For a positive integer n, let g(n) be the largest divisor of n not exceeding the square root of n (OEIS A033676), and group the integers into entry layers G_a = {n : g(n) = a}. The composite multipliers of a layer form the finite set X_a = {m composite : a a } the band invariant and d-(.) the predecessor divisor. This paper settles the first three-prime case. For squarefree a = pqr with p lambda, log r / log p -> mu, away from the boundary surfaces lambda = 2, mu = lambda + 2, mu = 2 lambda), |X_pqr| is of order a^2 / (beta_3(a) log^2 a), beta_3(a) = min(qr, p^2 r, p^2 q^2). The three regimes give cloud scales p^2 qr (when lambda 2 and mu max(lambda + 2, 2 lambda)). The upper bound follows from the divisor-pair majorant and refined fiber bound of the companion note. The lower bounds come from three explicit survivor families, one per regime, each shown to lie in X_a by a direct divisor check: am is a product of five distinct primes, and every divisor of am is either at most a or exceeds the square root of am, so the central window (a, sqrt(am)] is empty. The full beta_3 computation is given by enumerating all sixteen unordered admissible divisor pairs in both orderings of the divisor lattice (r pq). The result confirms, for three distinct prime factors, that the order of the count is controlled by the single extremal predecessor product beta(a), even though several divisor-pair fibers contribute and the most populous fiber is generally not the one that sets the scale. On this basis the paper states the general squarefree conjecture |X_a| is of order a^2 / (beta(a) log^2 a) for all squarefree a, supported numerically through four prime factors. The argument is elementary throughout, and all claims are verified by direct computation.