run 047Settled
../lab/runs/047-growth-axis-in-transport.mdPredictionodometer — β ∈ [0.95, 1.05] (its boundary is a 1,024-cycle; ballistic; the lattice row). IMG(z² + i) — β ∈ [0.30, 0.90] and PR/N ≤ 0.60 (intermediate growth, anomalous boundary transport, like Grigorchuk). Basilica — t_c ≥ 4 and β ∈ [0.30, 1.05], PR/N ≤ 0.90 (its Schreier graphs have polynomial growth as graphs though the group's is exponential; the walk spreads, ballistic or anomalous, and does not saturate like an expander). Aleshin — saturated, t_c < 4, and PR/N ≥ 0.60 (the bireversible free-group action's Schreier graphs are expander-like; the walk fills the boundary at once). Ordered by t_c and by β where defined: odometer ≥ Basilica ≥ IMG ≈ Grigorchuk on β, and Aleshin alone saturated — the growth class is visible in transport on the boundary as it was predicted for the bulk in run 043.
Falsifierany of — IMG β ≥ 0.95 (intermediate growth without anomalous boundary transport: the quasicrystal reading is Grigorchuk's alone, not the cell's); Aleshin t_c ≥ 4 or PR/N ≤ 0.30 (the free group's boundary at 1,024 leaves is not expander-like, and transport does not see the non-amenable end of the axis at this size); odometer β ≤ 0.90 (instrument fault); Grigorchuk calibration off by more than 0.005 (withdraw). Between prediction and falsifier is unanticipated.
VerdictUNANTICIPATED on the ordering clause, with every per-group clause hit and the class separation confirmed outright. Level-10 boundaries under the anisotropic Hecke operator: odometer β = 1.000 (ballistic, [0.95, 1.05]); Grigorchuk β = 0.800 (calibration, reproduces run 046 exactly); IMG(z² + i) β = 0.440, PR/N = 0.305 (anomalous, inside [0.30, 0.90] and ≤ 0.60); Basilica t_c = 43, β = 0.595, PR/N = 0.135 (spreads, anomalous, inside its bands); Aleshin saturated at t_c = 3.6 with PR/N = 0.949 and a boundary diameter of 6 on 1,024 leaves (expander-like). No falsifier fires. The sealed internal order "odometer ≥ Basilica ≥ IMG ≈ Grigorchuk" misses: the measured order is odometer 1.00 > Grigorchuk 0.80 > Basilica 0.60 > IMG 0.44 — IMG is twice as anomalous as Grigorchuk and Basilica sits between them. What holds is the class structure the seal was for: the lattice boundary is ballistic, every contracting group's boundary is anomalous, the non-contracting group's boundary is an expander — in transport the nucleus decides, as it did for expansion in run 043, and the growth class does not order the anomalous three. Basilica, which synchronised like an expander in the bulk (043), walks like a quasicrystal on its boundary. Unsealed extras: on Grigorchuk, β is a smooth monotone dial in the anisotropy — 0.96 at weights (1, t, t², t³) with t = 0.95, 0.80 at t = 0.5, a floor near 0.55 at t = 0.2 — stable across levels 10–12 to ±0.02; equal weights on b, c, d restore β = 1.000 exactly (the aperiodicity is which of the three); and the spectral measure's correlation dimension D₂ falls from 0.65 to ≈ 0.35 by t = 0.7 and then saturates near 0.3 while β keeps falling, with β ≥ D₂ in every row — the strong-anisotropy dial is in the eigenstates, not the spectrum.
Sealed 2026-09-28 after run 046 and before any group other than Grigorchuk had a boundary walk computed (codifier). Known at sealing: the five verified recursions (R27 README); that the odometer's level-n Schreier graph is a cycle; that Basilica's Schreier graphs approximate its Julia set and IMG(z² + i)'s approximate a dendrite (Nekrashevych's limit spaces, from memory); that Aleshin's automaton is bireversible and its Schreier graphs are expected to be expander-like (Glasner–Mozes, from memory, unverified — the falsifier carries the risk). The weights are fixed by the seal and not tuned; a weight-dependence sweep on Grigorchuk is an unsealed observation reported alongside.
What the ordering would mean. If it holds, the boundary of each group carries its growth class in the walk: ballistic for the lattice end, anomalous for the intermediate pair, saturated for the expander end, with Basilica — exponential group, polynomial-growth Schreier graphs — telling us whether transport follows the group or the graph. If IMG walks ballistically, the Cantor-spectrum transport of run 046 belongs to Grigorchuk's particular substitution and not to intermediate growth, which is the same question run 043 asks of the bulk. If Aleshin does not saturate, the boundary at this size is too small to see non-amenability, and the run repeats at level 12.
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| boundary | diameter | t_c | β | floor·N | γ | PR/N | distinct eigenvalues |
|---|---|---|---|---|---|---|---|
| odometer ℤ | 512 | 109 | 1.000 | 2.0 | 0.821 | 0.998 | 513 |
| Grigorchuk | 768 | 104 | 0.800 | 15.9 | 0.545 | 0.288 | 956 |
| IMG(z² + i) | 105 | 54 | 0.440 | 19.0 | 0.283 | 0.305 | 966 |
| Basilica | 74 | 43 | 0.595 | 23.0 | 0.441 | 0.135 | 475 |
| Aleshin | 6 | 3.6 | saturated | 4.3 | 0.892 | 0.949 | 513 |
Scored as written. Per-group clauses all hit; Aleshin alone saturated, as predicted; the internal β order missed; no falsifier. Unanticipated.
The anisotropy dial (unsealed, qwalk_dial.json). Grigorchuk, weights (1, t, t², t³):
| t | β (L10 / L12) | γ (L12) | D₂ (L10 / L12) |
|---|---|---|---|
| 0.95 | 0.96 / 0.96 | 0.70 | 0.65 / 0.59 |
| 0.90 | 0.94 / 0.94 | 0.56 | 0.55 / 0.47 |
| 0.80 | 0.92 / 0.91 | 0.43 | 0.42 / 0.37 |
| 0.70 | 0.91 / 0.90 | 0.38 | 0.37 / 0.32 |
| 0.50 | 0.80 / 0.79 | 0.40 | 0.37 / 0.28 |
| 0.30 | 0.62 / 0.60 | 0.41 | 0.35 / 0.31 |
| 0.20 | 0.56 / 0.54 | 0.44 | 0.38 / 0.33 |
| 0.10 | 0.58 / 0.58 | 0.52 | 0.39 / 0.38 |
Also: b, c, d weighted equally (½ each, a = 1) → β = 1.000, PR/N = 1.000 — periodic; a permuted assignment (⅛, ½, 1, ¼) → β = 0.72 with PR/N = 0.021, strongly localized. D₂ is the correlation dimension of the local spectral measure at v₀ from the two-point correlation sum over ε ∈ [10⁻³, 10⁻⁰·⁵]; at N = 1024–4096 it is a rough number, and it drifts downward with N.
What the dial says. "How quasicrystalline" is continuous, not binary: the spreading exponent moves smoothly with the anisotropy, and it is stable in N. But the Cantor spectrum's dimension does not follow it all the way — D₂ saturates near 0.3 for t ≤ 0.7 while β falls another 0.35. The Ketzmerick–Petschel–Geisel inequality β ≥ D₂ (from memory) holds in every row, and γ tracks D₂ as the time-averaged return probability should; so past t ≈ 0.7 the slowing of the walk is carried by the eigenfunctions' multifractality, not by the spectral measure. That is the next observation if this thread continues: the generalized dimensions of the eigenstates against t.
What run 047 adds to the cell. Transport on the boundary reads the nucleus coordinate: contracting → anomalous, non-contracting → expander, with the lattice ballistic. Run 043 found the same coordinate in bulk expansion (contracting 0.37–0.51, free 0.85). The growth class — intermediate versus exponential-amenable — orders neither. Basilica's split (043: amenable geometry, expander synchronisation) now reads: amenable in geometry and in boundary transport, expander only in bulk Kuramoto, and that instrument's stability at high degree is the run's own open question.
Eigenstates along the dial (unsealed, same afternoon; eigstate_dims.json). The correlation dimension of the eigenstates, D₂^ψ from I₂ = Σ|ψ|⁴ ~ N^(−D₂^ψ) fit across levels 10–12 and averaged over all states, falls monotonically with the anisotropy — 0.88, 0.80, 0.73, 0.72, 0.70, 0.64, 0.61, 0.59, 0.56 at t = 0.95 … 0.20 — in step with β (0.96 … 0.54), where the spectral measure's D₂^μ had stopped moving at t ≈ 0.7. The lower bound β ≥ D₂^μ/D₂^ψ holds in every row (the ratio runs 0.44–0.68) and is loose. So the dial past t ≈ 0.7 is carried by the localisation of the wavefunctions, not by the spectrum: "the rest is in the wavefunctions" is a measurement.
generated 2026-10-25the lab index