This paper develops a mathematically rigorous, limit-free operator-algebraic quantum mechanics built entirely from the discrete geometry of Universal Hyperbolic Geometry (UHG) and Rational Trigonometry, working strictly over finite prime fields and their quadratic extensions. No real numbers, transcendental functions, or infinite limits are used anywhere in the construction. The central objects are Cyclic Oriented Polygonal Splines (COPS): closed loops of orthic poles on a finite null conic. The finite set of all valid, non-degenerate COPS of a given length forms the basis of a finite-dimensional quantum state space over a quadratic field extension. A discrete kinetic Hamiltonian is constructed from the negative adjacency matrix of a single-vertex mutation graph on these configurations, which is proved to be isomorphic to a Johnson graph and to be invariant under the natural action of the projective linear group PGL(2) as a symmetry group. The loop-closure defect of each COPS — measuring how far the multiplicative holonomy of the loop deviates from unity — is identified as an exact finite-field analogue of a Wilson loop in lattice gauge theory, with the multiplicative group of the finite field playing the role of the gauge group. This yields a self-adjoint total Hamiltonian combining kinetic and potential terms. Unitary discrete-time evolution is implemented via the Cayley transform, replacing the standard matrix exponential with an exact rational construction. The unitarity of this propagator is proved rigorously using a Galois-theoretic adjoint operation grounded in the Frobenius automorphism of the field extension. A key result shows that in the compact field regime, the propagator is well-defined for all values of the coupling constant without exception. The paper classifies two physically distinct regimes depending on the prime characteristic: a compact regime with a positive-definite inner product and stable periodic orbits, and a non-compact regime with a split-signature form admitting null states. A concrete worked example is developed for the case of three-point loops over the field with seven elements, demonstrating a 56-dimensional state space, a connection to the Johnson graph J(8,3), a spectral collapse from four integer energy levels to three modular levels, and an exact quantum recurrence period of 8 time steps. A striking result of this example is that the path-integral ground state — the uniform superposition over all configurations — is simultaneously the Perron-Frobenius eigenvector of the kinetic operator and a null state under the modular inner product, giving it the character of a zero-norm topological ground state. The paper concludes with a conjecture on the non-uniform holonomy structure of four-point loop configurations, predicting non-trivial spectral mixing in the next sector of the theory, and identifying this as the subject of a companion paper.