Discrete Non-Hermitian Gauge Geometry and the Two-Sided Exceptional-Point Scaling Laws: Theory, Proof, and Empirical Verification

This preprint develops a complete framework for discrete non-Hermitian gauge geometry on simplicial complexes with link variables in GL(k,C), combining formal mathematical proofs with large-scale empirical verification. We rigorously establish three structural pillars for the gauge theory: (i) biorthogonal conservation of the information charge, (ii) singular-boundary exclusion via a log-determinant barrier which ensures the physical limit is well-defined without ad-hoc regularisation, and (iii) exponential convergence to a topologically protected Phoenix attractor with an explicitly derived spectral gap. From the fundamental topological identity ∂^2=0, the discrete Palatini variation naturally yields both the discrete Einstein-Yang-Mills equations and the Bianchi identity. Empirically, this geometric framework predicts a dual scaling law. Tested against 5.28 million millisecond-resolution trades from the 5 August 2024 BTC/USDT liquidity collapse, we confirm a +1/2 scaling exponent for the curvature spectral gap (Δcurv ~ |Δeff|+1/2). Together with the conjugate -1/2 susceptibility divergence, this forms a two-sided exceptional-point test. Both exponents are algebraically fixed by the same Jordan-block coalescence and cannot be simultaneously reproduced by mechanisms lacking this singularity. The renormalization programme yields a universal, rank-independent one-loop coefficient C_1(k) = -2/11. Empirical analysis of 585,505 time bins bounds non-constant background corrections to strictly negligible levels (< 2.3 × 10⁻³). Extending the transfer matrix to k=3 confirms that the primary +1/2 scaling exponent remains universal across dimensions.