Exact Symmetries in Discrete Gauge-Gravity Dynamics: A Unified Computational Framework

We present a computational framework unifying discrete exterior calculus, lattice gauge theory, and discrete-time quantum walks, where fundamental symmetries become exact algebraic identities at finite lattice spacing rather than approximate continuum limits. Fields are represented as cochains on oriented cell complexes with a discrete wedge product satisfying the graded Leibniz rule. Dynamics consists of holonomic shifts—permutations decorated by spatial and temporal link variables encoding space-time rectangle holonomies. We prove and verify numerically to∼10 ⁻¹⁵ residual - (i) sitewise probability continuity for split-step Dirac walks; (ii) time-dependent non-Abelian (SU(2)) gauge intertwiner identities; (iii) lattice U(1) Ward identities for vac-uum polarization; (iv) exact commutation of class-function plaquette dynamics with Gauss constraints; (v) palindromic Cayley integration preserving Einstein–Cartan vertex closure. The Whitney restriction operator commutes exactly with the coboundary (Rd= dR), enabling symmetry-preserving renormalization. This framework provides structure-preserving numerics for non-perturbative field theory and clarifies how continuum symmetries emerge from discrete structure