cube_cycle_rank
plain-language theorem explainer
The cycle rank of the 3-cube 1-skeleton is defined by E − V + 1, counting independent U(1) plaquette field strengths on Q₃. Anyone auditing the α-genesis U(1) normalization test cites this as the gauge-invariant photon degree-of-freedom count (equal to 5 once evaluated). The body is a one-line arithmetic definition from the hypercube edge and vertex counts at the forced spatial dimension D = 3.
Claim. Define the cycle rank of the spatial cube graph by $b_1 = E - V + 1$, where $E = D \cdot 2^{D-1}$ and $V = 2^D$ are the edge and vertex counts of the $D$-hypercube at the forced spatial dimension $D = 3$. This is the first Betti number of the 1-skeleton: the number of independent plaquette field strengths of a U(1) gauge field on $Q_3$.
background
Module M11 (U1Normalization) runs a make-or-break test: can the α seed $4\pi \cdot 11$ be promoted from a ledger channel-budget identification to a theorem about U(1) coupling normalization on the cube $Q_3$? A genuine Maxwell normalization would read inverse coupling off a gauge-invariant action, whose independent field strengths are the cycle rank of the 1-skeleton, not a passive-edge count.
Upstream geometry is fixed by the forcing chain: spatial dimension $D = 3$ (T8/T9), so the hypercube has $V = 2^D = 8$ vertices and $E = D \cdot 2^{D-1} = 12$ edges. The cycle rank $b_1 = E - V + 1$ is the standard first Betti number of a connected graph; equivalently, six faces minus one global Bianchi/closure relation. Gauge fixing removes $V - 1 = 7$ link phases, leaving the same five physical link modes.
The seed's $11 = E - 1$ only drops the single active edge. That is a ledger recognition-channel count, not the gauge-invariant photon stiffness.
proof idea
Pure definition: one arithmetic expression cube_edges D - cube_vertices D + 1. No tactics, no lemmas applied at the definition site. Evaluation to 5 is deferred to the sibling theorem that unfolds this def and decides by native computation; face-route and physical-link-route equalities are separate one-line rewrites against that evaluation.
why it matters
This is the combinatorial spine of the negative U(1) normalization verdict. Downstream, cube_cycle_rank_eq_5 pins the value at 5; gauge_dof_via_faces and physical_link_dof_eq_cycle_rank show two independent routes (faces−1 and E−(V−1)) agree with it; seed_channel_count_ne_gauge_dof records the sharp mismatch $11 \neq 5$. The gauge-invariant Maxwell seed is then $4\pi \cdot 5 = 20\pi \approx 62.8$, excluded from the RS α band near 137.
ForcedClosure packages the same facts as κ_γ-independent forced-closure invariants on $Q_3$. Framework landmarks in play: T8 forces $D = 3$, the eight-tick octave is the 8 vertices of $Q_3$, and the α inverse band sits far above $20\pi$. The declaration therefore closes the quarantine: the seed $4\pi \cdot 11$ remains a cross-consistent ledger number (also in $\Omega_\Lambda$, CKM, $\eta_B$), not a U(1) coupling normalization on the cube.
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