Identity-Preserving Local Holonomy Dynamics on Cochains: A Unified Discrete Framework For Guage Theory, Casuality Topology

Abstract We present a self-contained, discrete framework where fundamental statements of gauge theory, causality bounds, and topological identities hold exactly at nite resolution. Degrees of freedom are realized as cochains on an oriented cell complex; dynamics emerge from local, nite-depth updates; gauge structure enters through edge holonomies; and coarse- graining commutes with the incidence operator. From the single algebraic axiom d2 ≡ 0 we derive: discrete Bianchi identities, Wilson-loop gauge invariance, AharonovBohm 2π periodicity, Ward transversality invariant under local spin-frame transformations, Lieb Robinson causal cones, bulk-boundary holonomy protection, and min-cut entanglement for stabilizer states on trees. Each identity is veried numerically to machine precision (where exact) or tight tolerance (where dynamical), using only nite-dimensional linear algebra. Representative results include: (i) split-step QCA maximum speed equals matched tight- binding (vmax = 1.96945, |∆|< 10−12); (ii) AharonovBohm spectral mismatch at 2π is 2.44×10−15; (iii) SU(2) pure-gauge plaquette residual 5.05×10−16, jumping to ∼0.10 upon curvature injection; (iv) SU(2) Wilson-loop traces unchanged under local gauge transforms to 6.66 ×10−16; (v) Ward contraction residuals 2.58 ×10−19 (base) and 1.86 ×10−19 (after spin similarity); (vi) three-spin recoupling overlap unitary to machine precision; (vii) tree graph-state entropy equals GF(2) min-cut rank; (viii) spectral dimension ows with diusion scale (UV ≈0.240, mid ≈1.071, IR ≈1.012); (ix) entanglement fronts show inside/outside ratio ≈26.65 at distance d = 20; (x) persistent current identity: |Imicro−(−∂ΦEGS)|= 1.58 ×10−11. These results reveal a single algebraic structureidentity-preserving local holonomy dynamics on cochainsunder which lattice gauge theory, discrete geometry, spin- network recoupling, stabilizer entanglement, and LiebRobinson causality become dierent coordinate systems on the same invariants.