Quaternionic Wave Numbers and the Dirac Equation: Resolving the Nielsen--Ninomiya Barrier and Infinities

We exemplify the applicability of the rational wave numbers by formulating the relativistic Dirac equation on the discrete sequence space HZ over the quaternions H and demonstrating how this resolves fundamental issues relating to Dirac's equation. Generating HZ from the primitive basis P₀ = , E1̄\ as both a static closure and a dynamic enumeration, we show that coupling local quantum kinematics with global cosmological expansion leads to the following results: (1) Dirac 4-spinors and the Clifford algebra C1,3(R) emerge from the second Cayley--Dickson doubling (biquaternions H ⊗ C); (2) under dynamic generation on an integer clock lattice with causal horizon Λₘ, relativistic difference operators require temporal depth, establishing clock beat $m = 2$ as the universal Matter Genesis Threshold; (3) the Nielsen--Ninomiya fermion doubling barrier is circumvented on modular sequences Z/QZ because candidate zone-edge doubler modes ($p = Q/2$) are actively pruned at the generational boundary by the coprime sieve projector SQ ((Q/2, Q) > 1), preserving exact chiral symmetry without Wilson terms; (4) the discrete fermionic Green's function SQ(n, m) possesses no coincident-point singularity ($n = m$), converting ultraviolet loop integrals into finite Dedekind and Kloosterman character sums, realizing Dirac's demand for an infinity-free theory; and (5) Zitterbewegung is resolved as an observable stroboscopic beat against vacuum reference EZ, identifying rest mass m₀ as a vacuum coupling beat frequency for stationary periodic cycles on the saturated quotient graph.