Mersenne Primality, Sophie Germain Primes, the Modulo-4 Condition and a New Family of Prime Pairs

We introduce a geometric framework for Mersenne primality through the 6k±1 matrix. For any prime n, k(n)=(4^n-1)/3 satisfies 6k(n)+1=2^(2n+1)-1. We prove: (i) k(n) has alternating binary expansion 10101...01; (ii) the lower companion 2^(2n+1)-3 is divisible by 5 for odd prime n; (iii) a Sophie Germain prime n satisfies n=3 (mod 4) iff bin(n) ends in 11. This condition implies 2n+1 divides M_n=2^n-1 (Euler) but does NOT obstruct the primality of M(2n+1), as shown by n=3 where M(7)=127 is prime. The cyclotomic companion family F(n)=(3^(2n+1)+1)/4=Phi₄ₙ₊₂(3) has every prime factor satisfying p=1 (mod 4n+2). Using cubic reciprocity in Z[omega], we prove a refined congruence p=1 (mod 6(2n+1)) for factors with p=1 (mod 3), verified on 65 cases for n up to 2,000,000. Mersenne-cyclotomic SG pairs (both M(2n+1) and F(n) prime): known instances n=2 and n=3.