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 $\mathbb{H}^\mathbb{Z}$ over the quaternions $\mathbb{H}$ and demonstrating how this resolves fundamental issues relating to Dirac's equation. Generating $\mathbb{H}^\mathbb{Z}$ from the primitive basis $\mathcal{P}_0 = \{E_Z, E_{\bar{1}}\}$ 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 $\mathrm{C}\ell_{1,3}(\mathbb{R})$ emerge from the second Cayley--Dickson doubling (biquaternions $\mathbb{H} \otimes \mathbb{C}$); (2) under dynamic generation on an integer clock lattice with causal horizon $\Lambda_m$, 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 $\mathbb{Z}/Q\mathbb{Z}$ because candidate zone-edge doubler modes ($p = Q/2$) are actively pruned at the generational boundary by the coprime sieve projector $\mathcal{S}_Q$ ($\gcd(Q/2, Q) > 1$), preserving exact chiral symmetry without Wilson terms; (4) the discrete fermionic Green's function $S_Q(n, m)$ 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 $E_Z$, identifying rest mass $m_0$ as a vacuum coupling beat frequency for stationary periodic cycles on the saturated quotient graph.