Run 064 found that the ⟨b,c,d⟩-coset tower is not an exact renormalization under φ(μ) = μ(8 − μ)/2 (18 of 24 new level-4 eigenvalues orphans). Two things were never checked. (A) Does the classifier return zero orphans where exact decimation is a theorem — the level-n Schreier graphs Γₙ of the Grigorchuk group on the 2ⁿ vertices of level n, under their published map? (B) Since b, c, d all fix the vertex 1ⁿ, each coset graph covers Γₙ; is 064's negative an artifact of using φ rather than the Schreier map — does the coset tower decimate under the Schreier map?

Sealed 2026-09-29, after run 064 was scored and before any Schreier graph was built or any coset eigenvalue was classified by T (codifier). Known at sealing: run 064's counts and its φ-orphan values (2 ∓ √2, 3 ∓ √3, 4 ∓ √5, 6 ∓ √5 at level 4; 0.1492411 and 0.2052762 at level 5); the published Γₙ spectrum, read from a summary of arXiv:1503.06887 and not from the original paper — which is why clause A checks the formula rather than trusting it. Glimpse declared: while designing, the codifier evaluated T for one of 064's φ-orphans, μ = 2 − √2 (T = √2 − 1, T² − 2 ≈ −1.828); it does not know the level-3 coset spectrum, so no classification outcome was seen. T was evaluated on no other value. Reviewed before commit by a second reader (OPS.md); its three edits — one Laplacian 4I − P for both towers, this glimpse declaration, and clause B counted absolutely with the covering-forced eigenvalues removed — are folded in.

Why this run exists. Run 064's seal assumed the coset graphs sit on the finitely ramified side of P-048's line and used a map read off a pattern in run 039. Neither was checked. Clause A is the control that run lacked: a tower where the answer is a theorem. Clause B asks whether 064 failed because of the map. Both towers use the one matrix L = 4I − P, so loops (present in Γₙ, absent in 064's coset graphs) enter the same way in both.

The reductio this seal passed first. Invariance: inherited eigenvalues are automatic under a covering and are excluded, as in 064. By construction: in clause A the formula guarantees zero orphans if the classifier is right, so A is a control, not a finding — it is scored only to validate B. Null: for B, the T-map is quadratic in T(μ), so each level-(n − 1) eigenvalue has up to four preimages in μ; at 10⁻⁸ on a dense spectrum, coincidences remain negligible. Neighbour: if B is falsified, the coset graphs would renormalize under a map that belongs to the Schreier graphs they cover — a strong structural statement, so the prediction is set against it.

What the outcomes mean. A confirmed and B confirmed: the classifier is sound, the Schreier tower renormalizes as the theorem says, and the coset tower does not under either map — its new families come from representations the Schreier graphs do not see. A confirmed and B falsified: run 064's negative was the wrong map, and the coset tower is an exact renormalization after all. A falsified: the instrument is wrong and nothing in 064 or 065 stands until it is fixed.

Scored 2026-09-29 — confirmed on both clauses, with the seal's covering premise false

Scored by the codifier from R24_grig5_bisection/RESULTS065.md (7c5217e), as written. Run 064's coset matrices were checked to equal 4I − P entry by entry from the full Cayley graphs.

Clause A (the control): confirmed. Γ₃ … Γ₁₀ match the published spectrum to within 7×10⁻¹⁵, every eigenvalue simple, and the T-classification finds zero orphans at every level 4–10. The classifier is sound on a tower where the answer is a theorem.

Clause B: confirmed as written. Under the Schreier map the coset tower still has orphans: at level 4, 22 of 22 in the denominator (7 inherited, 2 decimated, 22 orphans); at level 5, 6 orphans and 0 decimated among the converged values. Run 064's negative is not a wrong-map artifact. Of 064's φ-orphans only 3 ∓ √5 become T-decimated, and 064's six φ-decimated values are T-orphans: the two maps explain disjoint pieces, and neither explains most of the spectrum.

The premise was false, and the seal should not have stated it. The seal said each coset graph covers Γₙ because b, c, d fix 1ⁿ; the second reader agreed. The covering check fails — 2 of 8 Γ₃ eigenvalues in the level-3 coset spectrum, 4 of 16 at level 4, 1 of 5 in range at level 5 — because the stored coset graphs are not Schreier graphs of the left action: they are the Cayley graph with each ⟨b,c,d⟩-coset contracted, so only a-edges survive between cosets (each coset meets four others through its four elements times a). Fixing 1ⁿ makes ⟨b,c,d⟩ lie in a vertex stabiliser, which gives a covering for the left-coset Schreier graph, not for this contraction. The clause-B denominator therefore removed Γ₄ eigenvalues for a reason that does not apply; it removed none that mattered (22 of 24 non-inherited were orphans either way). Both reviews missed it: an existence claim about a map between two graphs was checked in the abstract, not on the stored graph.

What it means. The finitely ramified Schreier tower renormalizes exactly (a theorem, reproduced here as a control); the carpets do not (run 061). That is consistent with probe P-048's ramification line, but it adds nothing new — the Schreier result was known. The coset graphs of runs 031–039 are a third kind of object, neither the Schreier graphs nor carpets, and runs 064–065 show they are not an exact renormalization under either map; whether they are finitely ramified at all has not been checked. Lesson for OPS.md's reductio pass: push 2 (invariance) should include "is the object the one the theorem is about?".

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). It now names what was written and committed: R24_grig5_bisection/schreier065.py. Found by sync_cases.py's record-field check (docs/REPO-AUDIT.md 5.4).