A quantum walk on the intermediate-growth cell — is the boundary a quasicrystal and the bulk a flat-band localizer, or is the analogy decorative?

The claim to test. Quantizing a self-similar group's action on its boundary gives the Hecke-type operators of Bartholdi–Grigorchuk, i.e. continuous-time quantum-walk Hamiltonians on the Schreier graphs, and Grigorchuk–Lenz–Nagnibeda (Math. Ann. 2018) show the Schreier-graph Laplacian is a Schrödinger operator with aperiodic substitution potential — a one-dimensional quasicrystal. If that reading is real, a quantum walk on the boundary spreads anomalously — an exponent strictly between ballistic (a path) and localized — and a quantum walk in the bulk (the coset graph) is partly localized by the flat bands run 039's spectrum found (multiplicities of 8–250 where a random graph has none). If it is decorative, the boundary walk is just ballistic on a decorated path and the bulk walk delocalizes like an expander's.

Statistics (all from one dense eigendecomposition per host, N ≤ 4096; the walk is ψ(t) = e^{−iLt} δ_{v₀} with L the graph Laplacian, multi-edges and loops as the graph has them): 1. Spreading exponent β: m(t) = Σ_v |ψ_v(t)|² d(v₀, v)² with d the graph distance; m_sat = the mean of m over t ∈ [100, 200]; t_c = first t with m ≥ m_sat/2; β = half the log–log slope of m on [0.5, t_c]; reported only if t_c ≥ 4, else "saturated". 2. Return-probability floor P̄_∞ = Σ over distinct eigenvalues λ of (Σ_{v ∈ E_λ} |⟨v|v₀⟩|²)² — exact from the spectrum — and the decay exponent γ of the time-averaged return probability on the window where it exceeds twice the floor. 3. Long-time participation ratio PR/N, PR = 1/Σ_v p̄_v², p̄_v the time-averaged occupation.

Hosts. Boundary: the Grigorchuk Schreier graph at level 10 (1,024 leaves, R27_tree_groups/schreier/grig_schreier10.edges), v₀ a leaf of median eccentricity. Bulk: the level-4 coset graph (1,024). Controls: the path P₁₀₂₄ from its middle (lattice; ballistic, β = 1); a Fibonacci hopping chain, N = 1024, hoppings 1 and 1/2 by the Fibonacci word (a quasicrystal — the positive control for "anomalous"); a random 4-regular graph, N = 1024, seed 1 (expander). Reported, not sealed: the same on the Cayley graph at 4,096 and on the other four groups' Schreier graphs from R27.

Sealed as run 044 — the predictions are in the run file, not here.

Outcome, same day. Runs 044 (Laplacian: boundary ballistic, falsified; bulk localized, confirmed), 045 (isotropic Hecke: falsified identically — one loop per leaf makes it periodic), 046 (anisotropic Hecke: confirmed, β = 0.800 beside the Fibonacci chain's 0.688). The reading is a transport signature. The operator that carries the quasicrystal is the anisotropic one, as Grigorchuk–Lenz–Nagnibeda's theorem says; the isotropic operators see a decorated path.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

The boundary question was answered by runs 044–047 (run 046: β = 0.800, anomalous; run 047 extends it to five groups). State left open for the owner to close; N-038 notes the bulk clause lacks an abelian twin control.

• Run 039 — falsified (2026-09-28, measured). lab/runs/039-two-band-skeleton-level5.md: FALSIFIED as written — the falsifier's clause "a ≤ 0.99" fires (agreement 0.97715) — with four of five prediction clauses hit and the cut exact.
• Run 044 — falsified (2026-09-28, measured). lab/runs/044-quantum-walk-on-the-cell.md: FALSIFIED on the boundary clause — the Grigorchuk level-10 Schreier graph walk is ballistic under the sealed Laplacian, β = 0.973 ≥ 0.95, with the return floor (1.5/N), decay (γ = 0.82) and participation (0.999) of the path control.
• Run 045 — falsified (2026-09-28, measured), via the builds_on thread. lab/runs/045-quantum-walk-hecke-operator.md: FALSIFIED 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 exactl…

Control sealed, 2026-10-01 (codifier). N-038's missing control for the bulk clause — an abelian twin of the same size — is sealed as run 088 (lab/runs/088-abelian-twin-control.md): the hypercube Q₁₀ (abelian, eleven bands) and the torus ℤ₃₂² (abelian, no flat bands beyond one accidental 62-fold one) through run 044's pipeline and thresholds. The codifier's lead is that the hypercube meets the bulk thresholds by arithmetic, which would rewrite the bulk clause as a degeneracy signature rather than a growth-class one; the probe closes on the owner's reading of 088 either way.

Run 088 scored, 2026-10-02 (codifier): confirmed. The hypercube reproduces the bulk signature (PR/N 0.015 against the cell's 0.096) and the torus does not (0.72). The bulk half of this probe's claim — "the bulk a flat-band localizer" — is true as a statement about spectral degeneracy and says nothing about intermediate growth; the boundary half stands on run 046. Ready to close on the owner's reading: boundary, a transport signature of the aperiodic potential under the theorem's operator; bulk, a degeneracy signature any flat-band graph shows. The one open item is PG-02's L5 null.