Rough and Smooth Numbers in Square-Centered Intervals: First-Order Structure and Second-Order Genericity

For n ≥ 1 let J_n = [4n²-n, 4n²+n], the interval of length 2n+1 centered at the perfect square (2n)², and set z = 2n+1 ≈ √x at x = 4n². We study the joint distribution of z-rough and z-smooth integers in J_n, and in the adjacent interval Q_n = [(2n+1)², (2n+2)²], computed for all n ≤ 10^5 with confirming spot samples at n ≈ 10^6. The organizing finding is a separation between scales: the square-centering imprints sharply on the first-order arithmetic of the interval, while the second-order statistics tested here show no detectable square-centering signature. On the first-order side, z-roughness and primality coincide in J_n; the adjacent interval Q_n contains a z-rough composite if and only if 2n+1 is prime; and the conjugate factorizations 4n²-a² = (2n-a)(2n+a) with a ≤ √n form a deterministic near-threshold smooth family. The measured z-smooth density is empirically consistent with approach to the Dickman value ρ(2) = 1-ln 2 through a correction c/log x with c ≈ 0.57. On the second-order side, the Poisson-scale second moment of π(J_n) is about one half of the Poisson benchmark — consistent with the Montgomery–Soundararajan prediction for windows of length √x — while the fourth-moment ratio is near its Gaussian benchmark and the tested arithmetic correlations are negligible. These findings are consistent with generic √x-window behavior, but do not establish an identity of distributions. We place J_n as the z = √x endpoint of the spectrum of rough-numbers-in-short-interval problems whose sub-barrier end was recently settled by Gafni and Tao, and record how the moment hierarchy separates the known almost-all statement from the substantially stronger fourth-moment control that would imply only finitely many prime-free J_n.