Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold

Abstract: A system with two periodic directions carries two commuting holonomies (a=(u,v)\in\R^2/\Z^2). We determine their distinguished values while separating lattice, arithmetic, and observable effects. Maximizing the lowest twisted eigenvalue places (a) at a deep hole of the momentum lattice. For every rectangular torus the maximizer is antiperiodic, so complex multiplication is sufficient for torsion optima but not necessary. Let (G_τ^{-1}) be the dual metric and (D_τ(a)) the normalized zeta determinant of the twisted Laplacian. At the rotation-fixed deep holes of the square and hexagonal lattices, symmetry gives the exact determinant response (-\operatorname{Hess}a\log Dτ=2π(\Imτ)G_τ^{-1}). With spectral wavenumber (κ=2π), the lowest manifolds are fourfold and threefold, with gaps (2κ^2) and (4κ^2/3); phase errors split them linearly while their centroids remain stationary. We then give a finite-device realization: an (8\times8) microring lattice closed by two phase-controlled seams. At a reported coupling scale of (16) GHz, its exact square-lattice spectrum has a (17.32) GHz shell gap and a (1.92) GHz doublet separation for a (0.1) holonomy error; a triangular-link configuration gives a threefold qutrit manifold with an (18.11) GHz gap. This is a quantitative spectroscopy proposal, not a claim of topological protection or a completed device.