Prime Pattern System: An Elementary Algebraic Framework for the Twin Prime Conjecture

This paper proposes an elementary algebraic framework---the Prime Pattern System---for the unified description of combinatorial structures related to the distribution of prime numbers. Its core is the exact product formula \(MH(p) = ∏q ≤ p (q - c_q)\), rigidly derived from the Chinese Remainder Theorem. I systematically develop the counting theory and structural properties of this framework: exact enumeration of large factors, candidate pairs (large-factor pairs of even spacing), quadruple prime candidate groups, and general \(k\)-tuple candidate groups; three core structural properties---nestedness, translational independence, and symmetry; the fixed prime pattern representations of \(P/2\) and \((P/2 ± 1)/2\); the constructive large factor family \(P/2 ± 2^k H\) and the algebraic origin of their spacings; the Sieving Step Theorem (\(d_1 + d_2 = q\)); and the complete spacing distribution statistics for the \(p=7, 11, 13\) levels together with empirical analysis of the safety zone interior. All conclusions are strictly derived from the Chinese Remainder Theorem, with no error terms or unproven conjectures.