Run 044's bulk clause was confirmed — on the Grigorchuk level-4 coset graph the quantum walker stays near its origin at long times (PR/N = 0.096, return floor 76/N) against a random 4-regular graph's PR/N = 0.99 — and P-045 reads it as a transport signature of the intermediate-growth cell. N-038's reductio audit pointed out that the only control is a random graph, so the instrument separates "Cayley-like with degenerate spectrum" from "random", not "intermediate growth" from "abelian". Does an abelian group with a flat-band spectrum show the same signature, and does an abelian group whose spectrum is not flat-banded (beyond one accidental band) not?

Sealed 2026-10-01, before either abelian host has been walked (codifier). Known at sealing: run 044's five hosts and numbers (coset4: PR/N 0.096, floor 76.4/N, γ 0.886, 31 distinct eigenvalues; rr4c: 0.992, 2.9/N), runs 045–047, and N-038's audit line: "the hypercube Q₁₀ (abelian) has return floor 180/N against the coset graph's 76/N" — the floor is exact arithmetic (C(20,10)/2²⁰ = 0.176), the PR/N is not, and PR/N is the clause run 044 said was "a claim about localization, not about degeneracy".

What this run is for. The owner asked that P-045 close clean. Its boundary clause was settled by runs 044–046 with a positive control (the Fibonacci chain) and a negative one (the path); its bulk clause was confirmed against a random graph only. The twins question (N-038's rule: does the control differ from the object only in the variable under test?) says the missing control is an abelian group of the same size, and there are two kinds: one whose spectrum is as degenerate as the coset graph's (the hypercube, 11 bands) and one whose is not (the torus: 145 distinct eigenvalues, mostly of multiplicity 4 and 8, with one accidental 62-fold band at λ = 4 where the two cosines cancel — an accidental degeneracy of the kind the record's note on accidental symmetries describes, named here so RESULTS088 does not discover it). Run 044's own body said PR/N could not be read off the multiplicities because degenerate eigenvectors might delocalize within a band as an expander's do. The hypercube is the exact case where that is checkable by hand.

The codifier's lead, held with conviction. On the hypercube the walk factorizes over the ten coordinates: the amplitude at Hamming distance k is cos^(10−k)(t)·(i sin t)^k up to a global phase, so the time-averaged occupation at the origin and at the antipode is C(20,10)/2²⁰ = 0.176 each and PR/N is near 0.015 — more localized by both measures than the coset graph (0.096). So A holds, and the bulk signature of P-045 is produced by the most abelian graph there is. On the torus the walker spreads ballistically; its PR/N is computable too — 0.72 from the Fourier projectors, the shortfall from 1 being the 62-fold band — so B holds by arithmetic as well, and this run is a control in both clauses: its content is the reading, that run 044's thresholds admit an abelian group on one side and reject one on the other, which is what P-045 lacked. Where the lead breaks: only if qwalk.py's numbers differ from the arithmetic, which control (iii) would catch first.

Power, stated. Both clauses are deterministic and both are known by arithmetic before the run (0.015 and 0.72 against thresholds 0.30 and 0.50); the thresholds are run 044's own, reused so that the twins are judged by the instrument that judged the cell. A control whose outcome is known is a control; what this run adds to the record is the reading, and the reported comparison of the cell's 0.096 with its own degeneracy null.

The reductio this seal passed first. By construction: the hypercube's localization is periodicity (the walk returns exactly at t = π), not disorder; that is precisely the objection — run 044's statistics cannot tell periodicity from localization, and the coset graph, whose spectrum is 31 integers' worth of bands, may be quasi-periodic in the same way. Negation: if A fails (PR/N ≥ 0.6 on q10), the within-band interference argument of run 044 is right and the coset graph's 0.096 is a cell property; P-045 closes as confirmed with its control. Cost: two eigendecompositions at N = 1024, under a second each; any machine.

Reported, not scored. - The torus's eigenvalue multiplicities as found, against the stated 2 / 44 / 98 / 1 (the accidental band). - The full table of all seven hosts (run 044's five from qwalk.json, plus q10 and torus32): t_c, β, floor·N, γ, PR/N, distinct eigenvalues; the degeneracy null Σ(m_λ/N)² beside the floor for q10, torus32 and coset4. - The time-averaged occupation profile on q10 by Hamming distance, against cos^(2(10−k)) sin^(2k) averaged. - What P-045's bulk clause becomes under each outcome, in one sentence each, for the owner to close it.

Reviewed before commit by a second reader (OPS.md); its edits are folded in: the torus's accidental 62-fold band and exact floor (10.57/N, above the first draft's 10/N margin) found by the reader, both floors moved to control (iii) and both clauses scored on PR/N; the torus PR/N 0.72 computed; the B-alone readings; P-024's rung-2 correction (S₂ sees the side but does not close).

Scored 2026-10-02 — confirmed

Scored by the codifier from RESULTS088.md and qwalk088.json, as written.

Controls first — all three pass. (i) q10: 1,024 vertices, 5,120 edges, degree 10, eleven Laplacian eigenvalues 0, 2, …, 20 with binomial multiplicities; torus32: 2,048 edges, degree 4, 145 distinct eigenvalues with the multiplicities the seal named, the 62-fold band at λ = 4 included. (ii) coset4 and rr4c_1024 reproduce run 044's PR/N and floor·N to four figures. (iii) q10's floor 0.17619705 = C(20,10)/2²⁰; torus32's 0.01032066 = 10,822/2²⁰, both to eight figures.

(A) Confirmed. q10, PR/N = 0.0152 ≤ 0.30. The hypercube — the most abelian graph there is — meets run 044's bulk threshold by a factor of six, with its floor 180/N against the coset graph's 76/N. The occupation profile by Hamming distance matches Γ(10 − k + ½)Γ(k + ½)/(π·10!) at every k.

(B) Confirmed. torus32, PR/N = 0.721 ≥ 0.50, β = 1.028 (ballistic). The shortfall from 1 is the accidental 62-fold band, as the lead said.

What it settles. Run 044's bulk clause — "the flat bands hold the walker near its origin at long times" (PR/N 0.096, floor 76/N on the level-4 coset graph) — is reproduced by an abelian group with flat bands and rejected by an abelian group without them. The signature is spectral degeneracy, not intermediate growth. P-045's bulk reading is rewritten as that; the boundary reading (run 046's anisotropic quasicrystal) is untouched. One further number worth having: on all three vertex-transitive hosts the floor equals its degeneracy null Σ(m_λ/N)² exactly (coset4 included: 0.074629 both ways), so the return floor carries no information beyond the multiplicities on any of them, and the only statistic that could still separate the cell from its degeneracy is PR/N against a null that is not yet defined — reported as the next object, not claimed.

What it does to PG-02. First run scored under the programme. L5: the bulk clause is a degeneracy signature; what remains open on the line is whether the cell's PR/N (0.096, between the hypercube's 0.015 and the torus's 0.72) exceeds what its own 31 bands would give on a graph with no group — the null PG-02's L5 names and no run yet builds. N-038's twins rule, which this run is the first use of, is the candidate for section 4 once adopted in OPS.md.

Lab notes. Deterministic; two eigendecompositions at N = 1,024; qwalk.py unchanged, v₀ taken as the vertex qwalk.py chooses (510) and recomputed at v₀ = 0 to the same PR/N on both vertex-transitive hosts. Results held by the lab database's quarantine until this scoring (the database chat reported qwalk088.json as sealed and unloaded), which is the mechanism working as designed.