A Structural Classification of Integers via Primorial Reduction: From Stable Numerators to Egyptian Fraction Representations

For each positive integer k and each primorial Pₙ = p₁ p₂ ⋯ pₙ, we study the reduced fraction obtained from k/Pₙ after cancellation. We prove that once pₙ exceeds the largest prime divisor of k, the reduction pattern stabilizes: the reduced numerator equals k / rad(k) and the reduced denominator equals Pₙ / rad(k), where rad(k) denotes the radical of k. This stable behavior yields a natural three-way classification of all integers k ≥ 2: primes (numerator $1$, denominator misses exactly one prime), squarefree composites (numerator $1$, denominator misses at least two primes), and non-squarefree integers (numerator $>1$). We determine the natural densities of the three classes to be $0$, 6/π², and 1-6/π², respectively. The classification is further refined by the stable missing-prime statistic, which equals the number of distinct prime divisors of k on the squarefree locus. We construct the associated Dirichlet generating series---expressed in terms of the Riemann zeta function ζ(s) and the prime zeta function $P(s)$---and a bivariate Euler product that uniformly encodes the refined squarefree strata. As a core application, we show that the framework generates a canonical family of unit fractions with squarefree denominators, and we prove that every positive rational with squarefree reduced denominator admits a constructive Egyptian fraction representation using only these building blocks, with an explicit bound on the number of terms. An algebraization of the framework via local valuation profiles and ideal-theoretic extensions to number fields is also initiated.