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 \(G_\tau^{-1}\) be the dual metric and \(D_\tau(a)\) 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 \(-\operatorname{Hess}_a\log D_\tau=2\pi(\Im\tau)G_\tau^{-1}\). With spectral wavenumber \(\kappa=2\pi\), the lowest manifolds are fourfold and threefold, with gaps \(2\kappa^2\) and \(4\kappa^2/3\); 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.