[E within this numerical program] Measured as the kernel of the combinatorial 1-Laplacian of the cubical complex (vertices, edges, faces of the level-K Menger approximant), with the chain property ∂₂∂₁ = 0 verified exactly and b₀ = 1 confirmed two independent ways:
| K | V | E | F | b₁ measured | formula |
|---|---|---|---|---|---|
| 1 | 64 | 144 | 96 | 5 | 5 |
| 2 | 896 | 2304 | 1728 | 81 | 81 |
| 3 | 15616 | 42048 | 33024 | 1409 | 1409 |
Third consecutive exact integer hit, not a fit. The formula's two terms have a
structural reading: a 3-dimensional bulk contribution 3(20^K−1)/19 and a
2-dimensional face-carpet contribution 2(8^K−1)/7 (8 of 9 face squares retained
per level).
Downstream: everything about flat U(1) connections on this complex —
see bar-cocycles-incomplete-basis. Any claim that counts "flux degrees of
freedom" on the complex must reconcile with this number.