physical_link_dof_eq_cycle_rank
plain-language theorem explainer
On the 3-cube, physical U(1) link modes equal the cycle rank: E − (V − 1) = b₁, both evaluating to 5. Anyone auditing the α seed quarantine or the gauge-invariant photon count cites this agreement of the two counting routes. The proof rewrites both sides to numerals via the closed equalities and finishes by native decision.
Claim. For the spatial 3-cube, the number of edges minus the U(1) gauge redundancy equals the cycle rank of the 1-skeleton: $E - (V-1) = b_1$. Explicitly $12 - 7 = 5$, so the two routes to the gauge-invariant photon degree-of-freedom count agree.
background
Module Alpha Genesis M11 quarantines whether the α seed $4\pi\cdot 11$ promotes from a ledger channel-budget identification to a genuine U(1) coupling-normalization theorem on the cube $Q_3$. A Maxwell-style gauge-invariant count on the cube graph is the number of independent plaquette field strengths, i.e. the first Betti number $b_1 = E - V + 1$ of the 1-skeleton.
Spatial dimension is fixed at $D = 3$ (T8/T9 forcing). Edge count is $E = D\cdot 2^{D-1} = 12$; vertex count is $V = 2^D = 8$. Gauge redundancy is one U(1) phase per vertex minus the global phase, $V-1 = 7$. Cycle rank is defined as $E - V + 1$, equivalently faces minus one Bianchi/closure relation ($6-1=5$).
Upstream equalities pin the numerals: cube_cycle_rank_eq_5 and gauge_redundancy_eq_7. The physical-link route is then $E - (V-1) = 12-7$, which must match $b_1=5$.
proof idea
One short tactic proof. Rewrite the left-hand gauge-redundancy factor by gauge_redundancy_eq_7 (so the difference becomes $E-7$) and the right-hand side by cube_cycle_rank_eq_5 (so the goal is a pure numeral identity). Finish with native_decide, which checks $12-7=5$ in the kernel. No geometric argument is re-derived here; the content lives in the two upstream equalities and the definitions of edge count, cycle rank, and redundancy.
why it matters
This is the agreement clause in the negative U(1) normalization verdict: the two legitimate gauge-invariant photon counts (physical link modes after gauge fixing, and independent plaquette strengths) coincide at 5, while the α seed uses the ledger passive-edge count 11. Downstream, forcedClosure_holds packages the triple (cube_cycle_rank_eq_5, physical_link_dof_eq_cycle_rank, seed_channel_count_ne_gauge_dof) as κ_γ-independent forced-closure facts. The module verdict record u1NormalizationVerdict stores this theorem as gauge_dof_two_routes_agree.
Framework role: it sharpens why $4\pi\cdot 11$ cannot be read as a Maxwell stiffness on $Q_3$. The genuine gauge-invariant seed would be $4\pi\cdot 5=20\pi\approx 62.8$, excluded from the RS α⁻¹ band $(137.030,137.039)$. The 11 remains a cross-consistent ledger number (also in $\Omega_\Lambda=11/16$, CKM, $\eta_B$), not a U(1) photon DOF count. Landmark contact is T8 ($D=3$) via the cube geometry, not the J-cost or RCL chain directly.
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