block 2-cycle compute the same quantity the Monte Carlo is estimating? Hypothesis named this precondition as unmet and it is the load-bearing one — the entire sign question depends on which estimator is the witness. It outranks the mechanism quarrel (Lateral/Bound's curvature story vs Framing/Legitimacy's crossover story), because both mechanisms are read off whichever estimator you trust. Evidence against equality is on the table and nobody used it: one-block parent values read high against MC by +0.0025 (X), +0.034 (Y), +0.065 (T), +0.045 (U′) — a per-pattern bias, not a constant offset.

FORCED

Forced: C_A·C_B and hence the cut exponent averaging exactly; fractal dimension likewise; order-independence of the infinite stacked object. Forced within any single-parameter real-space RG scheme: the stacked map's fixed point p* lies strictly between p*_A and p*_B when these differ (intermediate value), the composed log-derivative is f_A′(p*)+f_B′(p*) in logs at that one p*, and therefore the deviation from the parental average vanishes identically iff p*_A = p*_B and is second order in (p*_A − p*_B) otherwise. That much is structure, and it explains cut-matched averaging without ever invoking C₁.

Not forced: the sign of that second-order term. It depends on the curvature of f near p*, which no seat derived; Bound asserted one sign in round 1 and the opposite in round 2, both read off data. Lateral's quadratic law is fit on 3 points (X,T; U′,T; X,U′), coefficient k = surplus/Δp*² ranging 0.26–0.42 — scatter of ~60%, so call it k ≈ 0.3 fit on 3 points, usable for ordering, not for magnitude.

Inherited from measurement: all parent 1/ν, all stack values, the p* table, the one-block values.

Model artefact: the MC "deficit" for X,U′ and the MC "surpluses" for X,T and U′,T alike — all three are endpoint slopes over a window where local slopes still move and where order-independence measurably fails. Also a likely artefact: the one-block estimator's absolute values, which read high against MC by an amount that grows as p* grows.

My stance: exact averaging holds iff the parents share a crossing fixed point; deviation is +k·Δp*² with k ≈ 0.3 fit on 3 points; cut class is an accidental proxy. Both new pairs sit above the parental mean in the exact object (+0.0022, +0.0052, both computed not fitted), and the MC agreement in sign is partly luck, since R2-14 shows U′,T's MC reading is inflated.

TESTS

1. [lab] [cost: cheap — generators and the polynomial machinery are in hand] Compute the one-block 2-cycle and p* for a pair of cut-4 patterns from opposite ends of the 0.43–0.55 spread, with |Δp*| ≳ 0.15. Predicts a positive deviation of order +0.007 to +0.015 (k·Δp*², fit on 3 points) if fixed-point separation is the variable; predicts deviation ≲0.001 if matched C₁ is the variable. This single number separates the case's label from its rival outright and is the cheapest decisive test on the table.

1. [lab] [cost: free — all four pairs already computed] Regress one-block deviation on Δp*² across X,Y / X,T / U′,T / X,U′ (fit on 4 points) and on ΔC₁ and Δ(1/ν) separately. Predicts Δp*² wins on residual scatter if the curvature account holds; predicts no variable orders them if the deviations are scheme noise.

1. [lab] [cost: free — both columns already recorded] Tabulate one-block-minus-MC for the four parents against p*: +0.0025 (X), +0.034 (Y), +0.065 (T), +0.045 (U′). This tests the precondition Hypothesis named and nobody discharged — does the one-block scheme compute the same exponent? Predicts a flat offset if it is a good scheme with a constant bias (deviations then transfer cleanly to stacks); predicts p*-dependent bias if not, in which case one-block signs survive but its magnitudes do not, and Test 1's numeric range must be widened.

1. [lab] [cost: cheap] Re-run both orders of U′,T with fresh seeds at L = 6561 only. Predicts the 0.0227 order-gap reproduces within ~0.005 if it is genuine finite-size crossover (Ledger, Framing, Hypothesis all converge here); predicts it collapses if the bootstrap errors understate run-to-run spread, which would also soften every SE in the case.

1. [lab] [cost: cheap] Corrections-to-scaling fit, width = aL^(−1/ν)(1+bL^(−ω)) with ω pooled across all six stacks. Predicts U′,T's asymptotic value drops below 0.320 and X,T moves <0.005 (Hypothesis's prediction) if R2-14's bias direction is right; predicts both stable if the surpluses are structural.

1. [seat] Bound [cost: cheap] Evaluate f″ for X, T, U′ at the composed fixed points and report its sign, rather than inferring it. Predicts f″ > 0 at the relevant p* (surplus forced) or f″ < 0 (deficit forced) — this converts the only fitted piece of the leading account into a derived one.

1. [lab] [cost: infeasible here — L = 59049 is ~3.5×10⁹ sites per sample at 8,000 samples, two orders beyond the largest run completed] The clean convergence test: does the U′,T order-gap close and the value fall? Named because every seat asked for it; ranked last because it cannot be run, which is why Tests 1–3 have to carry the verdict.

RESIDUE

If Test 1 returns a large surplus for a cut-matched pair, the case as posed dissolves: there is no cut-class effect, and the sharper case is "when does a two-generator real-space RG average its parents' exponents?" — answered by p* coincidence, with the follow-on question being why p* clusters by C₁ at all in the small sample here (X 0.740, Y 0.782 both cut 4; T 0.832 cut 3; U′ 0.943 cut 2 — an ordering that does track C₁ across these four, which is exactly how the confound arose). If Test 3 shows p*-dependent bias, the sharper case is a two-parameter RG: the crossing state of a carpet block is not one number, and the whole one-block apparatus needs a second coordinate before any of its magnitudes mean anything.

BACKPROP

open. Write back: (a) an edge — "cut class C₁ → exponent averaging" is not an edge; replace with "fixed-point coincidence p*_A = p*_B → exact averaging," with C₁ marked as a confounded proxy in this sample; (b) a correction — my round-1 verdict's sign, and the estimator-direction rule it used, are retracted: a descending local slope makes the endpoint slope read high, and my X,T/U′,T predictions were falsified in sign by the lab table; (c) a probe — the order-gap between build orders is a free, forced-identity convergence diagnostic and should be reported for every stack in every future run, since it detects unconvergence without needing a larger size. Next case: the cut-matched, p*-separated pair of Test 1.

_Contract check: House adopted a flagged claim: H2-4 (from Hypothesis); House TESTS without a cost tag or reason: 4 (cost tag has no reason), 5 (cost tag has no reason), 6 (cost tag has no reason); House fitted numbers without a "fit on n points" label: 2 line(s)._

Checker — House final

30 claims: 20 ok, 1 mismatch, 0 sign conflict, 2 justify, 7 uncheckable

---

Codifier's note on this record (2026-09-30)

• Seat record not loaded. This panel ran with no record packets (the "off" half of the comparison with PN-003, which ran with them).
• Two House-final draws. After the House final above, Round 2's button was pressed again and then the House final's; the second House final's prose did not reach the export (the export kept the first draw's prose), but its checker card did. The record above is the first draw, internally consistent: Round 2, its checker, House final and its checker. The second draw's checker table is kept below for comparison only; its prose is not recorded. Whether Round 2 itself was redrawn cannot be told from the exports (both carry the same Round 2 text and checker).

Checker — House final, second draw (prose not captured)

### Checker — House final
30 claims: 17 ok, 2 mismatch, 0 sign conflict, 2 justify, 9 uncheckable

---

House final, second draw (prose recovered 2026-10-01)

Codifier. The continuation export of 2026-10-01 carries the stored run's House final, and it is the
second draw. Its checker table is the one filed above as "second draw (prose not captured)"; only the
summary line differs, by the newer tool's "0 derivation" count. Every other parent section matches the
record above, apart from an "Instrument notes" line the newer tool adds to the case. Continuation 1 was run
against this draw ("the House's latest diagnosis"), not against the first. The prose follows unchanged.

House final (second draw)

CHECKS

[R2-26] MISMATCH — Lateral's "2×√(0.0039²+0.0037²) ≈ 0.011". The flag is an extraction artefact: the checker pulled a single σ (0.0039) and compared it to the combined figure. The arithmetic is right (0.01075). Claim survives as arithmetic; it is not load-bearing for anything I adopt.

[R2-32] MISMATCH — Reductio, "House's [R1-28] mechanism conflicts with the lab's estimator note." The flag is correct and it is against me. Width ~ L^(−1/ν); the endpoint slope over 81→6561 is the mean of the two local slopes, so if the local slope is still falling, the reported number sits above the limit. My Round-1 direction was backwards. Consequence, which I take: crossover contamination in these runs pushes measured 1/ν high, not low. So X,U′'s −0.0147 cannot be a read-low artefact — the true deficit there is at least that large. My Round-1 "crossover explains the deficit" reading is retracted.

[R2-18] JUSTIFY — Hypothesis, "U′,T's +1.7 SE surplus reads high." Mechanism: still-descending local slope inside the fitting window; endpoint slope averages the window and therefore exceeds the limiting slope. Direction in the source series: supported — U′-over-T 0.3547→0.2874, T-over-U′ 0.3621→0.3255, both falling, no plateau. Survives. The corrected U′,T value lies below 0.3324 and may cross the parental mean 0.3249.

[H1-26] JUSTIFY — my Round-1 adoption of Legitimacy's "endpoint slope reads low when parents are far apart." Direction is now known to be wrong (see [R2-32]): descending slope ⇒ reads high. Does not survive. Nothing in this round rests on it.

[H1-27] JUSTIFY — Bound's "composed derivative at or below the parental average." The checker reads the one-block series as rising (0.4370→0.4485). Correct, and Bound has retracted. But I now refuse the one-block series as an adjudicator of sign at all, for a reason no seat stated: compare one-block to MC per parent — X +0.0025, Y +0.0341 (Y's one-block recovered exactly from R1's stated pair mean 0.5514), U′ +0.0451, T +0.0649. The one-block estimator carries a large, pattern-dependent positive bias that grows as cut number falls. An offset formed from biased parents against a biased stack inherits that structure. So the "exact one-block surplus" is not exact for this quantity, and the R1 sign conflict I built my verdict on is void in both directions. Ledger's premise that one-block tracks the same quantity is refuted.

[H1-16] MISMATCH — my own R1 remark about the cut-4 set. Extraction artefact again (U′ read into the cut-4 set); immaterial now, since the cut-class label is dead on other grounds.

VERDICT

Weight falls on Hypothesis, with my Round-1 reading corrected. The decisive fact is not a mechanism: order-independence is forced for the infinite object, and the new run violates it by 0.0227 — 4.1 SE — for U′,T, and by 0.0069 for X,T. The order-gap is a zero-assumption meter of finite-size error, and for U′,T it is three times the surplus being reported (+0.0075), for X,T comparable (+0.0115). Neither new offset is resolved above its own error floor. The lab's estimator note then fixes the correction's sign: falling local slopes ⇒ reported values read high, so both U′,T orders and X-over-T correct downward. The one clean result in the whole record remains X,U′: order-gap 0.0011, offset −0.0147, correction pushing it further negative. Matched-cut X,Y averages; the mismatched pairs do not average, and their asymptotic sign is not yet established.

DISAGREEMENT

The case's framing is what is in conflict, on two counts. (1) It asks why exponents average when cut classes match — but the sorting label is dead: the lab's p* table gives X 0.740, Y 0.782, i.e. the matched pair that averages has a fixed-point separation (0.042) of the same order as X,T's (0.092) which does not; and cut-4 patterns span 1/ν = 0.43–0.55, wider than every effect under discussion. Lateral conceded this outright. (2) It presupposes 1/ν is converged at L ≤ 6561. Hypothesis named that unmet precondition — single dominant scale, corrections negligible by L = 81 — and it is load-bearing for the claim actually in dispute (is the offset nonzero, and of which sign). It therefore outranks the mechanism quarrel between Bound/Lateral (structural subadditivity) and Hypothesis/Framing (crossover), both of which have now been withdrawn by their own seats. Legitimacy's error-bar complaint is the same objection, weaker; Ledger's pinned/free split is uncontested and ranks below both. Reductio's contradiction of my Round-1 direction is upheld and is the round's real correction.

FORCED

Forced: C_A·C_B and hence ln(C_AC_B)/ln81 = mean of parents' cut exponents; fractal dimension likewise; both hold for every pair, matched or not, by max-flow/min-cut plus exact self-similarity. Forced: order-independence of the stacked exponent in the limit — from which the measured 0.0227 gap is derived as finite-size error, not interpreted as one. Forced: the composed map's derivative is f_A′(p*)·f_B′(p*) at a single shared p*, and (Bound's surviving claim, from monotonicity + IVT) p*_AB lies strictly between p*_A and p*_B.

Not forced: that this product equals the parental arithmetic mean — that needs p*_A = p*_B, which the lab's p* table shows holds for no pair here, including X,Y. Bound's sign inequality was a curvature assumption, retracted.

Model artefact: the one-block 2-cycle offsets. Its per-parent bias (+0.0025 X, +0.0341 Y, +0.0451 U′, +0.0649 T — exact arithmetic on lab numbers, not a fit) is pattern-dependent and anti-correlated with cut number, so offsets built from it are not sign-informative.

Inherited from measurement: all parent and stack 1/ν, all local slopes, the 0.0045 spread.

Fitted: every predicted magnitude still on the table, including Ledger's −0.0061/−0.0086 and my own retracted R1 numbers — fit on 1 point; Hypothesis's and Lateral's orderings — fit on 2 points. I offer no new magnitude; the honest statement is an inequality (U′,T corrected value < 0.3324) with no fitted size.

TESTS

1. [lab] [cost: free — all numbers in hand] Tabulate one-block minus MC for each parent (X, Y, T, U′) and regress nothing; just display. Predicts a bias growing as C₁ falls if one-block is a biased approximation (then it cannot sign-check MC offsets, and the R1 sign conflict dissolves); predicts a flat near-zero bias if one-block is faithful (then the MC/one-block sign conflict for X,U′ is real physics).
2. [lab] [cost: free — in the report] Put order-gap beside |offset| for all four pairs: X,Y 0.0045 vs 0.0010; X,U′ 0.0011 vs 0.0147; X,T 0.0069 vs 0.0115; U′,T 0.0227 vs 0.0075. Predicts only X,U′ clears its own finite-size floor if the new pairs are unresolved; predicts all clear if the offsets are real at these sizes.
3. [lab] [cost: free] Window-resolved offsets from the 729→6561 slopes alone (report already gives X,T +0.014, U′,T +0.007 vs overall +0.0115, +0.0075). Predicts offsets shrink toward zero with window if pure crossover; grow if structural. Note the two new pairs already split on this — X,T grows, U′,T shrinks — which is itself evidence the window is too short to read either.
4. [lab] [cost: cheap] Re-run X,Y and X,U′ on the new machine, both orders, same seeds and counts. Predicts order-gaps near the old values (0.0045, 0.0011) if the U′,T 0.0227 is a genuine finite-size transient of that pair; predicts inflated gaps if it is a builder/estimator regression. The T-alone control cannot settle this — a single-pattern carpet has no build order to get wrong.
5. [seat] Bound [cost: cheap] Solve f_A∘f_B for p*_AB for all four pairs, check strict containment in (p*_A, p*_B), and report the sign of f′ curvature at p*_AB. Predicts containment always (forced); predicts a curvature sign that varies pair to pair if no directional bound exists, one sign if Bound's retracted claim can be rescued with an explicit convexity condition.
6. [seat] Ledger [cost: free] Power analysis on the X,Y null: with σ = 0.0039 and an order-gap of 0.0045, what offset would have been invisible? Discharges the ASSUMED premise that X,Y "averages exactly" rather than "averages to within an undetectable amount."
7. [lab] [cost: hours] U′,T both orders at 32,000 samples, L ≤ 6561. Predicts the order-gap falls ~2× toward 0.011 if it is sampling noise; predicts it holds near 0.0227 if it is a genuine finite-size boundary effect of that pair (which then also bounds the systematic on the offset).
8. [lab] [cost: hours] New stack: two cut-4 patterns from the extremes of the 0.43–0.55 spread — matched C₁, large exponent gap, unequal p*. Predicts a nonzero offset if p*-separation is the variable; predicts averaging if cut class is. This is the single measurement that separates the case's label from its rival.
9. [lab] [cost: infeasible here — L = 59049 is ~3.5×10⁹ sites per sample at 8,000 samples, two orders, four stacks; far beyond any run so far] The clean convergence test. Named because it is decisive, ranked last because it cannot be run.

RESIDUE

Not closed. The sharper case: for a two-generator hierarchical stack, derive the finite-L order asymmetry analytically from the boundary mismatch at level 1 (A-outer vs B-outer are genuinely different graphs at L = 81, identical only as L → ∞), and show that the reported offset from the parental mean is a function of that asymmetry divided by the window length — i.e. that the order-gap predicts the offset's contamination, pair by pair. If it does, the exponent question should be re-asked only of order-symmetrised, correction-to-scaling-fitted values. If Test 8 returns a nonzero offset for matched C₁, the case's title is simply false and the right question becomes: what function of (p*_A, p*_B, f′ curvature) sets the offset, with cut class demoted to a coincidence of the original four patterns.

BACKPROP

open. Write back two edges and one probe. Edge 1: in any construction where a symmetry or identity forces two nominally distinct builds to agree asymptotically, the measured gap between them is a zero-assumption meter of finite-size error — report it beside every exponent, and refuse any effect smaller than it. Edge 2: single-block (Migdal–Kadanoff-type) RG estimates on these carpets carry a pattern-dependent positive bias that grows as the cut number falls (+0.0025 to +0.0649 across X, Y, U′, T); they may not be used to sign-check Monte Carlo offsets. Correction: my Round-1 claim that a descending local slope makes the endpoint slope read low is wrong — it reads high; Reductio's [R2-32] is upheld and the Round-1 verdict built on it is withdrawn. Probe: Test 4, the new-machine re-run of X,Y and X,U′. Next case: the matched-C₁, far-apart-exponent stack of Test 8.

_Contract check: House adopted a flagged claim: H2-1 (from Lateral), H2-4 (from Hypothesis); House TESTS without a cost tag or reason: 3 (cost tag has no reason), 4 (cost tag has no reason), 5 (cost tag has no reason), 6 (cost tag has no reason), 7 (cost tag has no reason), 8 (cost tag has no reason); House fitted numbers without a "fit on n points" label: 5 line(s)._

Continuation 1 — Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why? · mode: continuation · 2026-10-01

_Continued panel. The same seats at the same versions (Ledger v1, Framing v1, Hypothesis v1, Legitimacy v1, Lateral v1, Bound v1, Reductio v1); each saw only the case, its own earlier TAKE and PREDICTION blocks, the House's latest diagnosis and the new lab report, not other seats' rounds. The House saw everything. Its forecasts are counted separately from first-round ones._

Record. not loaded (continuation) — no record packet and no seat record was sent to any seat.

Lab report (continuation 1)

LAB REPORT — run 076 (all two-map pairs), with run 077 on the same pairs; for continuation 1

All values below are of the EXACT ONE-BLOCK object (the 2-cycle of two patterns' crossing
polynomials), not of sampled lattices. No lattice was measured.

Scope: 63 census patterns = 49 distinct crossing polynomials; all 1,176 pairs. For each pair:
δ = p*_B − p*_A (fixed-point separation), D = y_AB − (y₉A + y₉B)/2 (deviation of the two-map
exponent from the parents' mean). Every pair has exactly one 2-cycle between the fixed points.
Controls: the four recorded pairs reproduce (X,Y 0.5503; X,U′ 0.4485; X,T 0.4984; U′,T 0.3851).

First-order term. With λ = R′(p*), a = R″(p*), s = a/(λ(λ−1)), the expansion of the 2-cycle gives
D₁ = δ(λ_A−1)(λ_B−1)(s_A−s_B)/[ln 81 (λ_Aλ_B−1)].
s over the 49 functions: −16.7 to −1.4; interquartile range 2.93; median |s| 3.21.
D/D₁, median by |δ|: ≤0.01: 0.79 · 0.01–0.02: 0.15 · 0.02–0.05: 0.06 · 0.05–0.1: 0.05 · 0.1–0.2: 0.07
For 0 < |δ| ≤ 0.01 and |s_A − s_B| ≥ 0.32 (46 pairs): median |D/D₁ − 1| = 0.23.
Two identical maps offset by h: D₁ = 0 and D = −1.85 h².

Sign of D: negative on 416 of 1,176 pairs (319 of the 976 with |δ| > 0.02).

Cut class at matched separation (|δ| quintiles; median |D|, equal C₁ vs unequal C₁):
| |δ| bin | n equal / unequal | median |D| equal | median |D| unequal | ratio |
| 0–0.023 | 154 / 82 | 0.000076 | 0.00072 | 0.11 |
| 0.023–0.050 | 144 / 91 | 0.00040 | 0.0026 | 0.15 |
| 0.050–0.086 | 104 / 131 | 0.00089 | 0.0034 | 0.26 |
| 0.086–0.123 | 60 / 175 | 0.0021 | 0.0063 | 0.33 |
| > 0.123 | 10 / 225 | 0.0112 | 0.0116 | 0.97 |

Rank correlations (Spearman) with |D|, all 1,176 pairs: |Δy₉| 0.81, |δ| 0.77, |ΔC₁| 0.74.
Partial Spearman of |D| with |ΔC₁| given |δ|: +0.60.
Where both parents have a measured 1/ν (378 pairs): Spearman of |D| with |Δ(1/ν)| 0.79.

Cut-4 pairs (both parents C₁ = 4): of 236 with |δ| > 0.03, none has D > +0.003.
Widest cut-4 separation: δ = 0.143, D = −0.0020. No cut-4 pair has |δ| ≥ 0.15.
Cut-4 pair with the largest |Δy₉| (0.095): δ = 0.042, D = +0.0010.
Largest |D| overall: cut 5 or cut 4 against cut 2, D = +0.017 to +0.021.

Run 077 (same pairs). At the wall, 1 − R(1 − q) = N₁·q^{C₁} exactly for every pattern, with C₁ the
minimum cut and N₁ the number of minimum cuts. Partial Spearman of |D|, controlling for |δ|:
with |Δ ln N₁|, equal-C₁ pairs (410, class also controlled): −0.26
with |Δ (ln N₁)/C₁|, unequal-C₁ pairs (704, |ΔC₁| also controlled): +0.015
with |ΔC₁|, unequal-C₁ pairs (704, amplitude also controlled): +0.72
with |Δy₉| in place of the amplitude: equal-C₁ −0.10; unequal-C₁ +0.54

Continuation 1 — round

Ledger v1
TAKE
Round 1 forecast (X,T and U′,T deficits of −0.0061 and −0.0086, both below parental mean) is falsified — House's record shows both landed above the mean (+2.7, +1.7 SE), and this report does not touch that: it is explicitly the exact one-block object only, "No lattice was measured." Round 2's forecast (order-gap shrinking at L=59049) is simply untested here; this report carries no new MC lattice data at all, so neither confirmed nor refuted.

What this report pins, exactly, is different from what the case is arguing about. D ≡ y_AB − (y_A+y_B)/2 is an exact, deterministic functional of the two crossing polynomials — no sampling, no error bars, reproducible to the digit (the four control values match my earlier record exactly: 0.5503, 0.4485, 0.4984, 0.3851). The identical-map check (D₁=0, D=−1.85h² for a perturbed-self pair) is itself a pinned consistency check on the expansion, not evidence about physics: it confirms D₁ is the correct leading term, nothing more.

The one new pinned fact worth stating plainly: controlling for |δ|, |ΔC₁| still carries independent explanatory weight on |D| (partial Spearman +0.60, and +0.72 in run 077 against amplitude). That is a bookkeeping fact about the one-block object's exact algebra, not a statement about the MC exponent — House already established the one-block estimator carries a pattern-dependent bias relative to MC and is not sign-informative for it. My axis cannot bridge that gap; it can only note that within the one-block object itself, cut-class separation is not a coincidence of these four patterns — it is structurally present across 1,176 exact pairs.

PREDICTION
A matched-C₁ pair drawn from the census at large |Δy₉| (Test-8-type, cut 4 vs cut 4, extremes of 0.43–0.55) will show exact one-block |D| ≤ 0.003, consistent with "no cut-4 pair with |δ|>0.03 has D>+0.003."
VALUE: 0.0005 to 0.0030, 80% interval, for |D| of such a pair.
SIGN: D negative or near-zero (|D|≤0.003), p=0.75 — consistent with the existing cut-4 census, not a prediction about the MC exponent.
NULL: if cut class carries no independent information beyond δ, |D| for this pair should instead track the unequal-C₁ curve at its δ (median |D| ≈ 0.003–0.006 depending on bin), i.e. several times larger than the equal-C₁ prediction above.

SIGN: see above
VALUE: see above
NULL: see above

ASSUMES
That "Test-8-type" pair (two cut-4 patterns at the 0.43/0.55 extremes) falls in the |δ| range already tabulated (0.05–0.15), since the case gives no p* value for those specific patterns; none of the case's actual stated numbers are used to fabricate the δ.

CHANGES MY MIND
A matched-C₁ pair at large |Δy₉| with |D| exceeding the unequal-C₁ median at the same |δ| bin — that would mean cut-class match no longer predicts anything even at the exact one-block level, collapsing the one structural regularity this report establishes.

Framing v1
TAKE
Neither of my prior predictions is adjudicated by this report, because run 076/077 measures a different object than the one my predictions were about. My Round-1/Round-2 forecasts concerned the sampled-lattice local slope at L=59049 and the order-gap of the Monte Carlo estimator; this report is exact algebra on the one-block map across 1,176 pairs, with "no lattice measured." So: not confirmed, not falsified — out of reach of the instrument used.

But the report sharpens my axis's complaint in a new way. It confirms that δ = p*_B − p*_A, not C₁, is the organizing variable for D (Spearman 0.77 vs 0.74, comparable, but the quintile table shows the cut-class effect collapses at large δ — ratio 0.97 at |δ|>0.123, meaning "equal vs unequal C₁" stops mattering once separation dominates). That is direct evidence the case's chosen label (cut class) is a correlate, not a cause: it tracks the curvature term s = a/(λ(λ−1)) because patterns of equal C₁ apparently cluster in curvature (median |s| 3.21, IQR 2.93), not because C₁ enters the exponent-composition law. The formula D₁ = δ(λ_A−1)(λ_B−1)(s_A−s_B)/[ln81(λ_Aλ_B−1)] names the real object: a curvature-mismatch-weighted fixed-point separation. The breakdown of D/D₁ at larger |δ| (0.79 → 0.07) shows that even this isn't the full object — nonlinear terms take over, so there is no single invariant "does it average" question; the answer is a continuously varying function of two parameters, asked as a binary (matched/unmatched class).

Also decisive: cut-4 pairs with δ=0.143 still show D=−0.0020, nonzero — matched class does not guarantee averaging even in this exact, non-lattice object. The case's title premise is false in its own idealized limit, independent of any finite-size argument.

PREDICTION
For the next cut-4 pair measured with |δ| in the 0.14–0.18 range (beyond the current maximum 0.143), the exact one-block D will be nonzero and negative, with |D| in 0.002–0.006 — not the ≈0 the case's "matched class averages" framing predicts.

SIGN: negative (D<0), p=0.75
VALUE: −0.0010 to −0.0060, 80% interval
NULL: if "cut class controls averaging" were true, D should be ≈0 (within the ±0.0004-scale noise floor seen at small-δ matched pairs) regardless of δ, for any matched-C₁ pair.
ASSUMES: that the trend in the given cut-4 table (δ rising from 0.042→0.143, D staying ≤0 and growing in magnitude, max observed −0.0020) extrapolates monotonically a bit further rather than reversing sign; no claim about any lattice/MC instantiation of this pair.
CHANGES MY MIND: a cut-4 pair at δ>0.14 returning D ≥ +0.001 (positive, not just near zero) would break the monotonic trend and remove my basis for calling cut-class matching a curvature-proxy rather than a direct cause.

Hypothesis v1
TOOL
First-order perturbative expansion of the 2-cycle, D₁ = δ(λ_A−1)(λ_B−1)(s_A−s_B)/[ln81(λ_Aλ_B−1)]
REQUIRES
(1) δ small enough that the Taylor expansion around the fixed point dominates the exact map; (2) λ, a (hence s) well-defined smooth derivatives at p*, i.e. the map is analytic there; (3) higher-order terms in δ genuinely subleading over the δ-range being used for inference.
SATISFIED
(1) no — report's own D/D₁ table shows median ratio 0.79 only for |δ|≤0.01, collapsing to 0.06–0.15 for |δ| in 0.02–0.1, i.e. the linear term is a small, often wrong-signed piece of D outside a narrow band. (2) yes — s is tabulated for all 49 functions, well-defined. (3) no for most of the 1,176 pairs, including — on the evidence that even at |δ|≤0.01 with large curvature mismatch (|s_A−s_B|≥0.32) the median deviation from D₁ is 23% — this tool cannot be trusted as more than a sign hint anywhere near the δ values implied by this case's patterns (Δy9 up to 0.28).

TOOL
Exact one-block (2-cycle) value as a proxy for the lattice-asymptotic MC exponent offset
REQUIRES
Per-pattern one-block bias must be either negligible or parent-independent so it cancels in the difference D = y_AB − mean(parents); the one-block flow must track the same fixed-point structure the finite-size MC estimator converges to.
SATISFIED
No — this was already falsified last round by the X,U′ sign conflict (one-block +0.0115 vs MC −0.0147) and the new report does nothing to repair it; it reconfirms the control values exactly (0.4485 for X,U′) and adds that "cut4 vs cut2" pairs generically show one-block D = +0.017 to +0.021, i.e. systematic positive one-block bias on exactly the pair whose MC result is the one clean deficit in the whole dataset. Precondition fails by direct demonstration, not inference.

TOOL
Partial Spearman correlation (control for δ, test ΔC₁; control for amplitude, test ΔC₁)
REQUIRES
Enough independent variation across pairs that δ and ΔC₁ are not collinear; only a monotone (not necessarily linear) relationship is needed since Spearman is rank-based.
SATISFIED
Yes, as far as checkable — N = 704 (unequal-C1) and 410 (equal-C1) pairs, broad δ-range per the quintile table, partial correlation of +0.72 (ΔC₁ given amplitude) and +0.60 (ΔC1 given δ) are large enough not to be artifacts of a thin or collinear sample. This is the one tool in this report whose preconditions actually hold.

TAKE
Confirmed: none of my earlier sign forecasts for X,T/U′,T is addressed by this report — it measures only the exact one-block object, no lattice. What it does confirm is my Round-2 magnitude intuition (|D| tracks |Δexponent|, Spearman 0.81) and directly falsifies House's "cut-class label is dead" verdict: partial correlations of +0.72/+0.60 show ΔC₁ has real explanatory power on |D| independent of δ, for this one-block quantity specifically. It also sharpens my precondition case against using one-block values to sign-check MC: the D₁ perturbative tool is shown, by the lab's own ratio table, to be invalid outside |δ|≲0.01, and the one-block proxy itself is shown, again by the lab's own numbers, to disagree in sign with MC on the one pair (X,U′) where both are known. So the new report strengthens cut-class as a real variable for the exact map, while leaving the finite-size-convergence objection to the lattice measurement completely untouched — these are two different objects and the report does not bridge them. My axis still finds no licensed tool connecting exact-map D to finite-L MC 1/ν.

PREDICTION
For the proposed matched-C1, exponent-extreme pair (Test 8: two cut-4 patterns near 1/ν = 0.43 and 0.55), the exact one-block D should stay small — consistent with the existing cut-4 table (|Δy9|=0.095, δ=0.042 → D=+0.0010; widest cut-4 δ=0.143 → D=−0.0020) — because cut-class match constrains D independent of δ.
SIGN: small magnitude, more likely negative than positive, p=0.6
VALUE: −0.004 to +0.001, 80% interval
NULL: no effect predicts D = 0 exactly (perfect averaging)
ASSUMES: the chosen Test-8 pair's δ falls within the existing cut-4 range (up to ~0.15), not beyond it.
CHANGES MY MIND: if that matched-C1 pair returns |D| > 0.01, cut-class would not be locking D independent of δ, and the partial-correlation result would need a different reading (e.g., confound via an unmeasured third variable like N₁).

Legitimacy v1
TAKE
Confirm/falsify first: my Round-2 prediction concerned a fourth lattice size (L=59049) for U′,T convergence — this report contains no lattice data at all, so it is neither confirmed nor falsified; it's orthogonal to that claim. What it does bear on is legitimacy of a different kind. Three concerns. (1) Sample-size inflation: "63 census patterns = 49 distinct crossing polynomials" means some patterns are duplicates under the polynomial — the 1,176 "pairs" are not 1,176 independent draws, and correlation/median statistics computed over them carry unstated clustering. (2) Severe collinearity: in the top |δ| quintile, equal-C₁ pairs number only 10 against 225 unequal — the quintile table's "cut class predicts small |D|" pattern is largely a selection artifact of same-class patterns clustering at small δ, only partially controlled by the partial-Spearman (0.60, not dispositive with n=10 in the extreme bin). (3) Most damning: the only existing calibration against real MC data — four pairs — already shows a sign mismatch (one-block D(X,U′)=+0.0115 vs MC-measured −0.0147). This whole 1,176-pair census is built entirely on the one-block instrument already flagged as pattern-dependently biased relative to MC. Its internal consistency is real; its claim to predict the lattice-measured quantity the case actually asks about is uncalibrated and, on the one check available, wrong in sign.

PREDICTION
SIGN: positive (D(Y,T) one-block computed later shows surplus), p=0.7
VALUE: 0.0005 to 0.006, 80% interval
NULL: no effect predicts D(Y,T)=0 (stack exponent exactly averages parents' one-block exponents)
SIGN: <the sign or direction you commit to, with a probability>
VALUE: <low> to <high>, <coverage>% interval
NULL: <one line>

ASSUMES
That Y,T has not yet been run through the one-block instrument (not listed among the four controls); that the X,T-type bias pattern (cut4/cut3, positive one-block D) generalizes to Y,T given Y and T's bias values implied by X,U′,T's recorded one-block-vs-MC offsets.

CHANGES MY MIND
A direct one-block computation of D(Y,T) landing negative or nonzero-opposite to the X,T analog, or a demonstration that the 1,176-pair census was deduplicated for shared crossing polynomials before computing correlations (removing concern 1), or a fifth MC-calibration pair where one-block sign matches MC sign — any of these would restore legitimacy to using this census for lattice prediction.

Lateral v1
TAKE
Both of my earlier forecasts are falsified outright. Round 1 predicted X,T and U′,T as deficits (0.452–0.458, 0.316–0.322); the controls in this report give X,T = 0.4984 (D = +0.0359) and U′,T = 0.3851 (D = +0.0602) — large surpluses, wrong sign, wrong magnitude by an order of magnitude. Round 2's Jensen/curvature claim ("D grows quadratically with |δ|, C1 is an accidental proxy, only s_A−s_B matters") is also falsified: the cut-4-only slice shows the widest separation (δ=0.143) gives D=−0.0020, tiny and negative, not the growing surplus my model demanded, while cross-class pairs at comparable or smaller δ reach D up to +0.021. The matched-separation table is decisive — at fixed |δ|, equal-C1 pairs sit 3–9× smaller than unequal-C1 pairs across every bin except the largest. And the partial Spearman of |D| with |ΔC1| controlling for δ is +0.60 — strong, independent of the continuous fixed-point data. So C1 is not merely a label for δ; it carries information my first-order expansion doesn't capture. Run 077's exact wall law, 1−R(1−q)=N1·q^{C1}, names the real shared object: C1 is the order of vanishing of the crossing generating function at the boundary, and two maps with mismatched vanishing order at a common fixed point don't compose smoothly — this is a resonance/normal-form obstruction (Poincaré–Dulac/Böttcher-type: conjugacy fails when local degrees differ), not a Jensen correction on a shared analytic map.

PREDICTION
For a new pair with |δ| < 0.01 (fixed points nearly coincident) but |ΔC1| = 2: |D| ≈ 0.0005–0.0015, sign negative, well above what the matched-δ, matched-C1 baseline (~0.00008) would give.
SIGN: D negative, p=0.65
VALUE: -0.0015 to -0.0003, 80% interval
NULL: if C1 only matters via δ (no independent resonance effect), |D| should be ≈0.00008 (the equal-C1 baseline at this δ bin), i.e. indistinguishable from matched-C1 pairs.
ASSUMES: the partial-correlation structure (ΔC1 effect at fixed δ) generalizes to an out-of-sample pair chosen specifically to isolate it; that "equal/unequal C1" bias direction (unequal always larger |D|) holds down to δ≈0 and not just in the bins reported.
CHANGES MY MIND: a newly measured near-δ-zero, ΔC1=2 pair landing at |D| ≤ 0.0002 (indistinguishable from the equal-C1 row) — that would mean the partial correlation in the big-n table is confounded by something correlated with both δ-binning and ΔC1 (e.g. s or N1) rather than a real order-mismatch effect, and I'd retract the resonance claim entirely.

Bound v1
TAKE
Two of my own forecasts are now checkable against exact one-block data and both fail. Round 1 predicted deficits for X,T and U′,T below their parental means — wrong: computing D from the per-parent one-block values (y9_X=0.5533, y9_T=0.4392, y9_U′=0.3206, from House's bias table) gives X,T D=+0.0022 and U′,T D=+0.0052, both surpluses, matching House's figures exactly. Round 2 predicted that a wide-separation matched-cut-4 pair would show positive surplus >+0.003, scaling with δ². This is now directly falsified by the census: of 236 cut-4 pairs with |δ|>0.03, none has D>+0.003, and the widest cut-4 separation (δ=0.143) gives D=−0.0020, negative.

What survives: the exact first-order identity D₁ = δ(λ_A−1)(λ_B−1)(s_A−s_B)/[ln81(λ_Aλ_B−1)] is forced by the chain rule applied to the unique 2-cycle (existence forced by stability, λ>1 at an attractive fixed point, so (λ_A−1)(λ_B−1)>0 and λ_Aλ_B−1>0). This forces sign(D₁)=sign(δ)·sign(s_A−s_B) — not a universal sign, since s ranges −16.7 to −1.4 and s_A−s_B flips sign across the census. So "below mean" and "above mean" are both genuinely free; what is forced is only this conditional rule plus the self-offset limit (identical map, p* shift h): D=−1.85h²<0, a strict, sign-definite quadratic floor. My axis overreached twice by asserting an unconditional sign; the corrected forced content is weaker but real.

PREDICTION
SIGN: for the next cut-5-vs-cut-3 pair with |δ| in the 0.05–0.09 range, D positive (surplus), p=0.65
VALUE: 0.0015 to 0.006, 80% interval
NULL: no effect predicts D=0 (exact averaging) regardless of cut class or δ

ASSUMES
That the next measured pair falls in the moderate-δ, unequal-C₁ regime where the partial-Spearman(|D|,|ΔC₁| | δ)=0.72 generalizes beyond the 704-pair sample it was computed on; no stronger structural guarantee than that correlation is assumed.

CHANGES MY MIND
A cut-5/cut-3 pair at that δ showing D<0, or |D| exceeding 0.006 — either would show the ΔC₁ partial-correlation isn't predictive at the pair level, only in aggregate, which would mean this axis has no forced content left beyond the already-conceded conditional sign rule and the self-offset floor.

Reductio v1
TAKE
My round-2 forecast (both new pairs read above, not below, the parental mean) was already confirmed by the round-2 lab report and is not re-tested by this one-block census — no new lattice numbers appear here, so neither confirmed nor falsified further.

Testing the current VERDICT: its DISAGREEMENT section calls cut class a "dead" label, coincidental to these four patterns, with fixed-point separation δ the real driver. Twins/Invariance push: the census gives exactly the twin pairs needed — matched δ, varied ΔC₁. The equal/unequal table contradicts VERDICT directly: at matched δ, equal-C₁ pairs have median |D| 3–9× smaller than unequal-C₁ pairs in four of five bins (ratios 0.11–0.33), and partial Spearman of |D| with |ΔC₁| controlling δ is +0.60 — strong, not coincidental. Neighbours push: cut-4 matched pairs (236, |δ|>0.03) never exceed D=+0.003 even out to δ=0.143, the exact "averages regardless of separation" signature the case asked about — contradicted by VERDICT's claim that this is spurious. Joint consistency: run-077's partial correlations (|ΔC₁| +0.72 vs |ΔlnN₁|/C₁ ≈0 among unequal-C₁ pairs) reinforce ΔC₁, not δ, as the controlling variable. VERDICT's dismissal of cut class is contradicted by its own new evidence.

PREDICTION
VERDICT's consequence most exposed: "cut class is a dead/coincidental label, not forced by anything beyond these four patterns." Test: regress |D| on |ΔC₁| and |δ| jointly across all 1,176 pairs (data already in hand); if |ΔC₁|'s partial effect survives at the stated magnitude, the dismissal fails outright.
SIGN: cut-class effect real and independent of δ, p=0.8
VALUE: partial correlation of |D| with |ΔC₁| given |δ|, 0.5 to 0.7, 90% interval
NULL: if cut class were truly coincidental, equal/unequal median-|D| ratios would cluster near 1 across all δ bins, not fall to 0.11–0.33
ASSUMES: that the one-block census's statistical regularities are informative about the sign/structure of the true (lattice) effect even though House has ruled the one-block magnitude itself biased — I take this as a structural, not quantitative, inference.
CHANGES MY MIND: if a direct regression on the 1,176-pair census shows |ΔC₁|'s partial effect controlling |δ| drops near zero once Δy₉ or Δ(1/ν) is also controlled — i.e., if ΔC₁ is proxying for exponent gap rather than driving it independently — then VERDICT's dismissal of cut class survives.

House — continuation 1

CHECKS

none flagged — the checker table extracted no checkable claims this round. Two arithmetic facts I recomputed myself, since nothing else did: (i) C(49,2) = 1,176 exactly, so the census's 1,176 pairs are pairs of distinct crossing polynomials — Legitimacy's "duplicate patterns inflate n" concern is already answered by the report's own arithmetic and is withdrawn; (ii) Lateral's continuation-1 claim that the controls give D(X,T) = +0.0359 and D(U′,T) = +0.0602 is a mis-subtraction: 0.4984 − (0.5533+0.4392)/2 = +0.0022 and 0.3851 − (0.3206+0.4392)/2 = +0.0052. Lateral's sign conclusion survives; its "wrong by an order of magnitude" does not, and I do not rest anything on that figure.

FORECAST SCORE

S-Ledger | X,T and U′,T both below parental mean (−0.0061, −0.0086), fit on 1 point | run 070 gave +0.0115, +0.0075; census controls confirm one-block surpluses +0.0022, +0.0052 | MISS
S-Framing | L=59049 X–U′ local slope falls below 0.38, no plateau | no lattice measured in run 076/077 | UNSCORABLE
S-Hypothesis | U′,T deficit <0.006, X,T deficit >0.008; deficit tracks |Δ(1/ν)| | both pairs came in as surpluses; census: |D| ranks with |Δy₉| 0.81 but sign is not a deficit | MISS
S-Legitimacy | fourth size raises X–U′ 1/ν by >0.008 | no lattice measured | UNSCORABLE
S-Lateral | X+T 0.452–0.458, U′+T 0.316–0.322, both deficits | run 070: 0.4740, 0.3324, both surpluses | MISS
S-Bound | both stacks strictly below parental mean, |deficit(U′,T)| > |deficit(X,T)| | both above; |offset(X,T)| > |offset(U′,T)| | MISS
S2-Ledger | order-gap shrinks at L=59049 | no lattice measured | UNSCORABLE
S2-Framing | U′,T orders converge at L=59049, mean drops ≥0.010 | no lattice measured | UNSCORABLE
S2-Hypothesis | correction-to-scaling pulls U′,T below 0.320, X,T stays ≥0.469 | no lattice, no fit reported | UNSCORABLE
S2-Legitimacy | U′,T orders agree at L=59049, mean ≤0.315 | no lattice measured | UNSCORABLE
S2-Lateral | cut-4/cut-4 wide-spread pair: one-block surplus +0.008 to +0.016 | of 236 cut-4 pairs with |δ|>0.03, none has D > +0.003; widest (δ=0.143) gives D = −0.0020 | MISS
S2-Bound | same pair, D > +0.003, scaling as (Δp*)² | same census row: no cut-4 pair reaches +0.003 | MISS
S2-Reductio | VERDICT's "both read below mean" already falsified | census controls reproduce X,T 0.4984, U′,T 0.3851, surpluses over one-block parental means | HIT

VERDICT

The census moves the weight to Reductio and Lateral, for one object only. [object: exact one-block] at matched fixed-point separation, equal-C₁ pairs carry 3–9× smaller |D| than unequal-C₁ pairs, and |ΔC₁| survives partialling on δ (+0.60) and on amplitude (+0.72) while |Δ ln N₁| dies (+0.015). The exact wall law 1−R(1−q)=N₁q^{C₁} makes C₁ the order of vanishing of the map at the boundary — a local degree, not a label. I withdraw "cut class is dead" for this object. [object: sampled lattice at the sizes measured] nothing changed: no lattice was measured, the U′,T order-gap (0.0227) still exceeds its own offset (+0.0075), and one-block conflicts in sign with Monte Carlo on X,U′, so the census cannot adjudicate the measured offsets. [object: limit] exact averaging is false even for matched parents: identical maps offset by h give D = −1.85h² < 0.

DISAGREEMENT

The case's framing is still the thing in conflict, but the conflict has moved. The case asks why matched cut classes make exponents average; the census says matched cut classes make |D| small, not zero, and the equal/unequal contrast collapses in the top separation bin (ratio 0.97). So "averages" is an approximation whose quality degrades with δ, asked as a binary.

Beneath that, the live fight is: does |ΔC₁| act independently (Reductio, Lateral: mismatched order-of-vanishing at the wall obstructs smooth composition), or is it a proxy for curvature clustering — equal-C₁ patterns having similar s, so that D₁ ∝ (s_A − s_B) cancels (Framing, Hypothesis, Ledger)? Run 077 pushes toward the first by killing the amplitude N₁ while keeping C₁; it does not yet control |Δy₉| and |ΔC₁| jointly.

Unmet preconditions, ranked. (1) Hypothesis's objection that the one-block object is not a licensed proxy for the lattice exponent is load-bearing for the claim the case disputes (the measured offsets), and it is confirmed by the X,U′ sign conflict — it outranks everything else about the lattice. (2) Hypothesis's second objection — D₁ perturbation theory is invalid outside |δ| ≲ 0.01 (D/D₁ median 0.06–0.15) — is real but sits in a part nobody contests: only Bound's retracted sign rule leans on D₁, and the ΔC₁ result is nonparametric. Ranked below the real disagreement.

FORCED

1. [object: limit] ln(C_A C_B)/ln 81 = mean of parents' ln C/ln 9; fractal dimension likewise — DERIVATION: max-flow/min-cut multiplicativity under series nesting plus exact self-similarity of the substitution.
2. [object: limit] The stacked exponent is order-independent — DERIVATION: A-over-B and B-over-A generate the same 2-cycle up to conjugation; the census confirms it (one value per pair, both orders). Hence the measured 0.0227 gap at U′,T is derived finite-size error, not interpreted as such.
3. [object: exact one-block] A unique 2-cycle exists strictly between p*_A and p*_B — DERIVATION: monotonicity of R plus intermediate value theorem; report confirms "exactly one 2-cycle between the fixed points" for every pair.
4. [object: exact one-block] sign(D₁) = sign(δ)·sign(s_A − s_B), with no universal direction — DERIVATION: in D₁ = δ(λ_A−1)(λ_B−1)(s_A−s_B)/[ln81(λ_Aλ_B−1)], λ>1 at the repelling crossing fixed point makes (λ_A−1)(λ_B−1)>0 and λ_Aλ_B−1>0; s spans −16.7 to −1.4, so s_A−s_B flips sign. Consistent with D<0 on 416/1,176 pairs.
5. [object: exact one-block] C₁ is the order of vanishing of the crossing map at the wall; N₁ is only its amplitude — DERIVATION: 1−R(1−q) = N₁q^{C₁} exactly, so differences in C₁ are degree mismatches, differences in N₁ are prefactors. Run 077's partials (+0.72 for ΔC₁, +0.015 for Δ ln N₁) match that split.
6. [object: exact one-block] Matched parents do not average exactly: identical maps offset by h give D₁ = 0 and D = −1.85h², strictly negative — DERIVATION: second-order term of the same expansion, computed exactly by the lab.
7. Inherited from measurement: all parent and stack MC 1/ν, local slopes, the 0.0045 spread, the 0.0227 order-gap.
8. Model artefact, [object: exact one-block vs sampled lattice]: the one-block per-pattern bias relative to MC (+0.0025 X, +0.0341 Y, +0.0451 U′, +0.0649 T) grows as C₁ falls; offsets built from it are not sign-informative for the lattice.
9. Fitted: D₁ as a predictor of D outside |δ| ≤ 0.01 — the lab's own ratio table (0.06–0.15) says it is not. Lateral's round-2 quadratic law was fit on 3 points; Ledger's −0.0061/−0.0086 and my own retracted round-1 magnitudes were fit on 1 point; any regression of one-block bias on C₁ is fit on 4 points.

ASSERTED

1. [object: exact one-block] The independent ΔC₁ effect is a degree-mismatch obstruction — two maps vanishing to different orders at the wall cannot be simultaneously conjugated to a common normal form, so the composite derivative acquires a term absent from any δ-and-curvature expansion. I believe it; I cannot derive it from the given expansion, which contains no C₁.
2. [object: sampled lattice at the sizes measured] The X,U′ deficit (−0.0147, order-gap 0.0011) is the one measured offset that is real, and the corrected value lies further below the parental mean, since both its local slopes fall and a falling slope reads high.
3. [object: limit] The lattice exponent and the one-block exponent need not share the sign of D at all; I expect them to disagree for at least one more pair.

ROAD NOT TAKEN

Ledger, Framing and Lateral all signed "down" (negative D for the next matched-cut-4 or extreme pair). The strongest alternative none took: that the sign is simply free, set by sign(s_A − s_B) as FORCED 4 requires, and that the whole equal-C₁ regularity is cancellation — equal-C₁ patterns cluster in curvature, so s_A − s_B ≈ 0 and D₁ vanishes by arithmetic, with C₁ entering nowhere. On that reading a wide-δ cut-4 pair whose two s values straddle the census median comes in positive; the census already contains one (|Δy₉| = 0.095, δ = 0.042, D = +0.0010). What would favour it: equal-C₁ pairs showing systematically smaller |s_A − s_B| than unequal-C₁ pairs at matched δ, and the ΔC₁ partial collapsing once |s_A − s_B| is controlled. Test 2 settles it and no seat proposed it.

FORECASTS

F1.
RESTS ON: FORCED 5
SIGN: positive — |ΔC₁| retains partial rank correlation with |D| after controlling both |δ| and |Δy₉| (p=0.68)
VALUE: 0.30 to 0.60, 80% interval
NULL: if ΔC₁ only proxies separation and exponent gap, the joint partial falls to 0.0–0.15.

F2.
RESTS ON: FORCED 6
SIGN: |D| small for the Test-8 matched-cut-4, extreme-Δy₉ pair; sign more likely negative than positive (p=0.6 negative)
VALUE: −0.0040 to +0.0015, 80% interval
NULL: if cut class carries nothing, |D| should match the unequal-C₁ median at that δ bin, 0.003–0.006.

F3.
RESTS ON: ASSERTED 2
SIGN: downward — a correction-to-scaling fit or a fourth size moves both U′,T orders and X-over-T down
VALUE: U′,T order-averaged 1/ν lands 0.310 to 0.330; its offset from 0.3249 falls below +0.004 in magnitude, 80% interval
NULL: no finite-size contamination predicts the order-averaged values stay at 0.3324 and 0.4740 with the order-gap unchanged at 0.0227.

F4.
RESTS ON: ASSERTED 1
SIGN: |D| for a |δ| < 0.01, |ΔC₁| = 2 pair exceeds the equal-C₁ baseline at that bin by at least 3× (p=0.7); its sign I do not commit to
VALUE: |D| 0.0003 to 0.0020, 80% interval
NULL: no resonance effect predicts |D| ≈ 0.00008, indistinguishable from matched-C₁ pairs.

TESTS

1. [lab] [cost: free — the 1,176-pair census is on disk] Joint partial Spearman of |D| with |ΔC₁| controlling |δ| and |Δy₉| simultaneously, then again adding |s_A − s_B|. Predicts ≥0.30 if cut class is a degree-mismatch effect (Reductio/Lateral); predicts ≤0.15 if it proxies curvature or exponent gap (Framing/Hypothesis). This is the decisive test and it costs nothing.
2. [lab] [cost: free] Tabulate |s_A − s_B| by equal vs unequal C₁ within each |δ| quintile. Predicts equal-C₁ pairs have systematically smaller curvature mismatch if the ROAD NOT TAKEN cancellation story holds; predicts no systematic difference if ΔC₁ acts independently.
3. [lab] [cost: free] List the 10 equal-C₁ pairs in the top |δ| bin individually with δ, D, ΔC₁, s-gap. Predicts the 0.97 ratio is an n=10 artefact if their |D| values straddle the unequal median; predicts a genuine collapse of the cut-class effect at large δ if they sit on it uniformly — which would bound the ΔC₁ claim to small separation.
4. [lab] [cost: cheap] Lateral's isolation pair: compute D for every census pair with |δ| < 0.01 and |ΔC₁| = 2 (the report says 46 pairs sit in the small-δ, large-s-gap cell, so such pairs exist). Predicts |D| an order above the 0.000076 equal-C₁ baseline if resonance; predicts baseline if not.
5. [lab] [cost: cheap] Compute one-block y₉ for every pattern with a measured MC 1/ν and regress the bias on C₁ — fit on 4 points, and label it so. Predicts a monotone bias decreasing with C₁, which forbids using one-block D to sign-check MC offsets at all; predicts a flat bias if the X,U′ sign conflict is instead real physics.
6. [lab] [cost: hours] Monte Carlo the Test-8 pair (two cut-4 patterns at the 0.43/0.55 extremes), both build orders, L ≤ 6561. Predicts a nonzero lattice offset if fixed-point or curvature separation drives the lattice effect; predicts averaging within the order-gap if cut class does. Report the order-gap beside the offset and refuse any offset smaller than it.
7. [lab] [cost: hours] U′,T both orders at 32,000 samples, L ≤ 6561. Predicts the order-gap falls ~2× toward 0.011 if it is sampling noise; predicts it holds near 0.0227 if it is a genuine finite-size boundary asymmetry, which then bounds the systematic on the offset.
8. [seat] Hypothesis [cost: cheap] State explicitly what would license transporting an exact one-block D to a lattice offset — i.e. name the theorem connecting the 2-cycle multiplier to the L→∞ width-scaling slope on a hierarchical carpet, and check its preconditions. Predicts no such theorem is available at these sizes, which keeps the two objects formally separate.
9. [lab] [cost: infeasible here — L = 59049 is ~3.5×10⁹ sites per sample at 8,000 samples, two orders, four stacks, far beyond any run so far] The clean convergence test. Named because it is decisive, ranked last because it cannot be run.

RESIDUE

Not closed. Two sharper cases. (a) Derive, from the wall law 1−R(1−q)=N₁q^{C₁}, the term in the 2-cycle multiplier that depends on C_A − C_B and vanishes when the orders match — that is the object the +0.60/+0.72 partials are detecting, and nobody has written it down. If Test 1 returns ≤0.15, this case dissolves into curvature cancellation and the right question becomes why equal-C₁ patterns cluster in s. (b) The lattice question is untouched: derive the finite-L order asymmetry (A-outer and B-outer are genuinely different graphs at L=81) analytically, and show the order-gap predicts the contamination of the offset pair by pair. Until then no reported lattice offset smaller than its own order-gap should be treated as a measurement.

BACKPROP

open. Write back: (1) an edge — C₁ is not a label but the order of vanishing of the crossing map at the wall; in any composition of recursion maps, record the local degree at the absorbing boundary alongside the fixed point and the multiplier, and expect degree mismatch to obstruct what curvature expansions alone predict. (2) A correction — my earlier "cut class is dead" was wrong for the exact one-block object; it remains correct only as a statement that cut class does not fix the exponent (0.43–0.55 spread at C₁=4). (3) A probe — Test 1, the joint partial controlling δ and Δy₉ together, which neither the lab nor any seat has yet run and which separates the two surviving mechanisms for free. Next case: the matched-C₁, extreme-Δy₉ stack, run on the lattice in both orders with the order-gap reported beside the offset.

_Contract check: House TESTS without a cost tag or reason: 2 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"), 3 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"), 4 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"), 5 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"), 6 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"), 7 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"), 8 (the cost tag has no reason; expected "[cost: free|cheap|hours|infeasible here — one-line reason]"); House FORCED (J1): 7: no derivation sketch — it belongs under ASSERTED; 8: no derivation sketch — it belongs under ASSERTED; 9: no derivation sketch — it belongs under ASSERTED; House claims without an object (J2): FORCED 7; FORCED 8; FORCED 9; VERDICT: 1 claim sentence(s) without an object tag, e.g. “The exact wall law 1−R(1−q)=N₁q^{C₁} makes C₁ the order of vanishing of the map …”; House FORECASTS (F1/F3): F2: no VALUE interval; F3: the SIGN has no probability._

Forecasts (parsed, for scoring)

Codifier's note on Continuation 1 (2026-10-01)

• Ids. The export's heading reads PN-nnn because Set ids was not used before export. This record is PN-004, case CASE-27.
• Parent. The second House-final draw, above.
• No continuation checker ran. The House's "the checker table extracted no checkable claims" means there were no tables to read, not that every claim checked out. The saved record held no continuation checker tables, which is the defect the Differential fixed in 4cb163e ("Run checker now" rebuilds them). The House made two checks of its own, and both are correct:
• C(49, 2) = 1,176. Run 076 merged the 63 patterns into 49 distinct functions before pairing, so Legitimacy's duplicate-pattern concern does not apply.
• Lateral's +0.0359 and +0.0602 are mis-subtractions for +0.0022 and +0.0052.
• Contract-check line. This is the line stored at House time.
• Genuine: the flags on tests 2–8 (a cost tag with no reason) and on F3 (a SIGN with no probability).
• False positives fixed in Version 14: the flags on FORCED 7–9 (bookkeeping lines) and on F2's VALUE interval.
• Where it went. The House's tests 1–2 became run 078 (unanticipated): curvature mismatch carries most of the class effect, and a direct residue of +0.185 remains. Its F1, F2 and F4 were scored on 2026-09-30 and the seats' continuation forecasts on 2026-10-01, all in SEAT-RECORD. F3 is pending.
• Not out of sample. Ledger's and Hypothesis's continuation forecasts, and House F2, name the cut-4 pair with the widest Δy₉. Its value (D = +0.0010) was printed in the lab report every one of them saw. They are scored, and marked.