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, yet is invisible to its second-order statistics. 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; the conjugate factorizations 4n²-a² = (2n-a)(2n+a) with a ≤ √n form a deterministic smooth family; and the smooth density tracks the Dickman value ρ(2) = 1-ln 2 with a first-order Θ(1/log x) correction of empirical constant ≈ 0.57. On the second-order side, the prime counts π(J_n) are Gaussian with variance about one half of the Poisson value — the Montgomery–Soundararajan prediction for windows of length √x — with vanishing excess kurtosis and no detectable correlation with the arithmetic of n; each of these is generic to a √x window and carries no trace of the center. 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 (an L² fact) from the open every-n statement (an L⁴ Gaussian-tail condition that the data shows holds at the required size).