Schrödinger-Dirac Formalism in Finite Ring Continuum

We present a strictly finitist formulation of Schr\"odinger-type and Dirac-type dynamics in the Finite Ring Continuum, together with exact information counts for reversible and compressive shell maps. The construction uses a symmetry-complete prime field, its quadratic extension, and the Frobenius involution to define finite Hermitian state spaces and finitist Hamiltonians. On Euclidean shells, continuum time evolution is replaced by a Cayley update that preserves the Hermitian form exactly and therefore produces periodic trajectories. On the Lorentzian extension, we construct explicit gamma matrices, a finitist Dirac operator, its associated Klein-Gordon factorization, and a covariant lifted boost action. To connect the formalism with entropy and information theory without leaving strict finiteness, we measure finite maps by their image counts and exact loss factors. This separates reversible transformations, which preserve distinguishability exactly, from shell power maps, which merge states by a computable arithmetic factor. All results are finite, algebraic, and exact; no limits, differential calculus, or continuum structures are used.