At level 4 the 176 minimum bisection of the ⟨b,c,d⟩-coset graph is exactly the sign of its projection onto the lowest 73 Laplacian eigenvectors — the constant, the λ₂ = 4−4cos(π/8) irrep (multiplicity 8, share 0.697), two bands it is orthogonal to (4−4cos(π/6), multiplicity 8, and 2−√2, multiplicity 16; together +0.0002), and the 4−4cos(π/5) band (multiplicity 40, +0.189) — with agreement 1.000 and the cut 176 on the nose. Run 038 showed that the first step of this at level 5 (λ₂ alone) gives 98,288 rather than the ¾-exact 98,304. Is the level-5 minimum still the sign of its projection onto two weighted bands, and is that exact or leading-order?

Sealed before the level-5 eigenvectors beyond the λ₂ irrep existed (2026-09-27, codifier). The level-4 table this is calibrated on was recomputed the same evening by dense diagonalisation; the block script had not yet been run on either level when the seal was written. Between the prediction and the falsifier is unanticipated.

What the level-4 table says, for the record. Cumulative dimension, share, and the cut of the sign-rounded projection:

Two bands carry the minimum; two bands in between carry nothing of it; everything above the fifth band changes no sign. That is the "spectral skeleton" of N-028, stated as a table for the first time.

Why the prediction says leading-order and not exact. Three small-level identities on this host have failed one level up today (N/log N, the fibre count, ¾ − 1/16), each by a part in 10³–10⁴. The share in the λ₂ irrep barely moves between levels (0.697 → 0.684), so the two-band structure is expected to persist; the exactness of the sign pattern is what is expected to go, because the first band's rounding already disagrees with the minimum on 2.7% of cosets at level 5 against 1.0% at level 4. The falsifier is the outcome that would kill the skeleton itself: weight spread beyond two bands, or a two-band rounding that is not the minimum to within 1%.

Cost. The π/10 band's expected multiplicity (twice the level-4 40, as λ₂'s 8 became 16) puts the eigenvector count near 150–200 on a 2²⁰-vertex graph; blocks of 32 with deflation, an hour or two on two cores. Eigenvectors are not committed (≈ 1 GB); the eigenvalue list and the table are.

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Dated note under the seal, 2026-09-27, written after the control passed and while the level-5 blocks were running, before any level-5 band beyond λ₂ had been seen. The prediction's "bands strictly between λ₂ and the π/10 band add at most 0.01" was calibrated on the level-4 list, where the two bands between are π/6 and 2−√2 and carry nothing. But the level-4 → level-5 correspondence is not "same list, doubled": the two bands already known at level 5, 4−4cos(π/16) and 4−4cos(π/12), are the level-4 bands 4−4cos(π/8) and 4−4cos(π/6) under θ ↦ θ/2, i.e. μ = 4 − 4cos(θ/2) with λ = 4 − 4cos θ, which is a spectral-decimation pullback. Under that map every level-4 band below 1.52 lands below 0.40 at level 5 — including 4−4cos(π/4), 1.268 and 1.387, which at level 4 sit above the π/5 band and carry 0.04 of the minimum's weight between them. If the level-5 minimum distributes its weight over pulled-back bands the way the level-4 one does over their sources, the ≤ 0.01 clause fails and the run scores unanticipated on that clause, whatever the rest does. The seal stands as written; this note records that the failure mode was seen before the data, not after. Pullbacks expected below 0.40, if the map is right: 0.0769, 0.1363, 0.149, 0.3045, 0.331, 0.363, 0.382 — seven bands, roughly 340 eigenvectors.

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Dated note under the seal, 2026-09-28 00:20, while block 5 of the M5 run was in progress and before any band beyond 0.20 had been reported. The seal contains an arithmetic error, caught by the Claude Code session running it: it writes "4 − 4cos(π/10) = 0.381966". The formula is right and the number is wrong — 4 − 4cos(π/10) = 0.195774; 0.381966 is 2 − 2cos(π/5), i.e. half the level-4 eigenvalue, which is what halving the value instead of halving the angle produces. The band is named by formula in the statistic ("the eigenvalue 4 − 4cos(π/10)"), the pullback map in the earlier note is μ = 4 − 4cos(θ/2), and the M5 run reports a band at exactly 0.195774; the formula governs, and the falsifier's "within 1e-4" clause is read against 0.195774. The sealed text is not edited. The same slip sits in the earlier note's pullback list (0.382 should read 0.1958, and it comes before 0.3045, not after) — which reverses that note's worry: the bands strictly between λ₂ and the π/10 band are 0.1363 and 0.1492 only, the pullbacks of the two level-4 bands that carried nothing. So the "≤ 0.01" clause is back in play as written. Two further facts from the M5 blocks, recorded before the table: the 0.149241 band has multiplicity ≥ 96 (blocks 3 and 4 fell entirely inside it — the level-4 2 − √2 band had 16), and one vector in block 1 converged to 3.7e-6 rather than the sealed 1e-7.

Literature, checked 2026-09-28 before any citation. Bartholdi–Grigorchuk, "On the spectrum of Hecke type operators related to some fractal groups", Proc. Steklov Inst. Math. 231 (2000) 1–41, arXiv:math/9910102: the spectra of the Grigorchuk group's leaf Schreier graphs (4-regular, on 2ⁿ leaves) are Cantor sets, "simple transforms of the Julia sets associated to some quadratic maps", computed by finite approximation and factorisation. That is the same type as the pullback observed here — y² = 2x + 8 between consecutive levels is inverse iteration of a quadratic map — but Q_n is a different Schreier graph (G on ⟨b,c,d⟩·St(n)-cosets, index 2^(5·2^(n−3)), not the leaves), so their map is not claimed to be Q_n's. Cite for the phenomenon, not for the result.

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Procedural amendment under the seal, 2026-09-28 07:55, before any sealed quantity has been reported. The M5 run reached 800 eigenvectors in seven hours and was thrashing (10 GB resident, swap full) with no end below 0.40 in sight: the sub-0.40 spectrum of the level-5 coset graph is far denser than the level-4 table suggested — bands at 0.149, 0.205, 0.217, 0.258 each carry roughly 100–200 eigenvectors, not 8–40 — and the eigenvectors were never checkpointed, so a kill would lose the projections. The stopping rule "until a block's smallest eigenvalue exceeds 0.40" was derived from the slip recorded above (0.382 plus a margin); every quantity the seal names lives at or below the π/10 band, 0.195774. The run is therefore stopped and restarted with the stopping rule 0.20 (a block whose smallest eigenvalue exceeds 0.20 certifies the π/10 band complete), with eigenvectors checkpointed after every block and the sealed table printed after every block. BLOCK = 32 and the tolerance are unchanged; no prediction, falsifier or statistic is touched. The 800-eigenvalue list from the stopped run is kept as an artifact (bands5_vals_to800.npy): it is the level-5 low spectrum, and the band multiplicities in it are an observation this run did not ask for — recorded, not scored.

Observation from the stopped run's 832 eigenvalues (bands5_vals_to800.npy), 2026-09-28 — recorded, not scored. Bands below 0.27 with their multiplicities (lower bounds where a block was still inside the band when the run stopped) and their level-4 source under the pullback μ = 4 − 4cos(θ/2), λ = 4 − 4cos θ:

Isolated values between the bands (0.1996, 0.2119, 0.2185, 0.2285, 0.2304, 0.2476, 0.2546, 0.2593–0.2615) are single vectors, 10⁻³ from their neighbours — too far to be tolerance, too isolated to be bands; most likely stragglers of a band that a deflated block caught before it converged. Not counted. So the low spectrum is the level-4 bands inherited, mostly doubled, plus bands born at level 5 that dwarf the inherited ones — the decimation shape, and the reason the seal's "roughly 340 eigenvectors" undercounted: it counted inheritance only. Between λ₂ and the π/10 band sit exactly the two inherited bands the corrected arithmetic named, 0.1363 (16) and 0.1492 (96), pullbacks of the two level-4 bands that carried none of the minimum's weight; the 96 makes the ≤ 0.01 clause a real test. The six-fold 0.149 band and the born bands are a probe of their own (why the quotient's irreps grow like this), after this run is scored.

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Measured 2026-09-28 (Claude Code, M5: 8 blocks of 32, 256 eigenvectors, 5,676 s, peak 5.9 GB)

The λ₂ row reproduces run 038 to the digit. The stopping block's smallest eigenvalue was 0.205276 > 0.20.

Scored as written. Falsifier clause "a ≤ 0.99" fires: falsified. Prediction clauses: m — hit; between-bands ≤ 0.01 — hit (0.0000); s — hit; c — hit, exactly the minimum; a — miss. Control — passed.

Why the clause fired, and what it was measuring. Agreement 0.97715 ↔ overlap 0.954, and run 042's twins sit at 0.953 = 1 − 2 × 0.0236: the two-band rounding is a twin of χ. The seal was written on 2026-09-27, before run 040 found the twins and run 042 showed the minimum is a valley of ~45,500 free spins; it took "agrees with χ" as the meaning of "determines the minimum". At level 4 that reading was exact (a = 1.000) because the level-4 rounding happened to return the same representative. At level 5 the skeleton returns a minimum, not the representative, and the a-clause cannot tell those apart. The codifier's error; recorded, not rescued.

What stands, and it is the lineage's strongest result. The sign of the projection of the 90,112 partition onto the 96-dimensional span of two irreducible bands is itself a 90,112 bisection. Two bands out of a million dimensions. The band between them — 96-fold, six times its level-4 parent — carries none of the minimum's weight, exactly as its level-4 parents carried none. So "write the construction down" (N-028) has a concrete form: project onto the sibling-odd irrep and the π/10 band, take the sign; the crystal is that, and everything the sign does not determine is a free spin. The level-up rule was written after three identities failed one level up; this is the first to pass, and it passed with the cut exact.

Recorded. The 0.199588 singleton: 0.0038 above the π/10 band, one vector, on blocks where scipy reported not reaching 1e-7; 79 + 1 = 80 fits the doubling of the level-4 40 and π/9.903 fits nothing — treated as a straggler, not a band. The full low spectrum to 0.52 from the stopped run (bands5_vals_to800.npy) is the observation the amendment promised.

Pointers corrected 2026-10-05 (codifier; no sealed field, score or number is changed). Both artifacts: lines named R24_grig5_bisection/bands4.json. That file is not on disk and no commit has ever held it (git log --all -- R24_grig5_bisection/bands4.json is empty). The lines now name the files that exist, and keep the absent name as a remark.