A constructive refinement of prime-type classification via nested primorial sievematrices. We introduce a primorial-based additive representation of integers (Mo numbers)and prove a residue separation property: for any prime q, primes in the arithmetic progressionq mod ps# cannot lie outside the nested Mo-type MTq. The Mo-types form a strictly descendinghierarchy MT2 ⊃ MT3 ⊃ MT5 ⊃ MT11 ⊃ ···, and we prove each deepest-type class containsinfinitely many primes (a constructive refinement of Dirichlet’s theorem without analytic densityestimates).We further classify twin prime candidates into coupled types such as MT(5,7) and MT(11,13),showing that the Twin Prime Conjecture is equivalent to the divergence of true twin pairs withinMT(5,7). As a rigorous corollary, Zhang’s bounded-gap theorem admits a Mo-type formulation:some even gap N ≤ 246 yields infinitely many prime pairs in a coupled Mo-type progression.The framework reframes classical distribution problems as explicit sieve-matrix counting,leaving open the quantitative density of twin pairs for future work