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 <= m <= a^2 and g(am) = a}, the rows of OEIS A163925, with sizes A163926. Companion notes determine the size |X_a| for prime, prime-power, and squarefree two-prime layers, where in each case |X_a| is of order a^2 / (beta(a) log^2 a) with beta(a) = min{ d-(d) d-(e) : d, e divide a, de > a } the band invariant and d-(.) the predecessor divisor. This paper settles the first three-prime case. For squarefree a = pqr with p < q < r, the band invariant is shown to be beta(a) = min(qr, p^2 r, p^2 q^2), and the main result is that, in fixed logarithmic shape (log q / log 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 < lambda + 2), q^2 r (when lambda > 2 and mu < 2 lambda), and r^2 (when 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 and 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.