Runs 044 and 045 walked the Grigorchuk Schreier graph under isotropic operators (the Laplacian, then the equal-weight Hecke operator) and found a ballistic path both times — every leaf carries exactly one loop, so equal weights make the loop potential constant and the operator periodic. Grigorchuk–Lenz–Nagnibeda (arXiv:1412.6822) prove the Cantor spectrum for the anisotropic Laplacians, with different weights on the generators, because the aperiodicity of the Schreier graph is which of b, c, d fixes each leaf. Last operator this thread will try: does the boundary walk spread anomalously under an anisotropic Hecke operator?

slug: quantum-walk-anisotropic
state: run
created: 2026-09-28
sealed: 2026-09-28
measured: 2026-09-28
statistic: For each host, one dense eigendecomposition of H = P_a + ½·P_b + ¼·P_c + ⅛·P_d on the level-10 Schreier graph, P_g the permutation matrix of generator g on the 1,024 leaves (symmetric, since each g is an involution; a fixed leaf is a loop on the diagonal), built from R23_ising_barrier/grig.py's perm(); plain adjacency on the loop-free controls and the walk ψ(t) = e^{−iHt} δ_{v₀}: (1) the spreading exponent β, half the log–log slope of the mean-square graph distance m(t) on [0.5, t_c], t_c the first time m reaches half its saturation mean over [100, 200], reported only if t_c ≥ 4; (2) the exact return-probability floor P̄_∞ = Σ_λ w_λ² with w_λ the weight of δ_{v₀} in the λ-eigenspace, and the decay exponent γ of the time-averaged return probability on the window where it exceeds 2·P̄_∞; (3) the long-time participation ratio PR/N of the time-averaged occupation. Hosts, N = 1024 throughout: the Grigorchuk level-10 Schreier graph from a leaf of median eccentricity (boundary); the level-4 ⟨b,c,d⟩-coset graph (bulk); the path from its middle (lattice control); a Fibonacci hopping chain, hoppings 1 and 1/2 by the Fibonacci word (quasicrystal positive control); a random 4-regular graph, seed 1 (expander control). Script R23_ising_barrier/qwalk.py with --aniso.
script: R23_ising_barrier/qwalk.py
artifacts: [R23_ising_barrier/qwalk_aniso.json]
outcome: confirmed
builds_on: [045]
concepts: [growth-of-groups, random-walks]
programme: PG-02
source: ../lab/runs/046-quantum-walk-anisotropic.md

PredictionPath — β ∈ [0.95, 1.05], PR/N ≥ 0.5, P̄_∞ ≤ 5/N. Fibonacci chain — β ∈ [0.30, 0.95] (anomalous, the positive control). Grigorchuk Schreier graph — β ∈ [0.30, 0.90], strictly anomalous like the Fibonacci chain and unlike the path, with γ ∈ [0.20, 0.95]. Random 4-regular — saturated (t_c < 4), PR/N ≥ 0.6, P̄_∞ ≤ 5/N. Coset graph — PR/N ≤ 0.30 and P̄_∞ ≥ 0.02: the flat bands hold the walker near its origin at long times, against an expander's PR/N ≥ 0.6.

FalsifierSchreier graph β ≥ 0.95 (the boundary walk is ballistic — a decorated path, and the quasicrystal reading is decorative) or β ≤ 0.10 (localized outright, which the Cantor spectrum does not predict either); coset graph PR/N ≥ 0.60 (the flat bands do not localize the walk and the bulk transport is expander-like); or the positive control failing — Fibonacci β outside [0.30, 0.95] — which withdraws the run, since the instrument then cannot see anomalous spreading at N = 1024. Between prediction and falsifier is unanticipated.

VerdictCONFIRMED. Under the anisotropic Hecke operator a + b/2 + c/4 + d/8 the level-10 Schreier graph walk is anomalous — β = 0.800 ∈ [0.30, 0.90], γ = 0.545 ∈ [0.20, 0.95], return floor 15.9/N, long-time participation PR/N = 0.288, 956 distinct eigenvalues in 1,024 dimensions — next to the Fibonacci chain on every statistic (β 0.688, γ 0.454, floor 9.7/N, PR/N 0.480) and away from the path (1.000, 0.836, 1.5/N, 1.000) and from its own isotropic operator (0.973, 0.816, 1.5/N, 0.999). Controls all inside their bands; the positive control passed; the bulk clause stands from run 044 (coset graph PR/N = 0.096, floor 76/N). The boundary of the intermediate cell transports like a quasicrystal under the operator whose spectrum Grigorchuk–Lenz–Nagnibeda prove is a Cantor set, and the two isotropic failures (044, 045) were the codifier's operators, not the host: every leaf carries exactly one loop, so equal weights erase the aperiodicity. The quantum reading of P-045 is a transport signature, not a decoration — boundary: quasicrystal; bulk: flat-band localizer.

Sealed 2026-09-28 after run 045 was scored and before any anisotropic walk was run. Hosts, statistics, predictions and falsifiers are 044's, unchanged; only the boundary Hamiltonian is new, and it is the class of operator whose spectrum Grigorchuk–Lenz–Nagnibeda prove is a Cantor set of Lebesgue measure zero. The weights ½, ¼, ⅛ are arbitrary and fixed here so the run cannot tune them; the isotropic weights 1, 1, 1 gave β = 0.973 in run 045. The Fibonacci chain remains the positive control; the path and the random graph do not depend on the weights and their 044 numbers are the calibration. This is the third and last operator: if β ≥ 0.95 again, the quasicrystal reading of the boundary is recorded as decorative at N = 1,024 and the thread stops walking. If β lands with the Fibonacci chain's, the boundary of the intermediate cell transports like a quasicrystal under the operator the literature uses, and 044 and 045 were the codifier learning which operator that is.

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Measured the same hour (one eigendecomposition per host, under a second each)

host	β	floor·N	γ	PR/N	distinct eigenvalues
path	1.000	1.50	0.836	1.000	1024
Fibonacci chain, hoppings 1 and ½	0.688	9.69	0.454	0.480	1024
Grigorchuk boundary, a + b/2 + c/4 + d/8	0.800	15.93	0.545	0.288	956
Grigorchuk boundary, isotropic (run 045)	0.973	1.50	0.816	0.999	1024
random 4-regular	saturated	2.88	0.873	0.992	1024

Scored as written. β and γ inside their bands; controls inside theirs; confirmed.

The three-run arc, for the record. 044 sealed the Laplacian, which cancels loops; 045 sealed the equal-weight Hecke operator, which the one-loop-per-leaf structure makes periodic; 046 sealed the anisotropic operator the literature's theorem is about. Two falsifications, each with its cause counted out of the graph, then a confirmation with the Fibonacci chain as the positive control at every step. The weights ½, ¼, ⅛ were fixed in the seal and not tuned; the walk is anomalous at the first weights tried, which is what a Cantor spectrum of measure zero should give for any generic anisotropy. Whether β depends on the weights is an observation for later, not a clause here.

What it adds to the cell. The growth trichotomy now has a transport reading measured on one tree: lattice (the odometer's boundary is a path — ballistic, β = 1), intermediate (Grigorchuk's boundary — anomalous, β ≈ 0.8, with a Cantor-spectrum operator), expander (a random regular graph — saturated at t ≈ 2). And the bulk of the intermediate host localizes a walker on a tenth of the graph by its flat bands, which is the spectral skeleton of runs 038–039 seen dynamically.

generated 2026-10-25the lab index