Diabatic Gauge Theory (Part I): Variational Principles, Topological Obstructions,\\ and Discrete Canonical Bases on Manifolds

For an isolated $N$--state electronic subspace over a nuclear configuration domain $\Omega\subset\mathbb{R}^{D_\Omega}$, with degeneracy locus $\Sigma$ removed, and with distances interpreted in the chosen coordinate chart, the freedom to choose an electronic basis is a non-Abelian $U(N)$ gauge freedom. First-order nonadiabatic couplings define a $\mathfrak{u}(N)$-valued Berry connection $\mathcal{A}$, while position-dependent unitaries act as gauge transformations. This paper develops a unified theoretical framework for diabatization that (i) characterizes the geometric and topological obstructions to \emph{strict} diabatization with vanishing first-order couplings in terms of curvature and holonomy, (ii) introduces a gauge-theoretic variational principle whose minimizers define \emph{canonical} (quasi-diabatic) gauges even when strict diabatization is obstructed, and (iii) formulates discrete ``lattice'' realizations on graphs, including (a) an isotropic edge-only action---for which we derive closed-form Riemannian gradients---and (b) a reconstruction-based action consistent with anisotropic coordinate-index weights with gradients obtained by automatic-differentiation or adjoint methods applied to the reconstruction map. A solvable one-dimensional real two-state limit is used for validation: strict diabatization reduces to the ODE $d\theta/dx=\tau_{12}(x)$ up to a sign convention, and we provide both the analytic solution and a simple discrete minimization experiment on a one-dimensional grid recovering the same gauge. A lexicographic two-stage selection rule is recorded that enforces strict gauges whenever they exist.