The Geometry of Dirichlet Classes: Formalization of the Quadridimensional Prime Configuration Space M4

We introduce and formalize a novel discrete geometric framework, denoted as the Quadridimensional Prime Configuration Space M4. By mapping the four arithmetic progression classes of prime numbers modulo 10 (namely p≡1,3,7,9 (mod 10)) into mutually orthogonal coordinates in R4, we structure a discrete lattice populated exclusively by prime vectors. This paper formalizes the topological, metric, and algebraic axioms governing M4, tracking sequential prime trajectories as discrete dynamical particle motions characterized by a persistent unities-based torsion. We investigate geometric embeddings of special prime subfamilies—such as Sophie Germain, Mersenne, Fermat, and Wagstaff primes—demonstrating that classical unproven number-theoretic conjectures reduce to deterministic locus constraints within our space. Finally, we explore advanced non-abelian operations, curvature bounds, and cryptographic projections.