Math Mapgenerated 2026-10-25 · a354234
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run 049Settled

The dictionary's §45 claim that phase transitions need infinite ramification is a literature theorem (Gefen–Mandelbrot–Aharony 1980–84), retrodicted, not found. The part that is not settled is §43's reading of it — that on a fractal the universality class is set by the room (lacunarity, how the holes are arranged) and not by the Hausdorff dimension. Two Sierpinski carpets of identical dimension log 8/log 3 but opposite lacunarity: do they percolate in the same class?

slug
carpet-room-or-dimension
state
run
created
2026-09-28
sealed
2026-09-28
statistic
Bond percolation on the nearest-neighbour graph of the kept cells, spanning left–right, open-boundary. Carpet H (high lacunarity) — the standard carpet, generator 3×3 with the centre removed, generations 4–7 (L = 81, 243, 729, 2187). Carpet U (uniform) — generator 9×9 keeping 64 of 81 cells, removing the 17 singletons {(i, j) : i, j ∈ {1, 4, 7}} ∪ {(2,2), (2,6), (6,2), (6,6), (2,4), (4,2), (6,4), (4,6)} (0-indexed; D₄-symmetric; no hole larger than one cell; dimension log 64/log 9 = log 8/log 3 exactly), generations 2–4 (L = 81, 729, 6561). Controls: the square lattice at L = 81, 243, 729, 2187; the Sierpinski gasket at generations 6–9. For each graph, spanning probability R_L(p) on a p-grid of spacing 0.0025 over the crossing region, ≥ 2,000 samples per point (union–find); p*(L) = the p with R_L = ½; width w(L) = p(R = 0.8) − p(R = 0.2). Exponent 1/ν from the log–log slope of w(L) against L over the sizes listed (H: four sizes, U: three; the L = 81, 729 pair is common to both and is reported alone as well). Reported, not sealed: p*(L) sequences and extrapolated thresholds (thresholds are non-universal even on regular lattices), the fractal dimension of the spanning cluster at p*.
script
R27_tree_groups/carpet_perc/ (Lab_Tree_Model)
artifacts
[R27_tree_groups/carpet_perc/perc.json, R27_tree_groups/carpet_perc/RESULTS.md]
measured
2026-09-28
outcome
withdrawn
concepts
[fractal-dimension]
source
../lab/runs/049-carpet-room-or-dimension.md

Prediction(1) Controls: the square lattice gives 1/ν = 0.75 ± 0.05; the gasket has no crossing — p*(L) increases monotonically with generation toward 1. (2) Room decides: 1/ν(H) and 1/ν(U) differ by at least 0.10, with the uniform carpet U closer to the square lattice's 0.75 than H is; on the common pair L = 81, 729 the difference has the same sign. (3) Both carpets have a crossing: p*(L) converges on each (successive differences shrinking), to thresholds that differ from each other.

Falsifier|1/ν(H) − 1/ν(U)| ≤ 0.03 on the full size ranges and on the common pair — the two carpets are one class at equal dimension, and the dimension, not the room, labels it; §43's "classes labelled by the room" is then wrong on fractals and the §24 dichotomy (ramification) decides existence only. Either control failing withdraws the run. Between prediction and falsifier is unanticipated.

VerdictWITHDRAWN — the square-lattice control fails on the sealed statistic: w(L) interpolated on the sealed p-grid (spacing 0.0025) gives 1/ν = 0.640 over L = 81–2187, outside 0.75 ± 0.05, because at L = 2187 the true width (0.0021) is narrower than one grid step. The gasket control passes (p*(L) = 0.751, 0.772, 0.788, 0.800, rising). Unsealed, on the grid-free quantile estimator that passes the control (0.753 against 3/4; D_f 1.905 against 91/48): carpet H 1/ν = 0.546, carpet U 0.526 — Δ = 0.020 on the full ranges and 0.030 on the common pair L = 81, 729, at the falsifier's edge; bootstrap errors 0.005–0.011. The two carpets are one class at equal dimension to this resolution, and U is not closer to the square lattice than H (both near 0.53). Thresholds differ (0.6845 against 0.7528, reported only).

Sealed 2026-09-28 before any percolation on carpet U was computed (codifier). Known at sealing: the dictionary's §45 numbers (carpet H generations 3–4, crossing near 0.68; gasket crossing drifting toward 1); the Gefen–Mandelbrot–Aharony theorems on finitely vs infinitely ramified fractals (titles and abstracts only); that percolation thresholds are non-universal. Not read: any study of percolation or Ising exponents on carpets of equal dimension and different lacunarity — deliberately, so that the prediction is the lens's and not the literature's. Machine: the M5, after run 048.

Why these two carpets. Equal Hausdorff dimension is exact, not approximate: 64 of 81 at scale 9 is 8 of 9 at scale 3, squared. Generation 2 of carpet H is itself a 9×9 generator keeping 64 cells — with one 3×3 hole and eight singletons. Carpet U keeps the same count with seventeen singletons. Everything but the arrangement of the holes is held fixed; the matched sizes 81 and 729 compare the two at identical L.

What the outcomes mean. Confirmed: on fractals the room labels the class at fixed dimension, as §43 says, and "universality = symmetry + room" survives its first test where the room varies alone. Falsified: dimension labels the class; ramification decides only whether a class exists (the literature's statement) and §43's room reading is a restatement of dimension on regular lattices, where the two cannot be separated. Unanticipated in (0.03, 0.10): the classes differ weakly — crossover or slow convergence at these sizes — and the run repeats at carpet U generation 5 on BernoullisBook or a cluster.

Note under the seal, 2026-09-28 (codifier; before any carpet-U result was reported). The seal's description of carpet U is wrong in one respect and its operational definition stands. The explicit list of 17 removed cells is the definition; the gloss "no hole larger than one cell" is false: (2,4)–(1,4), (4,2)–(4,1), (6,4)–(7,4) and (4,6)–(4,7) are edge-adjacent, so U has four domino holes and nine singletons (found by Lab_Tree_Model's generation-2 checks). Dimension is unaffected (64 of 81 kept) and the contrast the run is built on survives — U's largest hole is 2 cells against H's 9-cell block — but U is less uniform than the seal says. The run proceeds on the sealed list. A D₄-symmetric 17-singleton set exists (Lab_Tree_Model found 12); if the chat runs one, it is an unsealed observation reported beside the result, not a substitute for it.

Scored 2026-09-28 (codifier; R27_tree_groups/carpet_perc/RESULTS.md, Lab_Tree_Model). Withdrawn as written: the sealed instrument cannot resolve the square lattice at its largest size, and the control was built to catch exactly that. The grid-free numbers are an unsealed observation.

What they show, held with conviction and marked unsealed: H and U — one 9-cell hole against four dominoes and nine singletons — percolate in the same class (1/ν 0.546, 0.526; D_f 1.841, 1.831). On this contrast the dimension, or something both carpets share, labels the class, and §43's "room" reading gets no support. But U′ does not fit. Lab_Tree_Model's all-singleton set, also exactly 64 of 81 per block and connected, gives p* = 0.866, 0.877, 0.881 — rising with generation like the gasket's — and 1/ν = 0.28. At equal dimension, rearranging seventeen single-cell holes so that they nearly wall off blocks along diagonals moves the threshold from 0.75 to near 0.88 and still drifting, and the fitted exponent to half the others'. That is the room acting — but on whether a transition exists at these sizes, the literature's statement (Gefen–Mandelbrot–Aharony), not on which class it belongs to. The honest one-line reading: ramification decides existence; at equal dimension, among carpets that do percolate, no class difference was seen.

What would settle it. A reseal with a p-grid of spacing 0.0005 in the crossing window (or the grid-free estimator named as the statistic), H and U as sealed, and U′ promoted from observation to arm with its own prediction: p*(L) → a limit below 1 or → 1. U′ is the arm that can still move the dictionary's §43.

generated 2026-10-25the lab index