A Geometric Reformulation of Mersenne Primality via the 6k±1 Matrix, Sophie Germain Primes and the Binary Lock

v3 (May 2025): Added §7 heuristic analysis of Bessaoudi super-pairs(E < 5×10⁻⁴ under Wagstaff and Bateman-Horn conjectures). AddedCyclotomic Theorem 6.1. Corrected base-3 table. n=53 identified asterm of A396249 but not a super-pair. Two new references.OEIS sequence A396249 submitted. We introduce the Bessaoudi-Germain conjecture, a geometric reformulation ofMersenne primality through the 6k+/-1 matrix. For any prime n, the indexk=(4^n-1)/3 places 6k+1=2^(2n+1)-1 in the right column as a Mersenne candidate.Three results are proved: (i) k always exhibits a pure binary wave 10101...01; (ii)the left column 2^(2n+1)-3 is systematically divisible by 5 for all odd primes n;(iii) Euler's theorem (1750) provides the fifth criterion n=1 (mod 4). NEW: theBinary Lock Corollary shows that a Sophie Germain prime n is Euler-locked if andonly if its binary representation ends in '11', giving an O(log n) eliminationtest. A base-9 family k=(9^n-1)/8 with right column (3^(2n+1)+1)/4 is proposed asan unstudied quasi-Mersenne family. Computational verification for n<=1000 confirmsonly n=2 and n=3 satisfy all criteria.