Probe P-048 argues that block renormalization fails to converge on the Sierpiński-type carpets because they are infinitely ramified (blocks share whole edges). The Grigorchuk ⟨b,c,d⟩-coset graphs are self-similar hosts on the other side of that line. Is their Laplacian spectrum an exact renormalization — every eigenvalue new at level n a preimage, under the one fixed map φ(μ) = μ(8 − μ)/2, of an eigenvalue at level n − 1 (the half-angle rule 4 − 4cos θ ↦ 4 − 4cos(θ/2))? Eigenvalues inherited from level n − 1 are excluded from the test: the level-n graph covers the level-(n − 1) graph, so their presence is automatic.

Sealed 2026-09-29 before any level-3 or level-4 spectrum was computed for this run and before bands5_vals_to800.npy was opened (codifier). Pre-seal glimpse, from run 039's verdict text only: the four level-5 band values quoted there map under φ onto level-4 bands to six places — φ(0.0768589) = 0.304482 = 4 − 4cos(π/8), φ(0.1362967) = 0.535898 = 4 − 4cos(π/6), φ(0.149230) = 0.585785 ≈ 2 − √2, φ(0.195774) = 0.763932 = 4 − 4cos(π/5) — and the one straggler, 0.1996 (multiplicity 1, which run 039 counted into the π/10 band "by the doubling"), maps to 0.77848, which matches no level-4 value the codifier knows. The glimpse is what makes the claim worth sealing; the whole-spectrum test, the level-3 → 4 step and the controls are what it has not seen. Reviewed before commit by a second reader (OPS.md); its four edits — the loop convention, a second (abelian 2-power) control, small-denominator wording, and iterates of φ reported for orphans — are folded in.

The reductio this seal passed first (OPS.md, the reductio pass). Invariance: eigenvalues of level n − 1 reappear at level n by the covering alone, so "old bands land on new bands" would pass by construction; only non-inherited eigenvalues are tested. Twins: a random cover has the same size, degree and covering relation to level 3 but no self-similarity; an abelian 2-power cover shares the doubling structure too. Both are controls, and if either shows no orphans the test is empty. Null: at tolerance 10⁻⁸ in [0, 8] a stray coincidence is negligible; at 10⁻⁵ on level 5 it is not negligible for dense spectra, which is why level 5 is restricted to the converged low eigenvalues and the level-3 → 4 step carries the weight. Neighbour: on the carpets the analogous composition is not exact — run 059's composition defect D₂(p*) ≠ 0 — which is the contrast this run is for.

Why it should hold, and why it might not. Exact spectral decimation is known for the Schreier graphs of this group acting on the level-n vertices (Bartholdi–Grigorchuk, as the codifier recalls it; not checked). The coset graphs here are much larger quotients of the group's Cayley graph, where the codifier knows of no such theorem. Against: the straggler; and the level-5 bands born at each level (runs 038–039) may include families that come from no level-4 eigenvalue.

What the outcomes mean. Confirmed: on a finitely ramified self-similar host the spectrum is an exact renormalization, as the carpets' crossing probability is not — the ramification line of P-048 has a second, independent thread behind it, and the spectral side of runs 038–039 is closed-form at every level. Falsified: the orphans are where the self-similarity breaks; the level at which they first appear is the next object. The minimum cut (11/16 at levels 4 and 5, 16/16 at level 3) is not part of this test: "spectra recurse, cuts don't" is scored only on its spectral half.

Scored 2026-09-29 — falsified

Scored by the codifier from R24_grig5_bisection/RESULTS064.md and decim064.json (8417131), as written. The compute chat left the prediction and falsifier lines unread.

Controls pass, so the run stands. C1 (random lift): 992 of 992 non-inherited distinct eigenvalues are orphans; C2 ((ℤ/2)⁵ voltage lift): 932 of 932. Neither shows a single decimated eigenvalue, so the classification separates the level tower from both covers.

The falsifier fires at both levels. Level 4 against level 3: 18 orphans of 24 non-inherited distinct eigenvalues (75%). Level 5 against level 4, converged eigenvalues only: 2 of 6 (33%; a small denominator, accepted as sealed). The orphans are not a wrong map: none has φ^k(μ) in the level-below spectrum for any k ≤ 5. The level-5 orphan 0.1492411 sits 1.1×10⁻⁵ from the decimated 0.1492303 and misses its image by 4.2×10⁻⁵, four times the tolerance the error budget justified; it is an orphan as sealed. Coverage at level 5 was thin — 178 of 832 eigenvalues checkable — so level 4, with its full dense spectrum, carries the verdict.

What is left standing (reported, not scored). The decimated part is exact and orderly: every decimated eigenvalue has twice the multiplicity of its image (level 4: 8 against 4; level 5: 16 against 8), and run 039's four quoted level-5 bands are all decimated. The orphans at level 4 are 2 ∓ √2, 3 ∓ √3, 4 ∓ √5, 6 ∓ √5 — roots of quadratics, multiplicities 16–40 — and 2 − √2, an orphan at level 4, is the image of a decimated level-5 family. The picture the data suggest: each level births new families that no map from the level below produces, and from then on decimates them with doubling multiplicity. That is a lead, not a result — at level 5 the birth-then-decimate reading can be checked only once the eigenvalues in [0.136, 0.419] converge.

What it means for P-048. The seal's contrast — finitely ramified hosts renormalize exactly, carpets do not — is not supported on this host. Caveat the seal did not state: the ⟨b,c,d⟩-coset graphs are quotients of Cayley graphs, not the level-n Schreier graphs for which exact decimation is classical, and the codifier did not check that these coset graphs are finitely ramified. So the run falsifies "this tower is an exact φ-renormalization", and leaves open whether the ramification line is wrong or was tested on the wrong object. Next exact test: the same classification on the level-n Schreier graphs (2ⁿ vertices), where the classical result says zero orphans — the control the seal should have had.

Pointers corrected 2026-10-05 (codifier; no sealed field, score or number is changed). The script: line was written with the seal, before the script existed, and read: script: R24_grig5_bisection/ (to be written by the Code chat; reads grig3_q., grig4_q., bands5_vals_to800.npy). It now names what was written and committed: R24_grig5_bisection/decim064.py. Found by sync_cases.py's record-field check (docs/REPO-AUDIT.md 5.4).