Quantum Curvature within Finite Reversible Closure: Discrete Gauge Dynamics from Local Unitary Structure (Paper 6)

Quantum Curvature within Finite Reversible Closure: Discrete Gauge Dynamics from Local Unitary Structure (Paper 6) Abstract Within the finite reversible closure framework developed in Papers 3–5, we construct a fully dynamical gauge curvature sector consistent with finite local Hilbert spaces, strict locality per primitive tick, global unitarity and the invariant causal slope c=ℓp/tp. A finite-dimensional link Hilbert space with conjugate electric operators is introduced, and plaquette holonomy defines curvature in the discrete setting. A finite-depth local unitary update generates electric and magnetic dynamics while preserving exact local gauge symmetry. The resulting theory supports a discrete Gauss constraint and charge conservation. In the weak-field, long-wavelength regime, gauge excitations exhibit relativistic-form dispersion with calibrated group velocity saturating the structural slope c. A strict causal bound is proven, ensuring no superluminal signalling under any parameter choice. This work embeds discrete gauge curvature into the finite reversible closure programme. It does not claim derivation of continuous gauge symmetry from closure alone, but demonstrates structural compatibility between finite local reversible dynamics and dynamical gauge curvature. Introduction The finite reversible closure programme seeks to identify which large-scale physical structures are forced by strict locality, finite Hilbert dimension per site and global unitarity under discrete primitive updates. Paper 3 established that quadratic relativistic dispersion emerges from locality and reversibility alone. Paper 4 identified the invariant structural slope c=ℓp/tp as a causal bound intrinsic to the closure architecture. Paper 5 demonstrated that compact gauge-compatible matter sectors can be embedded while preserving finite local structure and exact Gauss constraints. The present paper extends the framework by introducing a fully dynamical curvature sector. Using finite-dimensional link Hilbert spaces and Weyl-type clock and shift operators, we construct electric and magnetic energy terms analogous to those of Hamiltonian lattice gauge theory. Plaquette holonomy defines curvature and a finite-depth split-step unitary update generates gauge dynamics while preserving strict locality per tick. The central question addressed is structural: can dynamical gauge curvature coexist with finite local reversible closure while maintaining exact gauge symmetry, a strict causal cone and relativistic infrared behaviour? The answer is affirmative. This work therefore demonstrates that discrete gauge curvature is structurally compatible with finite reversible closure and that relativistic-form propagation of gauge modes arises naturally in the infrared regime without violating the closure constraints established in earlier papers.