run 044Falsified
../lab/runs/044-quantum-walk-on-the-cell.mdPredictionPath — 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 — 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. Controls all pass — path β = 1.000, PR/N = 1.000, floor 1.5/N; random 4-regular saturated at t_c = 1.7, PR/N = 0.992, floor 2.9/N; Fibonacci chain β = 0.763, the positive control for anomalous spreading, inside [0.30, 0.95]. The bulk clause is CONFIRMED and strong: on the coset graph the long-time participation ratio is PR/N = 0.096 (≤ 0.30) and the exact return floor 0.0746 = 76.4/N (≥ 0.02) — the walker occupies a tenth of the graph where the expander's occupies all of it; the flat bands localize. The reason the boundary clause failed is the codifier's operator, not the host: the seal named the graph Laplacian, and in the Laplacian every loop cancels — a loop adds to the diagonal and to the degree equally — while the Schreier graph's aperiodicity lives entirely in its loops (which of b, c, d fixes a leaf follows the substitution). Bartholdi–Grigorchuk's Hecke operator and Grigorchuk–Lenz–Nagnibeda's Schrödinger operator are the adjacency operator with loops on the diagonal. The seal measured a path with alternating hoppings 1 and 2 and found a path. Scored as written; the corrected operator is sealed as run 045 with the same predictions, before any Hecke walk was run.
Sealed before any walk was run (2026-09-28, codifier). What was known at sealing: the level-4 coset graph's band structure (runs 038–039), the level-5 low spectrum, and the two literature facts in P-045 (Bartholdi–Grigorchuk's Hecke-type operators; Grigorchuk–Lenz–Nagnibeda's aperiodic Schrödinger reading of the Schreier Laplacian — the second verified by search on 2026-09-28, the first the day before). No walk on any of these graphs had been computed.
Why the boundary prediction is the one that matters. A Cantor spectral measure with fractal dimension D bounds the spreading (Ketzmerick–Petschel–Geisel, from memory: β ≥ D₂-type bounds) between localization and ballistic transport; on the Fibonacci chain the exponent is known to sit well inside (0, 1). If the Schreier graph's walk gives the same, the boundary of the intermediate cell is a quasicrystal in the operational sense — the same class of transport as R5's fractal Středa hosts, from a group rather than a substitution. If it is ballistic, the aperiodicity is in the potential but not in the transport at this size, and the reading stays a sentence.
Why the bulk prediction is not a tautology. The return floor P̄_∞ on a vertex-transitive graph is Σ(d_b/N)² over bands and can be read off the multiplicities — it will be large because the bands are. The participation ratio cannot: it depends on how the degenerate eigenvectors interfere at the origin's neighbourhood, and an expander-like delocalization within each band would give PR/N near 1 even with a large floor. PR/N ≤ 0.30 is a claim about localization, not about degeneracy.
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| host | diameter | t_c | β | floor·N | γ | PR/N | distinct eigenvalues |
|---|---|---|---|---|---|---|---|
| path, from the middle | 512 | 109 | 1.000 | 1.50 | 0.836 | 1.000 | 1024 |
| Fibonacci chain, hoppings 1 and ½ | 512 | 100 | 0.763 | 11.58 | 0.456 | 0.373 | 1024 |
| Grigorchuk Schreier graph, level 10 | 768 | 99 | 0.973 | 1.50 | 0.816 | 0.999 | 1024 |
| coset graph, level 4 | 12 | 1.9 | saturated | 76.42 | 0.886 | 0.096 | 31 |
| random 4-regular, seed 1 | 8 | 1.7 | saturated | 2.88 | 0.873 | 0.992 | 1024 |
Scored as written. Boundary: β = 0.973 ≥ 0.95 — falsifier fires. Bulk: PR/N = 0.096 ≤ 0.30 and floor = 0.075 ≥ 0.02 — confirmed. Controls: path, random and Fibonacci all inside their bands; the positive control passed, so the instrument can see anomalous spreading at this size and the boundary result is a result, not a blind spot.
Why the boundary walk is a path. The Schreier graph file carries 2,562 edges on 1,024 leaves including loops; the combinatorial Laplacian discards every loop identically, and what remains is a path with hoppings alternating 1 (a-edges) and 2 (the two of b, c, d that move the leaf) — a period-2 chain, absolutely continuous spectrum, ballistic. The 1,024 distinct eigenvalues and the path-like floor, decay and participation all say the same. The aperiodicity of the Grigorchuk Schreier graph is which of b, c, d is the loop at each leaf — a diagonal potential the Laplacian cannot carry. The codifier chose the wrong operator; the physics literature's operator is the adjacency operator with loops. Run 045 seals that one, with 044's predictions unchanged.
What stands. The bulk half is a transport signature of the crystal: the coset graph's 31 distinct eigenvalues in 1,024 dimensions — the flat bands of runs 038–039 — keep a walker on a tenth of the graph forever, against an expander's full delocalization at the same N and degree. The flat bands are not only where the minimum bisection lives; they are where a quantum walker gets stuck.
generated 2026-10-25the lab index