run 043Falsified
../lab/runs/043-growth-axis-on-one-tree.mdPredictionRelator-driven — u(N/8)/u_random ≥ 0.5 for IMG, Basilica and Aleshin alike: no group but Grigorchuk has the square obstruction, because no other has a Klein-four subgroup to contract, and (σβ)⁴ = 1 in IMG yields 8-cycles, not squares. Growth-driven, on the Cayley graphs — expansion ratio ρ_Φ: IMG ∈ [0.25, 0.60], Basilica ∈ [0.40, 0.90], Aleshin ∈ [0.85, 1.05], ordered IMG < Basilica < Aleshin with Grigorchuk's 0.374 at or below Basilica; cut-fraction ratio: IMG ≤ 0.50, Basilica ∈ [0.40, 0.90], Aleshin ≥ 0.85; barrier as % of cut: IMG ≥ 20%, Basilica ∈ [8%, 25%], Aleshin within a factor of 1.5 of its control's; Kuramoto f: IMG ≥ 0.90, Basilica ∈ [0.20, 0.95], Aleshin ≤ 0.10. The two intermediate groups agree in class — both at least a factor of two from their controls on ρ_Φ and the cut fraction, both above 20% on the barrier — while their numbers differ; the two exponential groups separate by amenability, Basilica between the intermediate pair and Aleshin on every growth-driven quantity.
Falsifierany of — Aleshin's ρ_Φ ≤ 0.60 (the free group's level-5 quotient is not expander-like at N = 1024, and the growth axis is not visible at this size); IMG's barrier ≤ 10% of its cut (intermediate growth without the glass — the barrier is a Grigorchuk fact, not a growth fact); Basilica's ρ_Φ ≤ 0.374 (exponential growth with worse expansion than the intermediate host — the axis reversed); Aleshin's f ≥ 0.50 (an expander that does not synchronise; either the graph is not an expander or Abdalla et al. do not apply at this size); or u/u_random ≤ 0.10 for any of IMG, Basilica, Aleshin (the square obstruction is not specific to the Klein-four contraction). Between prediction and falsifier is unanticipated. Control: the Grigorchuk row's reproduction at 432c51e is the control and passed before sealing; the random d-regular controls must be connected or the row is withdrawn.
VerdictFALSIFIED — the relator clause's falsifier fires: Basilica's unique-neighbour minimum at N/8 is 1 of its control's 101 at best-of-20 and 0 of 94 at best-of-40, u/u_random ≤ 0.10, so the square obstruction of N-034 is not specific to the Klein-four contraction; a torsion-free exponential-growth group with no squares has zero unique-neighbour expansion by another route. IMG (0.38) and Aleshin (0.19) also miss the ≥ 0.5 prediction without firing it. Growth-driven clauses: ρ_Φ inside every band — IMG 0.508, Basilica 0.474, Aleshin 0.853 — but not in the predicted order: Grigorchuk 0.374 < Basilica 0.474 < IMG 0.508 < Aleshin 0.853, the three contracting groups clustered and the free group apart, which is the nucleus coordinate ordering (amenable vs not), not the growth class; cut-fraction ratios 0.506 / 0.46 / 0.83 (IMG and Aleshin miss their edges by 0.006 and 0.02); barriers 53% / 40% / 24% of the cut against controls of 42% (random 3-regular is itself glassy) / 8.0% / 8.4% — IMG's clause hit, Basilica's and Aleshin's missed high, and every Cayley graph including the free group's has walls three to five times its control's, so barriers separate Cayley graphs from random graphs rather than growth classes; Kuramoto — IMG 1.000 (hit), Basilica 0.020 (miss: it synchronises like an expander), Aleshin invalid because kuramoto.py's fixed dt = 0.2 is unstable at degree 6 (dt·λ_max = 2.4 > 2), so that falsifier clause is unmeasured. Controls connected; calibration passed. Basilica is amenable-like in geometry (expansion, cut, barrier) and expander-like in dynamics (synchronisation) — the split the run did not predict and the one worth a probe. IMG's cogrowth κ = 0.797 against Grigorchuk's 0.665, shortest prime (ac)⁴ at length 8, confirming the relator.
Sealed 2026-09-28 after the control row and before any of the four other groups was measured (codifier). What was known at sealing: the five recursions verified against Bux–Pérez, Bartholdi– Virág and Vorobets–Vorobets by Lab_Tree_Model; the nuclei ({1, a, a⁻¹}, {1, a, b, c, d}, {1, a, b, c}, seven elements for Basilica, none for Aleshin to length 12) and depth-1 contraction ratios (0.50, 0.50, 0.83, 1.00, 1.00); the levels, orders and degrees; that only Grigorchuk has a Klein-four subgroup among its generators; and the Grigorchuk Cayley-graph row with its random control. Nothing about the other four groups' graphs beyond their sizes.
What the seal corrects in P-043. The probe predicted zero unique-neighbour expansion for IMG(z² + i) from (σβ)⁴ = 1. The relator is real (Bux–Pérez Lemma 20) but the squares of N-034 come from the relator and the contraction of ⟨b,c,d⟩ — the 8-cycle alternating a- and d-edges becomes a 4-cycle only when the d-edges vanish inside cosets. IMG's pairs of generators generate dihedral groups of order 8 and 16, not a Klein four-group; there is nothing to contract, and the relator leaves 8-cycles in the Cayley graph. So the square obstruction is predicted to be Grigorchuk's alone. That is a sharper claim than the probe's, and the falsifier's last clause is its test.
Why the odometer is reported and not sealed. Its Cayley graph at N = 1024 is a 1024-cycle and its degree-matched control is a random 2-regular graph, a union of cycles. Every instrument is degenerate there — Φ(s) = 2/s, bisection 2, barrier 2, no free spins, and a Kuramoto flow that mostly ends in twisted states. It is the lattice row's occupant; its numbers are recorded as the trivial end of the axis, and the one non-trivial expectation — the cut's share in the first spectral band, 8/π² ≈ 0.81 for a cycle — is noted here as an observation to check, not a clause.
What the outcomes mean. Confirmed: the growth class orders the growth-driven instruments on one tree, amenability separates the exponential pair, and N-034's obstruction is a property of one group's Klein-four structure — the two-family split of P-043 holds with the relator family narrower than the probe said. Falsified on Aleshin: the axis needs larger N to appear, and the run repeats one level up. Falsified on IMG's barrier: the glass belongs to Grigorchuk, and the "classical memory" reading of the intermediate cell is a reading of one group. Falsified on the unique-neighbour clause: squares are not the only route to zero, and N-034 is incomplete.
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R27_tree_groups/RESULTS043.md, 9b1c43d)| graph | u at N/8 (best 20 / 40) | ρ_Φ | best cut (fraction) | barrier % | free spins | first-band share (mult) | κ | Kuramoto f |
|---|---|---|---|---|---|---|---|---|
| IMG(z²+i), N 4096, d 3 | 18 / 18 | 0.508 | 256 (0.0417) | 53.1 | 0.125 (2+1 split) | 0.808 (8) | 0.797 | 1.000 |
| random 3-regular | 47 / 47 | 1 | 506 (0.0824) | 41.9 | 0.247 | 0.321 (1) | — | 0.220 |
| Basilica, N 4096, d 4 | 1 / 0 | 0.474 | 512 (0.0625) | 39.6 | 0.041 | 0.804 (8) | — | 0.020 |
| random 4-regular | 101 / 94 | 1 | 1114 (0.1360) | 8.0 | 0.054 | 0.0006 (1) | — | 0.000 |
| Aleshin, N 1024, d 6 | 8 / 6 | 0.853 | 512 (0.1667) | 24.0 | 0 | 0.908 (16) | — | invalid |
| random 6-regular | 42 / 42 | 1 | 620 (0.2018) | 8.4 | 0.050 | 0.368 (1) | — | invalid |
| odometer, N 1024, d 2 | 6 / 6 | 1.000 | 8† (0.0078) | 0 | 0.016 | (merged) | — | unconverged |
| random 2-regular (a cycle) | 8 / 6 | 1 | 6† (0.0059) | 0 | 0.012 | (merged) | — | unconverged |
| Grigorchuk Cayley, N 4096, d 4 (control row) | 12 | 0.374 | 354 (0.0432) | 32.8 | 0 | 0.257 (8) | 0.665 | 0.995 |
† The Kawasaki anneal does not find a cycle's true bisection (2); the odometer rows are recorded, not read. Instrument limits found by the run: kuramoto.py's fixed step is unstable above degree 5; the band grouping merges λ = 0 with a cycle's first pair; a degree-3 vertex cannot split its neighbours evenly, so "free spins" there is the 2+1 count.
Scored as written. Falsifier fired (Basilica u/u_random ≤ 0.10): falsified. Clause tally — ρ_Φ three hits; cut-fraction one hit, two misses at the edges; barrier one hit, two misses high; Kuramoto one hit, one miss, one unmeasured; ordering miss; relator clause falsified.
What it found. 1. Zero unique-neighbour expansion has a second route. Basilica has it with no squares and no torsion. N-034's theorem is Grigorchuk's; its "the reason is the relators, and squares are how" is now one mechanism of at least two. What Basilica's is — its Cayley graph's structure at s = N/8 — is the next thing to look at. 2. Expansion orders by the nucleus, not by growth. Contracting groups 0.37–0.51, the free group 0.85. Within the amenable side, intermediate and exponential do not separate. 3. Barriers are a Cayley-graph property. Every group, the free one included, has walls three to five times its random control's. Symmetry makes degenerate crystalline minima; the growth class does not enter. Run 040's 43% at level 5 is the intermediate host's value of a quantity every Cayley graph here has. 4. Basilica splits. Amenable-like on every static instrument, expander-like on the one dynamic instrument that ran validly. Probe candidate: which of the two does a quantum walk follow? Run 047 asks exactly that on its boundary.
Addendum, same afternoon (N-036). The mechanism behind Basilica's zero is a construction — unions of cosets of ⟨a², b²⟩ (order 8, index 512) have no unique neighbours, every outside neighbour hit 2 or 4 times — and the same construction gives 0 on Aleshin (⟨a², b², c²⟩ of order 64, index 16; unions of two cosets at s = 128), so the annealer's 8 was an instrument shortfall and Aleshin's row satisfies the falsifier as well. IMG(z² + i), whose generators are involutions, has neither this route nor N-034's, and its 0.38 stands. The relator clause was wrong twice over.
generated 2026-10-25the lab index