run 054 — Lab

Math Mapgenerated 2026-10-25 · a354234

PUBLIC BUILD

slug: min-cut-count-orders-class
state: run
created: 2026-09-28
sealed: 2026-09-28
measured: 2026-09-28
builds_on: [053, 052]
concepts: [fractal-dimension]
statistic: R27_tree_groups/carpet_perc/ (Lab_Tree_Model), run 052's cut geometry exactly (nearest-neighbour bonds between kept cells; terminals the left and right boundary columns; planar dual with cost 1 across a kept–kept bond and 0 along a hole or the top and bottom boundary). N(g) = the number of DISTINCT minimum left–right cuts, counted as distinct sets of bonds, at generation g (L = 9^g: L = 9, 81, 729, 6561 for g = 1–4). Because zero-cost dual edges make many dual paths name the same bond set, count on the dual after contracting every zero-cost dual edge (holes, top and bottom strips) into one node, then count shortest source–sink paths in that contracted multigraph with parallel edges counted separately; carry the count as a logarithm (log-sum-exp over the shortest-path DAG) or as an exact big integer where it fits. The statistic is s(g) = ln N(g) / C(g), the cut entropy per cut bond, with C(g) the exact minimum cut already verified in runs 052–053. Required for H, U and V at g = 1, 2, 3; g = 4 where memory allows, stated. Control: at g = 1 (L = 9), for all five carpets (H, U, U′, T, V), N(1) by brute force — every bond subset of size C(1) whose removal disconnects left from right — must equal the contracted-dual count exactly; and at g = 2 for H, the dual count must equal an independent enumeration (all minimum cuts via the residual graph of one max-flow, e.g. Picard–Queyranne). Reported, not sealed: N(g) and s(g) for T and U′, and the growth ratio ln N(g+1) / ln N(g).
script: R27_tree_groups/carpet_perc/ (Lab_Tree_Model)
artifacts: [R27_tree_groups/carpet_perc/RESULTS054.md, R27_tree_groups/carpet_perc/mincount054.json]
outcome: confirmed
next: Does s predict 1/ν, not just order it? Across all five carpets the unsealed extras fall on the same monotone order (s: H 0.693 < U 1.058 < V 1.153 < T 1.528 < U′ 1.993; 1/ν: 0.547 > 0.524 > 0.477 > 0.374 > 0.276) — fit nothing to them; seal a numeric 1/ν for a new carpet chosen by its s alone, at a cut exponent none of the five has | builds_on: 054
source: ../lab/runs/054-min-cut-count-orders-class.md

PredictionAt the largest generation computed for all three (at least g = 3), s(V) > s(U) > s(H) — the carpet with more ways to be cut at the same cut size has the smaller 1/ν, matching 053's order 1/ν(V) = 0.477 < 1/ν(U) = 0.524 < 1/ν(H) = 0.547.

Falsifiers(H) ≥ s(V) at that generation, or the two within 1% of each other (the pair with the largest 1/ν gap is not ordered by the count: minimum-cut entropy is not the second number). The control failing withdraws the run. V above H but U out of place is unanticipated. Between prediction and falsifier is unanticipated.

VerdictCONFIRMED 2026-09-28. Both controls passed — brute force equals the contracted-dual count at g = 1 on all five carpets (H 16, U 48, V 64, T 36, U′ 16), and Picard–Queyranne enumeration equals it for H at g = 2 (65,536) — and every shortest path length equals the verified C(g). At g = 4 (L = 6561), the largest generation computed for all three: s(V) = 1.1534 > s(U) = 1.0579 > s(H) = 0.6931, the predicted order, the same at g = 2 and g = 3; V and H are 66% apart against the 1% tie clause, so the falsifier did not fire. The count of distinct minimum cuts orders the three carpets that share a cut exponent exactly as their 1/ν does. An ordinal prediction on three items has a 1-in-6 chance by luck; the body records what that does and does not settle.

Sealed 2026-09-28 after run 053 was scored and before any minimum cut was counted (codifier). Known at sealing: every number in runs 051–053, including the three 1/ν values this prediction orders, and one minimum cut per carpet drawn at generation 2 by run 052 ("minimum cuts need not be unique"). Not known: any count N(g), for any carpet, at any generation. The direction is a bet made before the counts exist, so this is out of sample in the way 053's ratio was not.

Why this direction. The spanning cluster fails when some minimum cut is fully open. With more distinct minimum cuts of the same size there are more independent ways to fail near threshold, so the crossing probability should sharpen more slowly with L — a wider window, a smaller 1/ν. V's central 3×3 block with eight separate singletons should allow many more routes for the dual path than H's arrangement, at the same cost.

What the outcomes mean. Confirmed: at fixed dimension and fixed cut exponent, a room's class tracks how many ways it can be cut, and the dictionary's "the room decides the class" (§43, §45) gains its second measurable number. Falsified: the count is not the missing variable; the next candidates are the geometry of the cut (how the minimum cuts are spread across generations) or finite-size corrections that differ between rooms. Unanticipated with V above H but U misplaced: the count separates the extremes and something else sets the middle.

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Scored 2026-09-28 — `confirmed`

Results: R27_tree_groups/carpet_perc/RESULTS054.md, mincount054.json (commit b5f4bda, Lab_Tree_Model). Method as sealed: zero-cost dual edges contracted, shortest paths counted on the contracted multigraph by layered DP, counts exact to 128 bits through g = 3 and as logarithms at g = 4 (U′ exact throughout). All five carpets reached g = 4 in 1–2 s at under 0.8 GB.

s(g) = ln N(g) / C(g):

carpet	x	g = 1	g = 2	g = 3	g = 4	1/ν (051, 053)
H	0.631	0.6931	0.6931	0.6931	0.6931	0.547
U	0.631	0.9678	1.0365	1.0536	1.0579	0.524
V	0.631	1.0397	1.1264	1.1480	1.1534	0.477
T (unsealed)	0.5	1.1945	1.4256	1.5026	1.5282	0.374
U′ (unsealed)	0.316	1.3863	1.7329	1.9062	1.9928	0.276

What it settles. At fixed dimension and fixed cut exponent, the three rooms are ordered in 1/ν exactly as they are ordered in minimum-cut entropy, at every generation from 2 to 4. That is the second number run 053 said was missing, as far as an ordering of three can show it.

What it does not. Three items in a predicted order is one of six permutations; this is real but thin evidence. And the fit to all five carpets, which looks clean, is in-sample: T and U′ were reported as extras, and they also differ in cut exponent, so for them s and x are confounded. No functional form is claimed. The next: asks for a numeric prediction on a new carpet, sealed from s before its percolation is sampled.

Structural observations, unsealed. H's count is exactly N = 2^C at every generation: each cut bond is an independent binary choice, and its minimum cuts factor completely. The growth ratio ln N(g+1) / ln N(g) tends to C₁ on every carpet (4, 4, 4, 3, 2 by g = 4), so ln N grows like s∞ · C(g) and s(g) converges; H converges at g = 1, the others from below.

generated 2026-10-25the lab index

Scored 2026-09-28 — confirmed

Scored 2026-09-28 — `confirmed`