gauge_invariant_seed_eq_20pi
plain-language theorem explainer
The gauge-invariant Maxwell seed on the cube graph equals $20\pi$. Anyone arguing the U(1) coupling-normalization quarantine cites this identity: cycle rank 5 forces the seed $4\pi\cdot 5$. The proof unfolds the seed definition, substitutes the proved cycle-rank value 5, and closes by ring arithmetic.
Claim. The gauge-invariant Maxwell seed on the cube, defined as $4\pi$ times the first Betti number (cycle rank) of the cube 1-skeleton, equals $20\pi$.
background
Module AlphaGenesis.U1Normalization runs a make-or-break test: can the $\alpha$ seed $4\pi\cdot 11$ be promoted from a ledger channel-budget identification to a theorem about U(1) coupling normalization on the cube $Q_3$? Foundation.GaugeFromCube already yields the U(1) group as a parity quotient of $\mathrm{Aut}(Q_3)$, but never touches the $\alpha$ pipeline.
A genuine Maxwell normalization would read inverse coupling off gauge-invariant plaquette field strengths. Those are counted by the cycle rank of the 1-skeleton: $b_1=E-V+1=12-8+1=5$ (equivalently 6 faces minus one Bianchi closure). The definition gauge_invariant_seed is exactly $(4\pi)\times$(that rank). Upstream, cube_cycle_rank_eq_5 fixes the rank at 5 by native decision on the combinatorial count.
By contrast the seed's 11 is $E-1$ (passive edges), not the gauge-invariant photon stiffness 5. The module's negative verdict rests on that mismatch.
proof idea
One short term-mode calculation. Unfold the seed definition to $4\pi\cdot(\mathrm{cube_cycle_rank}:\mathbb{R})$. Rewrite with cube_cycle_rank_eq_5 to replace the rank by the numeral 5. push_cast aligns the natural-number coercion with the real scalars, and ring closes $4\pi\cdot 5=20\pi$.
why it matters
This identity is the numerical hinge of the U(1) quarantine. Downstream, gauge_invariant_seed_excluded rewrites through it and shows $20\pi<63<137.030<\alpha^{-1}$, so a genuine gauge-invariant Maxwell seed on the cube cannot source the electromagnetic coupling. The seed's 137-scale value therefore requires the ledger channel count 11, not the gauge count 5.
In the broader AlphaGenesis program the result keeps $4\pi\cdot 11$ as a cross-consistent ledger number (also appearing in $\Omega_\Lambda=11/16$, CKM structure, and $\eta_B=\varphi^{-44}$) while blocking its promotion to a derived U(1) stiffness. It does not touch the forcing chain T0–T8 or the RCL; it only polices how $\alpha^{-1}$ may be read off cube geometry. The infrared CODATA match $\alpha^{-1}(0)=137.035999$ remains an open boundary condition.
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