Statistical Forecast of the 53rd Mersenne Prime Exponent

This supplement applies the empirical methodology developed in the series to a forecasting problem: estimating the exponent of the 53rd Mersenne prime. Three independent statistical approaches — empirical growth-rate extrapolation over recent discoveries, log-linear regression across all 52 known exponents, and a calibrated Wagstaff-type heuristic — yield a central forecast of approximately 175 million (plausible range 145–250 million), corresponding to a Mersenne prime of roughly 52–53 million decimal digits. We emphasize that this is a statistical extrapolation constrained by verified GIMPS coverage facts, not a constructive formula: no known method predicts which exponents yield Mersenne primes. A cross-connection with the original study is established: Mersenne primes are extreme anti-random outliers in bit-transition statistics, having C(T) = 0, the minimal possible value.