How far is each carpet from a pure composition — is the drift in its pair slopes, and the misplacement of carpet B, a measurement of the commutator between crossing a block and crossing the arrangement of blocks?
no run lists this probe in builds_on:
Why this exists. Filed 2026-09-29 as a lead so it is not lost; nothing here is sealed. It came from reading runs 053–058 side by side rather than along one arc, and it waits on run 058's output.
The object. For a 9×9 generator, R(p) is the probability it is crossed left to right under bond percolation. Its failure polynomial F(q) = 1 − R(1 − q) vanishes to order C₁ at q = 0 with leading coefficient N_min — the minimum-cut size and count of runs 052–055. The null case is a single bond, R₀(p) = p. If blocks were independent, generation g would be R composed with itself g times.
Three readings of one polynomial. 1. Coefficients are root statistics. Factor F(q) = N_min q^C₁ ∏(1 − q/q_j). As in Euler's Basel argument, the next coefficient — cuts one bond above the minimum — is N_min times a power sum of reciprocal roots, and each further coefficient is the next symmetric function of the roots. 2. Derivatives at the fixed point. p* = R(p*) is the threshold; λ = R′(p*) gives the approximate exponent ln λ / ln 9 (run 058 tests its order). 3. The iteration. Near the repelling fixed point the composition is linearised by one function Φ with R(Φ(z)) = Φ(λz). Under pure composition, the crossing probability at every scale is Φ(λ^g Ψ(p)), so Φ is the finite-size scaling function, and the pair slopes lock onto ln λ / ln 9 within a generation or two. The zeros of the composites accumulate on R's Julia set, which meets the real line at p*; the nearest complex zero approaching p* as L^(−1/ν) is the root-statistic (Fisher-zero) reading of the same exponent.
The claim to test: the defect is a commutator. A real carpet is not a composition. "Cross each block, then cross the arrangement of blocks" and "cross the whole" are two operations, and on a carpet they do not commute: neighbouring blocks share edges, so their crossings are correlated. The defect D_g(p) = R_true,g(p) − R^{∘g}(p) measures the failure. Two in-sample signs point at it: the measured cut entropy falls short of the composition value s_comp = ln N₁ / (C₁ − 1) by ratios H 0.750, U 0.820, V 0.832, T 0.853, U′ 0.719, A 0.909, B 0.567 (run 058's pre-compute note); and carpet B, far the lowest, is the one whose 1/ν sat out of place in run 055.
Provisional — not sealed, not scored
Provisional predictions — not sealed, to be sealed after 058. - Pure composition of each carpet's exact R gives pair slopes flat to at least three decimals from the second generation on, so none of the drift seen in runs 053–055 comes from the iteration itself. - The size of the defect at p*, |D₂(p*)| (true two-generation crossing by Monte Carlo at L = 81 against R(R(p*))), orders the carpets as the cut-entropy ratio does: B largest, A smallest. - The complex zeros of R and R∘R cluster toward p* on the real axis, and their approach gives an exponent within the rough accuracy of 058's y.
What would make it worthless. If the composition's own pair slopes drift as much as the carpets' do, the defect is not what the drift measures. If |D₂| does not rank B worst, the cut-entropy ratio and the commutator are different quantities, and B's misplacement needs another explanation.
Attributions to verify before sealing. Russo's formula; the Koenigs–Poincaré linearisation at a repelling fixed point; Fisher-zero finite-size scaling; the Brown–Colbourn conjecture on reliability-polynomial roots and its refutation in general. All are from memory, unverified.
Back-links (2026-09-29)
Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.
- Run 054 — confirmed (2026-09-28, measured).
lab/runs/054-min-cut-count-orders-class.md: CONFIRMED 2026-09-28. - Run 055 — unanticipated (2026-09-28, measured).
lab/runs/055-cut-entropy-predicts-class.md: UNANTICIPATED 2026-09-28. - Run 057 — confirmed (2026-09-29, measured).
lab/runs/057-bsd-euler-product-slopes.md: CONFIRMED 2026-09-29, scored as written. - Run 058 — confirmed (2026-09-29, measured).
lab/runs/058-reliability-polynomial-rg.md: CONFIRMED 2026-09-29.
69 source lines, no addenda.