This manuscript develops a prime-specialized multiplicative geometric approach to the Mersenne prime problem. For prime exponents q, the obstruction to the primality of 2^q − 1 is encoded by the exact terminal divisor trace D(q,k), where every possible proper prime divisor has the form 2kq + 1. The proof is formulated as a one-sided survivor-transfer argument. After fixing the final visible quotient, the observed survivor ledger is compared with a promoted survivor main term on the same prime-exponent fibre. The resulting discrepancy is controlled through stopped identities, compensator estimates, wall-flux bounds, Bonferroni transfer, and controlled ledger estimates. The manuscript concludes a logarithmic lower bound for the number of Mersenne primes with prime exponent q ≤ X, implying the infinitude of Mersenne primes.