A Deterministic Generative Grammar for Mersenne Prime Exponents

We present a deterministic generative grammar for the exponents of Mersenne primes. The grammar expresses every Mersenne exponent p (for p > 5 )) as the product of two earlier exponents plus or minus the difference of two additional earlier exponents: . We demonstrate that this formula, using only the largest product of earlier exponents less than the target, successfully generates 47 of the 48 analyzable exponents (M5 through M49, M51, M52). One exponent (M50) remains an exception that may require extension. All exponents can be generated by an additive Matryoshka recurrence with small integer coefficients . The nesting depth satisfies , forming a strictly linear chain from seeds {2, 3}. We identify a convergent seed ratio governing the asymptotic frequency of the seeds in full expansions. The grammar is supported by a binary ladder tree structure, modular fingerprint constraints, and a coefficient multiplier sequence based on M12=  127.