This manuscript provides the rigorous mathematical foundation for a GL (k, C) non-Hermitian gauge theory defined on finite oriented simplicial complexes. While motivated by recent empirical observations of exceptional-point criticality in macroscopic information flows, the focus here is strictly on the underlying formal geometry and topological proofs. We establish three main structural results without relying on commutativity assumptions. First, we prove the conservation and reality of the biorthogonal information charge under non-Hermitian dynamics. Second, we show that a log-determinant barrier naturally prevents gradient-flow trajectories from collapsing into singular metrics, rendering the physical limit well-defined without ad-hoc regularisation. Finally, we demonstrate the exponential convergence of the system to a topologically protected fixed point, deriving its explicit spectral gap using Weyl's Eigenvalue Inequality. Beyond these stability proofs, we show that the discrete Palatini variation naturally yields both the discrete Einstein-Yang-Mills equations and the Bianchi identity directly from the topological boundary condition (²=0). Within this framework, the macroscopic onset of an exceptional point is formally mapped to the holonomy defectiveness of the plaquette operator.
Chao Ma (Sat,) studied this question.