The Cubic Lattice as Unique Global Zero of a Variational Functional on Six-Port Graphs

We prove two complementary theorems on the geometry of discrete unitary networks. Theorem 0 (dimensional selection from $S^2$ target): among discrete substrates that (i) carry an $S^2$-valued local field, (ii) require infrared stability, and (iii) admit an integer topological-charge spectrum ($\pi_3(S^2)=\mathbb{Z}$), the spatial dimension is uniquely $d=3$; criteria (ii) and (iii) are substantive inputs, not consequences of the $S^2$ target alone. The further claim that the local valence is $6=2\times 3$ requires an additional postulate aligning the three internal modes with the three spatial axes (the Mode-Axis Alignment postulate). Main Theorem (conditional lattice uniqueness): on a connected 6-regular port graph with antiparallel-paired frame field, the dissonance functional $\mathcal{F}=\mathcal{F}_E+\mathcal{F}_P$ (antiparallel-pairing edge term plus Wilson-holonomy plaquette term) admits $G\cong\mathbb{Z}^3$ as its unique global zero under the two-clause faithfulness axiom \textup{(F$'$)}: (i) every closed loop has vanishing net frame-displacement (well-definedness of $\Phi$); (ii) for any $v\neq w$ there exists a path with non-zero displacement (injectivity of $\Phi$). The axiom \textup{(F$'$)} is a substantive geometric faithfulness condition, not a technical lemma: clause (i) excludes toroidal quotients and the Heisenberg discretization, clause (ii) excludes the Cayley graph of $F_3$. Corollaries (under the same conditions) are: $O_h$ symmetry per node; the decomposition $\mathbf{6}=A_{1g}\oplus E_g\oplus T_{1u}$; the algebraic identity $I^2+R^2+C^2=1$, which appears both as a state-space postulate and as a scattering corollary of unitarity; and the defect classification $\pi_1(SO(3)/O)=2O$ of the polycrystalline regime. The proofs combine standard algebraic topology, graph-theoretic arguments, and variational reasoning, conditional on the stated postulates and on (F$'$). This manuscript was drafted in May 2026 and is uploaded here in May 2026 as Version 1.0 of the preprint.