[E within this numerical program] The mechanism that survived falsification of φ. At K=4 (Menger approximant, 40 spectra):
| arm | n | spacing_std | exact degeneracies |
|---|---|---|---|
| organized φ | 1 | 5.903 | 1 |
| N1 (keeps magnetic point group, scrambles values) | 8 | 5.78 ± 0.94 | ≥3 |
| N2b (permutes level values) | 23 | 8.84 ± 2.22 | ≥2 |
| shuffle (breaks symmetry) | 8 | 2.99 ± 0.57 (max 4.04) | 0 |
Organized φ sits dead center of the symmetry-keeping cloud (z = +0.13) and +5.1σ above every symmetry-breaking draw. Degeneracies appear only in symmetry-keeping arms — direct sector-structure evidence. Symmetry audit: 6 unitary + 6 antiunitary operators, exact match (third independent confirmation at K=4).
Mechanism statement: a connection whose magnetic point group is nontrivial
splits the Laplacian into non-repelling symmetry sectors; superposed independent
sectors inflate eigenvalue spacing variance, force exact degeneracies, and
reduce ⟨r⟩. The geometry (Menger hierarchy, tunnel levels), the value c = φ, and
the level ordering are all inert — the group theory does all the work. The same
mechanism explains the complement cocycle's partial ordering
(bar-cocycles-incomplete-basis): its stabilizer is only the C₃ subgroup, so it
forces zero degeneracies and only partial ordering.
Level-stable K=2 → 3 → 4.
Open gaps: magnitude not derivable from group order alone
(gap-spacing-magnitude-from-group-order); why complement ordering is only
partial (gap-complement-partial-ordering).