Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 052Settled

Run 051 found the all-singleton carpet U′ in a different percolation class from carpets H and U at the same dimension (1/ν = 0.276 against 0.547 and 0.524), but its exponent sits beside the gasket's (0.263), and the gasket is finitely ramified. Is U′ effectively finitely ramified — a gasket crossover — or does it carry a cut set that grows with size, as an infinitely ramified carpet must?

slug
carpet-cut-sets
state
run
created
2026-09-28
sealed
2026-09-28
builds_on
[051]
concepts
[fractal-dimension]
statistic
R27_tree_groups/carpet_perc/ (Lab_Tree_Model). For each graph, C(L) = the exact minimum number of nearest-neighbour bonds whose removal disconnects the left boundary column from the right boundary column (max-flow, or equivalently the shortest top-to-bottom path in the planar dual where holes and the outside cost 0 and each crossed bond costs 1 — either method, the other as a check on the two smallest sizes). Graphs as in run 051: carpet H at L = 81, 243, 729, 2187, 6561; carpet U and U′ at L = 81, 729, 6561; square lattice at L = 81, 243, 729, 2187 (control: C = L exactly); gasket at generations 6–10 corner to corner, C = the minimum cut separating the two base corners (control: bounded — the same value at every generation). Cut exponent x = log(C(6561)/C(81)) / log(81) on the common sizes, and the per-generation ratios C(L·9)/C(L). Reported, not sealed: the location of one minimum cut drawn at generation 2 (which cells it passes), and C for a single generation-(g−1) copy separated from the rest.
script
R27_tree_groups/carpet_perc/ (Lab_Tree_Model)
artifacts
[R27_tree_groups/carpet_perc/cuts052.json, R27_tree_groups/carpet_perc/RESULTS052.md]
measured
2026-09-28
outcome
confirmed
next
Is 1/ν a function of the cut exponent alone? On the three carpets 1/ν/x = 0.867 (H), 0.830 (U), 0.873 (U′). Seal a generator whose generation-1 cut is 3 (x = 0.5, predicted 1/ν ≈ 0.43) and a second cut-4 generator arranged unlike U | builds_on: 052
source
../lab/runs/052-carpet-cut-sets.md

Prediction(1) Controls: square C = L at every size; gasket C constant across generations 6–10. (2) U′ is infinitely ramified in the cut sense: C_U′ grows at every step, with ratio C(9L)/C(L) ∈ [1.5, 3] (about 2 — its generator's two diagonal hole-walls leave crossings only around a ring, and each ring cell is itself a U′ block), so x_U′ ∈ [0.18, 0.50]. (3) H and U grow faster: x_H, x_U ∈ [0.50, 0.80], both above x_U′ by at least 0.15. The ordering of the cut exponents (H ≈ U > U′ > gasket = 0) is the ordering of 1/ν in run 051 — the class follows how fast the cut set grows, which is the room, not the dimension.

Falsifierany of — C_U′(6561) = C_U′(729) (bounded cut: U′ is effectively finitely ramified at these sizes, its 1/ν is the gasket's crossover, and run 051's "own class" reading falls); x_U′ ≥ x_H − 0.05 (U′'s cuts grow as fast as H's, so the cut set does not explain the exponent gap and the room acts some other way). Either control failing withdraws the run. Between prediction and falsifier is unanticipated.

VerdictCONFIRMED on every clause; both controls exact (square C = L at 81–2187; gasket C = 2 at generations 6–10), dual and max-flow agree wherever both ran. U′ is infinitely ramified with a slowly growing cut — C = 4, 8, 16 at L = 81, 729, 6561, ratio 2 per factor 9, x = 0.3155 = log 2/log 9. H and U grow twice as fast in exponent — C = 16, 64, 256 (H also 32, 128 between), ratio 4 per factor 9, x = 0.6309 = log 2/log 3 for both. The cut exponents order as run 051's 1/ν: H = U > U′ > gasket = 0.

Sealed 2026-09-28 before any cut was computed (codifier). Known at sealing: run 051's numbers; the U′ generator's hole set, read by hand — its holes include both full diagonals of the 9×9 block ((1,1)…(7,7) and (1,7)…(7,1)), which for nearest-neighbour bonds are walls, so inside one block the left and right halves meet only through the kept boundary ring. That reading shaped clause (2)'s ratio; it is a glimpse of structure, not of a cut value. Not known: any C(L). Minutes of compute — an exact cut on 16.7M cells is one max-flow or one planar shortest path.

What the outcomes mean. Confirmed: U′ is infinitely ramified but with a slowly growing cut, and the percolation class orders by cut-set growth — the "room" of §43 made into a number. Falsified on boundedness: U′ is a gasket in disguise at these sizes, and ramification decides existence and class together. Falsified on the gap: the cut set is not the variable, and the exponent difference needs another explanation (correlated connectivity through the ring, not its width).

Scored 2026-09-28 (codifier; R27_tree_groups/carpet_perc/RESULTS052.md, Lab_Tree_Model). Confirmed, and more exactly than the seal asked: every carpet's cut is a pure power of its generation-1 cut — C(g) = C₁^g with C₁ = 2 (H, at scale 3), 4 (U, at scale 9), 2 (U′, at scale 9) — so the cut exponents are log 2/log 3 and log 2/log 9 exactly. U′'s generation-2 minimum cut crosses two bonds in the top block row and two in the bottom, running free along the corner-touching removed blocks between: the diagonal walls read by hand at sealing are what halves its exponent.

Unsealed, and the reason for the next seal. Against run 051: 1/ν ÷ x = 0.867 (H), 0.830 (U), 0.873 (U′) — the three carpets' percolation exponents sit at a near-constant fraction of their cut exponents, while their dimensions are identical. If that holds on a fourth room, the class of a carpet at fixed dimension is set by its cut exponent: §43's "room" becomes one number.

generated 2026-10-25the lab index