run 038Falsified
../lab/runs/038-level5-spectral-rounding.mdPrediction(verbatim from N-028, dated 2026-09-27, before the level-5 eigenvectors were computed) the sign-rounding of the 90,112 partition's projection onto its 16-dimensional λ₂ irrep is a balanced cut of 98,304 = (3/4)·131,072, agreeing with the minimum on ≈ 99% of cosets, and 90,112 = 98,304 − 8,192 is the same 3/4 − 1/16
Falsifiernot written separately in N-028 — the prediction is an exact constant, so any other cut size falsifies it. The "≈ 99%" clause is an open constant and would not pass the seal validator; it is recorded as an observation, not scored.
VerdictFALSIFIED as an identity. The λ₂ eigenspace has dimension 16 with eigenvalue 0.0768589 (= 4 − 4cos(π/16)) to seven digits, and the next band is 0.1362967 = 4 − 4cos(π/12), the second irrep on the same ladder — the representation-theoretic picture from N-028 §1 holds exactly at level 5. But the sign-rounded cut is 98,288 (98,284 at tolerance 1e-6), sixteen edges short of 98,304; the 66 cosets whose projection lies within 1e-5 of zero are the only ambiguity, and flipping all of them together in either direction gives 98,302 — two edges short, and only by resolving every tie the same way. The rounding is balanced exactly (Σ sign = 0 without rebalancing, at both tolerances), which is a structural hit. Agreement with the minimum is 97.3%, not ≈ 99%. So 3/4 is the leading term (0.74988 of the unit) and the bands take 0.06238 rather than 1/16 = 0.0625; the two-term identity 11/16 = 3/4 − 1/16 is a level-4 accident. The constant 11/16 itself is untouched — it is exact at both levels because it is measured. What the host has now done three times in one day: hand over a clean identity at the small level (N/log N, the fibre count of run 031, 3/4 − 1/16) that fails by a part in 10³–10⁴ at the next.
Sealed by N-028's dated paragraph, written after the level-4 rounding (192 = (3/4)·256, exact) and before any level-5 eigenvector existed. The seal is the note's own text, quoted verbatim in the prediction: field; nothing was reworded.
What was computed. Claude Code's run-031 files: grig5_q.adj (the coset graph, 1,048,576 vertices, 4-regular), grig5_q.map (Cayley vertex → coset), best5.bits (the 90,112 bisection). Laplacian L = 4I − A; LOBPCG for the 20 lowest eigenpairs from a random start with the constant vector pinned; the 16 eigenvectors within 2·10⁻³ of 4 − 4cos(π/16) orthonormalised by QR; the ±1 partition vector projected; the sign taken.
| tol 1e-6 | tol 1e-9 | |
|---|---|---|
| eigenvalues in the λ₂ band | 16 | 16 |
| ‖P₂ χ‖² / N (share of the partition in the irrep) | 0.6844 | 0.6844 |
| cut of sign(P₂ χ) | 98,284 | 98,288 |
| Σ sign(P₂ χ) | 0 | 0 |
| agreement with χ | 97.34% | 97.34% |
| wall time | 437 s | 776 s |
Ties. Distribution of |P₂ χ| near zero at tol 1e-9: 2 cosets below 10⁻⁶, 66 below 10⁻⁵, 728 below 10⁻⁴, 6,658 below 10⁻³ — a smooth tail, not a cluster of exact zeros with a gap. Flipping every coset below a threshold to −1, or to +1, gives the same cut both ways (the ambiguous set is symmetric under the swap): 98,288 at 10⁻⁶, 98,302 at 10⁻⁵, 98,300 at 10⁻⁴, 98,332 at 10⁻³. Setting them to the minimum's own side instead gives 98,288 / 98,260 / 97,756 / 93,548. The prediction 98,304 is two edges beyond the uniform-flip band at the credible threshold.
Scored as written. 98,288 ≠ 98,304: falsified. Balance: hit. Agreement: 97.3% against an unsealed "≈ 99%": recorded, not scored.
What it means for N-028 §2. The handle stays a handle. The sibling-odd irrep carries three quarters of the minimum's cost to leading order, the higher bands carry the rest, and 90,112 = 98,288 − 8,176 is a decomposition of a measured number, not a formula for it. "Write the construction down" is still: the best sign pattern of a sibling-odd function, then a correction — but the correction is not 1/16 of anything, and the first term is not exactly 3/4.
Named, not sealed. The one instrument that can rule 98,304 in or out is exact arithmetic: a rational basis for the sibling-odd irrep built from the tree structure (grig5_q.fibre), the projection in exact rationals, ties reported as ties rather than resolved. If the exact rounding has a tie set whose two resolutions bracket 98,304, the identity is "true up to a convention"; if not, this run's verdict is final. Prediction, if that run is ever sealed: it is final — the tail of |P₂ χ| is smooth, which is what a generic vector does, not what an exact identity does.
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The named follow-up, done 2026-09-28 (round_l5_polish.py). The twenty block-1 eigenvectors were polished by LOBPCG from their saved values to the float64 limit: the sixteen λ₂ eigenvalues sit within 3 × 10⁻¹⁶ of 4 − 4cos(π/16), residuals ‖Lv − λv‖ between 6 × 10⁻¹⁴ and 1.5 × 10⁻¹³. The projection of the 90,112 partition onto that eigenspace has no entry below 10⁻⁷ in magnitude (two below 10⁻⁶, sixty-six below 10⁻⁵, none smaller), against a numerical error near 10⁻¹². There are no ties: every sign is determined with at least five orders of margin. The rounding is 98,288, balanced, agreement 0.97336, share 0.68438 — the same numbers as the 1e-9 run, now certified. 98,304 is not a resolution of ties; it is simply not the value. The verdict stands and the recount is closed; exact rational arithmetic could only repeat it.
generated 2026-10-25the lab index