Run 058's one-block polynomial ordered all eight lattices' exponents but not their values (y reads carpets high by 0.009–0.065, the square low by 0.020), and put the thresholds p* 0.03–0.06 above the measured ones. Run 056 found the three-size estimator reads the square 0.009 low. Two questions, one run. (A) The composition defect — the failure of "cross each block, then the arrangement" to commute with "cross the whole" — has a zeroth-order part at the fixed point, D₂(p*), and a first-order part, D₂′(p*). Does the zeroth-order part account for the threshold shift and the first-order part for the exponent offset? (B) Does the square's estimator bias shrink when the fit uses larger sizes?

Sealed 2026-09-29 after runs 056 and 058 were scored and before any two-generation crossing or new square replica was sampled (codifier). Known at sealing: every number in runs 051–058 and the reference values above. Not known: any P₂, D₂, y_c, or the windowed square fits.

Why this split, stated because it goes against P-047 as filed. P-047 guessed that the composition defect explains the exponent offset. Looking at 058's numbers before sealing, B has the largest threshold gap (0.060) and the smallest exponent offset (0.009): whatever moves the threshold is not what moves the exponent. That matches the Taylor picture at the fixed point — the zeroth coefficient is the threshold, the first is the exponent — so the defect is split the same way: D₂(p*), its value at the fixed point, should shift the threshold (A1) and say nothing about the exponent (A2); its slope D₂′(p*) should carry the exponent's error (A3). A2 is the contrarian bet against the lead as written.

What the outcomes mean. Confirmed: the renormalization's error has the same Taylor structure as the object it approximates — zeroth order to threshold, first order to exponent — and the commutator is a usable correction, not only a diagnosis; B's oddity is a zeroth-order effect. Part B confirmed: larger sizes shrink the estimator's bias, and future fits should drop L = 81. Falsified on A1: the defect at the fixed point is not the threshold shift, and P-047's framing needs a different defect. Falsified on A3: the exponent's error is not first-order in this commutator, and y's offset has another source (the estimator's own bias, Part B, being the first candidate).

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Scored 2026-09-29 — falsified

Results: R27_tree_groups/carpet_perc/RESULTS059.md (commit ec6e82e, Lab_Tree_Model). Seeds 5901–5931 and bootstraps 59101–59105, all checked unused.

Square, windowed (fresh replicas): (81, 243, 729) 0.7313 ± 0.0080; (243, 729, 2187) 0.7335 ± 0.0073; all four sizes 0.7293 ± 0.0047. H, five sizes 81…6561: 0.5504 ± 0.0037 (three-size 0.5468; pair slopes 0.543, 0.577, 0.553, 0.514).

Why A3 failed, and whose fault it is. The corrected exponent was defined from the slope of the 81×81 crossing probability at the one-block fixed point p*. But the 81×81 lattice is not at its own threshold there: D₂(p*) is +0.13 to +0.18 on the carpets, so P₂(p*) sits near 0.9, on the flat shoulder of the crossing curve, where every slope is small. y_c was therefore bound to read low on every carpet, and it did. The seal measured the first-order commutator at the wrong point; the codifier specified it. The falsification stands as written — the sealed correction does not work — but it is a verdict on the construction, not on the idea that the first-order term carries the exponent's error.

The lead the data show, post-hoc. The size of that same first-order term ranks with the exponent offset: Spearman ρ(D₂′(p*), y − measured) = 0.88 over the eight lattices, with B (−8.24, smallest offset) and the square at the extremes. That is found after the fact, in-sample, and it is exactly the kind of pattern this project exists to test rather than believe. The next: asks for the out-of-sample version with the construction fixed.

The rest. A2 held (the zeroth-order defect does not track the exponent offset, ρ = −0.12), which is the half of the Taylor split that came out as bet. A1 was weaker than predicted: the defect at the fixed point only loosely tracks the threshold shift (ρ = 0.48; 0.21 among carpets). Part B: dropping L = 81 barely moves the square (0.7313 → 0.7335), and the four-size fit (0.7293) is no closer — the estimator's low bias of roughly 0.01–0.02 persists at these sizes, so it is not mainly the smallest size's fault; its source is open. H's five-size fit (0.5504) sits 0.004 above its three-size value, with pair slopes that oscillate, as 057 found for pair slopes generally.