On the Additive Structure of Pure Hurwitz Quaternions :Rigorous Proofs, the Ternary Goldbach Problem, Sieve Theory, and Ten New Conjectures with Complete Supporting Theorems

We investigate the additive decomposition of admissible pure Lipschitz quaternions $q$ into the sum $q = p_1 + p_2$, where $p_1$ is a pure prime quaternion and $p_2$ is a pure almost-prime quaternion. Specifically, we require $\Norm(p_1)$ to be prime and $\Norm(p_2)$ to be in $\Ptwo$ (having at most two prime factors). By recasting the problem as a sieve over ternary quadratic forms and employing the weighted linear sieve with the Chen-Iwaniec Switching Principle, we prove that every sufficiently large admissible $q$ admits such a decomposition. Our result is conditional on a Bombieri-Vinogradov type hypothesis for prime vectors in lattice arithmetic progressions.