Gauge-Compatible Matter Sectors in Finite Reversible Closure

Gauge-Compatible Matter Sectors in Finite Reversible Closure (Paper 5) Abstract This paper develops the matter coupling sector of the finite reversible closure research programme. Building on Paper 3 (Relativistic Dispersion from Finite Reversible Closure) and Paper 4 (Local Special Relativity from Finite Reversible Closure), which established emergent relativistic dispersion and local Lorentz symmetry from bounded discrete dynamics, the present work examines admissible matter sectors and their local deformations. Within a finite-dimensional Hilbert space per lattice site and strict nearest-neighbour causal radius per discrete tick, scalar and fermionic sectors arise as finite dimensional unitary representations of the closure algebra. If one further requires local phase covariance of nearest-neighbour propagation, multiplicative link variables valued in compact unitary groups naturally arise. In the long-wavelength limit this reproduces minimal gauge coupling. The result establishes kinematic compatibility between finite reversible closure and gauge structure. No gauge field curvature or dynamics are assumed here; those are developed in Paper 6 (Quantum Curvature from Finite Reversible Closure). Introduction This work forms Paper 5 in the finite reversible closure programme. The preceding papers established that: Paper 3 (Relativistic Dispersion from Finite Reversible Closure) derived Lorentz-type dispersion relations from strictly bounded reversible discrete dynamics without assuming a pre-existing continuum spacetime. Paper 4 (Local Special Relativity from Finite Reversible Closure) demonstrated that local Lorentz symmetry arises as the isotropic quadratic limit of such discrete closure models with invariant slope c=ℓp/tp. The present paper asks a natural structural question: given a closure-derived causal framework with strict nearest-neighbour propagation and finite dimensional local Hilbert spaces, is gauge structure compatible with such a setting? Unlike conventional lattice gauge theory, which discretises continuum field theories, this framework begins from finite reversible closure as primitive. Gauge covariance is therefore analysed as a structural requirement placed upon discrete nearest-neighbour transport rather than as an imported continuum symmetry. The central result is that if local phase covariance is required for nearest-neighbour propagation, multiplicative edge (link) variables valued in compact unitary groups necessarily arise. In the long-wavelength limit this reproduces minimal coupling while preserving the causal cone and invariant slope established in earlier papers.