Schrödinger-Dirac Formalism in Finite Ring Continuum

We extend the algebraic framework of the Finite Ring Continuum (FRC) to the domain of quantum and relativistic dynamics. Building on the previously established results on symmetry-complete finite fields Fp and their quadratic extensions Fp2 , we show that both the Schrödinger and Dirac equations admit exact finite-field realizations. Within a single Euclidean shell Fp, the discrete Schrödinger equation governs reversible, scale-periodic evolution of framed wavefunctions under finite-difference operators. Upon extension to the Lorentzian shell Fp2 , the finite-field Clifford algebra Cl(1, 3; Fp2 ) and the corresponding orthogonal group O(Qν, Fp2 ) of split type (Witt index 1) support a fully covariant finite-field Dirac equation, in which the Minkowski metric and causal structure emerge algebraically from the separation of square classes. The two regimes—Euclidean and Lorentzian—together form a consolidation-innovation cycle: powering operations within Fp compress information by merging residue classes, while the quadratic extension Fp2 restores completeness by generating new algebraic degrees of freedom. This alternation of compression and expansion provides an intrinsic algebraic origin for the coexistence of reversible quantum evolution and irreversible causal propagation. The resulting formalism reproduces the essential dynamical structure of relativistic quantum mechanics within a purely finite and relational arithmetic, eliminating any dependence on the continuum hypothesis.