Run 044 sealed the graph Laplacian as the walk Hamiltonian, and on the Schreier graph the Laplacian cancels every loop — which is exactly where the graph's aperiodicity lives (which of b, c, d fixes a leaf follows the substitution). Bartholdi–Grigorchuk's Hecke-type operator and Grigorchuk–Lenz–Nagnibeda's Schrödinger operator are the adjacency operator with loops counted on the diagonal. Same question as 044 with the right operator: does the boundary walk spread anomalously under the Hecke operator?

slug: quantum-walk-hecke-operator
state: run
created: 2026-09-28
sealed: 2026-09-28
measured: 2026-09-28
statistic: For each host, one dense eigendecomposition of the adjacency operator H = A with loops counted on the diagonal (the Hecke operator Σ_g g on the Schreier graph; plain adjacency on the loop-free hosts) 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 --hecke.
script: R23_ising_barrier/qwalk.py
artifacts: [R23_ising_barrier/qwalk_hecke.json]
outcome: falsified
builds_on: [044]
concepts: [growth-of-groups, random-walks]
programme: PG-02
source: ../lab/runs/045-quantum-walk-hecke-operator.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.

VerdictFALSIFIED on the boundary clause, identically to run 044 — β = 0.973 under the equal-weight Hecke operator, floor 1.5/N, γ = 0.816, PR/N = 0.999, the path's numbers — and the reason is structural, found by counting: every one of the 1,024 leaves carries exactly one loop (1,022 with one, the two path ends with three), so with equal weights the loop potential is a constant and the Hecke operator is the loop-free adjacency plus the identity, a period-2 chain. The aperiodicity of the Schreier graph is which generator loops at each leaf, invisible to any isotropic operator. Grigorchuk–Lenz–Nagnibeda (arXiv:1412.6822, abstract) prove the Cantor spectrum for the anisotropic Laplacians — unequal generator weights — which is the operator run 046 seals. Controls unchanged: path β = 1.000; Fibonacci β = 0.688 (positive control inside [0.30, 0.95]; adjacency instead of Laplacian shifts it from 0.763); random saturated, PR/N 0.992; coset graph PR/N 0.096, floor 76/N (the bulk clause, not re-tested, identical since the coset graph is regular and loop-free). Scored as written.

Sealed 2026-09-28, minutes after run 044 was scored, before the Hecke-operator walk was run on any host. Everything in 044's seal is carried over unchanged — hosts, statistics, predictions, falsifiers, the positive control — with one substitution: the Hamiltonian is the adjacency operator with loops on the diagonal instead of the Laplacian. On the path, the Fibonacci chain and the random 4-regular graph the two differ only by a constant shift (regular, loop-free) or by the Fibonacci chain's site potential (which 044's Laplacian also carried, as the degree), so their 044 numbers are the calibration; on the Schreier graph the loops are the aperiodic potential and the operator is a different one. What 044 established and this run does not re-test: the bulk clause (coset graph PR/N = 0.096, floor 0.075) — the coset graph is loop-free and 4-regular, so its Hecke operator is 4 − L and the walk is identical.

Why the boundary prediction is the same. The aperiodic potential is the thing the quasicrystal reading rests on; 044 measured the walk without it and found a path. If the potential makes the walk anomalous, β lands where the Fibonacci chain's does; if the potential is too weak at level 10 to matter — the modulation is one unit of on-site energy against hoppings of 1 and 2 — the walk stays ballistic and the reading is decorative at this size, which the falsifier records.

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Measured the same hour

host	β	floor·N	γ	PR/N
path	1.000	1.50	0.836	1.000
Fibonacci chain	0.688	9.69	0.454	0.480
Grigorchuk Schreier, level 10, H = a + b + c + d	0.973	1.50	0.816	0.999
coset graph	saturated	76.42	0.886	0.096
random 4-regular	saturated	2.88	0.873	0.992

Loop census of the level-10 Schreier graph: 1,024 vertices with loops; multiplicity 1 on 1,022 of them, 3 on the two ends; 1,028 loops in all. One loop per leaf means one of b, c, d fixes it — always exactly one — so the isotropic diagonal is flat. The codifier's second wrong operator, and the count that shows why both were wrong. Run 046 is the anisotropic one and the last.

generated 2026-10-25the lab index