Factor Shift Deterministic Twin Primes Detection Mechanism

We present a complete deterministic framework for isolating twin prime indices withinthe 6k±1 domain. The method maps composite-carrying indices to four algebraic formsk =6ab±a±b derived from (6a±1)(6b±1). We prove (1) the 6k±1 domain exhaustivelycaptures all primes > 3, (2) the four forms exhaustively capture all composites in thatdomain, (3) the Factor Shift Operator k(a,b + 1) = k(a,b) + (6a ± 1) provides additivenavigation of the lattice Z+ ×Z+, (4) an odd-even-odd inductive parity proof establishesexhaustive coverage of the upper-triangular lattice L = {(a,b) | b ≥ a} with zero gaps,(5) boundary leakage beyond any finite frontier is closed by arithmetic progressions La ={pa + da · i}. Empirical validation up to Kmax = 1,000,000 (Nmax = 6,000,001) shows962,085 composite indices and 37,915 twin indices with 100% precision and recall againstprime sieve ground truth.