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

v4.2 (June 2026): Critically revised edition. Corollary 4.2 corrected: the modulo-4 condition characterizes n via Euler's theorem on M_n = 2^n - 1, but does NOT obstruct the primality of M(2n+1) = 2^(2n+1) - 1, as shown by the counter-example n=3 (M(7)=127 is prime). Theorem 7.2 (Eisenstein cubic obstruction in Z[omega]) refines the Birkhoff-Vandiver bound from mod (4n+2) to mod 6(2n+1) for factors with p = 1 (mod 3); verified computationally on 65 cases for n up to 2,000,000. Sequence of SG primes n with M(2n+1) prime: n = 2, 3, 53.