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Established The bar cocycles are exactly flat but span a shrinking minority of H¹; the complement is spectrally active

id: bar-cocycles-incomplete-basis · filed: 2026-10-01 · depends: b1-formula-exact · falsifier: a level at which bar cocycles span all of H¹, or a complement cocycle that leaves the spectrum unchanged

[E within this numerical program] Context: all spectra previously run on this complex threaded flux only through "bar cocycles" — phases on edges crossing each removal bar's half-plane, one cocycle per bar. The probe measured where those cocycles sit inside the true H¹ (b1-formula-exact):

Bar cocycles are exactly flat (max face flux 0.0 at all K) and independent, but span 60% → 33% → 15.5% of H¹. The complement — harmonic 1-cocycles orthogonal to the bar span — is not dead gauge freedom: threading the canonical complement cocycle moves the spectrum smoothly and monotonically (spacing_std of lowest 200 eigenvalues: K=2: zero 2.069 → complement 1.870 → bar-φ 1.150; K=3: 1.716 → 1.611 → 1.465), flat to 1e-10 in phase.

The canonical complement cocycle is 64% (K=2) / 70% (K=3) interior by weight. "Face stratum" was the working name and is a misnomer: the complement is bulk-threading flux freedom that the bar parameterization silently discards.

Connector relevance: this is the construction's native supply of "incomplete pieces" — more independent flux classes than the obvious menu, growing like ~(20/8)^K relative to the bars. Any mapping of the construction onto particle content must account for where these classes live (gap-hol3-particle-content-connector).