Limit laws for prime multiplicities under the reciprocal-exponent weight

On each open interval where a single exponent m maximizes 1/a - t/a^2, the measure given by J(n) to the power t concentrates on integers that have at least one prime of exponent m. The number of those primes has mean and variance log log x plus a bounded term, and the normalized count converges to the standard normal law. The other multiplicities converge to a product of independent prime laws, with finite means and negative covariances. A typical integer factors as Q to the power m times a defect that converges in distribution to a finite integer. Mod-Poisson convergence is used only in the sense of Kowalski and Nikeghbali. There is no claim about prime gaps or the Goldbach conjecture.