Runs 054–055 found that the number of distinct minimum cuts orders the carpets' percolation exponents. That count is the leading coefficient of the generator's reliability polynomial, R(p) = the probability that one 9×9 generator is crossed left to right under bond percolation, since 1 − R(p) ≈ N_min · (1 − p)^C₁ near p = 1. Iterating p ↦ R(p) is a renormalization whose fixed point and slope give an approximate exponent, y = ln R′(p*) / ln 9, with R′(p*) the expected number of pivotal bonds at the fixed point (Russo). Does y, from one generator's polynomial and no sampling at all, order the measured exponents of all seven carpets and the square?

Sealed 2026-09-29 after run 057 was scored and before any crossing polynomial was computed (codifier). Known at sealing: every measured 1/ν above, the minimum-cut counts used as control (i), and the idea's origin — found by looking back across runs 053–057 for the object behind the cut count, not by following one thread forward. Not known: any coefficient of any R(p), any p* or y.

Why it can fail. This renormalization treats each 9×9 block as a single bond whose state is independent of its neighbours. A carpet is not a hierarchical lattice: adjacent blocks share edges, and their crossings are correlated. For the square lattice such small-cell renormalizations are known to be rough, so y is not expected to equal 1/ν; the bet is only that the errors are similar across rooms, so the order survives. Nine of the eight lattices' 28 pairs sit within 0.1 of each other; a rough approximation can easily scramble those.

What the outcomes mean. Confirmed: the class of a room is organised by one computable object — its generator's reliability polynomial — and the minimum-cut count mattered because it is that polynomial's leading term; the next test is whether y's values, not only their order, can be corrected into predictions. Falsified: the cut count is a correlate that the one-block polynomial does not explain, and the correlations between blocks carry the class.

Pre-compute note 2026-09-29 (codifier), committed before any R(p) was computed. Not part of the sealed prediction; an unscored side bet, recorded so it cannot be claimed afterwards. The renormalization treats blocks as independent, which makes the cut counts compose: N(g+1) = N₁ · N(g)^C₁, so the cut entropy would converge to s_comp = ln N₁ / (C₁ − 1). Against the measured s(4) of runs 054–055, the ratio s / s_comp is H 0.750, U 0.820, V 0.832, T 0.853, U′ 0.719, A 0.909, B 0.567 — B is far below every other carpet, i.e. its blocks' cuts are the most correlated and the independence approximation is worst there. Side bet: if any of the 27 scored pairs comes out discordant, every discordant pair involves B. These ratios are in-sample, computed from numbers already known; they are a lead, not a result.

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Scored 2026-09-29 — confirmed

Results: R27_tree_groups/carpet_perc/RESULTS058.md, rg058.json (commit 521f120, Lab_Tree_Model). Exact integer crossing counts a_k by an edge-by-edge column transfer matrix over connectivity states; weights summed to (1 + x)^M on every generator; 30 s for all eight.

What it settles. One generator's exact crossing polynomial, iterated with no sampling at all, orders all eight lattices' percolation exponents without a single inversion — including the five carpets that share a cut exponent, and B, which minimum-cut entropy had misplaced in run 055. The cut count worked in 054–055 because it is this polynomial's leading coefficient (control i, exact on all seven); where it failed, on B, the rest of the polynomial corrects it. The higher coefficients — cuts above the minimum — carry real information about the class.

How strong. A random order of eight would match with probability 1/40,320, but the order is not random: cut exponent already separates U′, T and the rest. The sharper test is the five rooms at one cut exponent (H, U, B, V, A), where 1/5! = 1/120 by chance, and where cut size alone predicted nothing. B–U was predicted unscored, yet came out concordant.

What it does not settle. The values. y reads the carpets high by 0.009–0.065 and the square low by 0.020, and the offset does not follow the cut-entropy shortfall recorded in the pre-compute note (A has the least shortfall but a large offset; B the most shortfall but the smallest offset). The independent-block approximation orders the classes; something else sets how far off it is. That is P-047's question, now with numbers to test against.

Unsealed: the thresholds do not order the same way. p* sits 0.03–0.06 above the measured p∞ on every carpet and ranks B above U and A, where the measured thresholds put it below both. The polynomial gets the exponent's order right and the threshold's order wrong — so the two are not one quantity seen twice, and B's oddity lives in its threshold, not its exponent.