gauge_dof_via_faces
plain-language theorem explainer
On the 3-cube, faces minus one global Bianchi identity equals the cycle rank of the 1-skeleton: 6 − 1 = 5. Anyone counting independent U(1) plaquette field strengths on Q₃ cites this face-based route to the same b₁. The proof rewrites the cycle rank to 5 and finishes by native decision on the numerals.
Claim. With spatial dimension fixed at $D=3$, the face count of the $D$-cube minus one equals the cycle rank of its 1-skeleton: $2D-1=E-V+1$. Explicitly $6-1=5$, matching the single global Bianchi/closure relation $\prod_{\mathrm{faces}} F=1$.
background
Module Alpha Genesis M11 quarantines a make-or-break test: whether the ledger seed $4\pi\cdot 11$ promotes to a genuine U(1) coupling-normalization theorem on the cube $Q_3$. A gauge-invariant Maxwell action counts independent plaquette field strengths, i.e. the first Betti number of the 1-skeleton.
Upstream, spatial dimension is fixed by the forcing chain at $D=3$. Face count is $2D$ (six square faces). Cycle rank is defined as $E-V+1$; for $Q_3$ that is $12-8+1=5$, already proved equal to 5. The face route subtracts the single global closure $\prod F=1$ from the six faces, giving the same 5.
Gauge fixing removes $V-1=7$ link phases, leaving $12-7=5$ physical link modes. All three counts agree on five independent U(1) degrees of freedom.
proof idea
Term-mode one-liner. Rewrite the right-hand side with the already-proved equality that the cycle rank equals 5, then native_decide evaluates the closed numeral identity $2\cdot 3-1=5$. No further graph theory is invoked at this step.
why it matters
This identity is the face-based half of the combinatorial spine of M11's negative verdict. Independent plaquette strengths equal 5 whether counted as $E-V+1$ or as faces minus Bianchi. That 5 is the gauge-invariant photon stiffness on $Q_3$, not the ledger channel count 11 ($E-1$).
The mismatch $11\neq 5$ blocks promoting $\alpha^{-1}=4\pi\cdot 11$ to a U(1) Maxwell normalization. The genuine gauge-invariant seed would be $4\pi\cdot 5=20\pi\approx 62.8$, excluded from the RS $\alpha^{-1}$ band $(137.030,137.039)$. The declaration therefore anchors the quarantine: 11 remains a cross-consistent ledger number (also in $\Omega_\Lambda=11/16$, CKM, $\eta_B$), not a cube-gauge DOF count. No downstream consumers are wired yet; siblings carry the redundancy and exclusion steps.
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