We introduce a geometric framework for Mersenne primality through the 6k±1 matrix. For any prime n, k (n) = (4ⁿ-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 Mₙ=2ⁿ-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=Phi₄₍+₂ (3) has every prime factor satisfying p=1 (mod 4n+2). Using cubic reciprocity in Zomega, 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.
BESSAOUDI HAMZA (Wed,) studied this question.