The Locked Window. A Controlled Map of Squarefree Pair Structure.

Abstract

This note re-runs the discipline of the companion paper ‘Harmonic Architecture of Prime Pairs’ [1] — construct, control, classify — on a second deterministic territory: the squarefree numbers. The instrument is not imported. The construct is rebuilt from the domain's own exclusion law, and every control is run from zero. Five results. (i) The corridor audit is exact: squarefree numbers fill the three live residue classes mod 4 in exact thirds, class density , at every scale measured. (ii) The pair law is exact: the density of squarefree pairs follows the closed product rule , with when divides and 2 otherwise. Maximum residual over gaps 1–120: . The fingerprint is a rational step spectrum — release factors of exactly 3/2 at , 8/7 at , 24/23 at , composing multiplicatively to 12/7 at . (iii) The correlation is repulsive, the reverse of the prime domain: against a baseline of 0.6079. The independence model predicts 0.3696 and is falsified. (iv) The window does not breathe. Density, pair density, and conditional correlation are flat from to to four decimals. We state and prove the Regime Dichotomy: breathing is the signature of thinning sequences, locking the signature of positive-density ones. (v) The pair pivot is confined to a single parity sublattice of the step-6 lattice, with profiles 2:3:2 (odd gaps) and 3:2:2 (even gaps) fixed exactly by the pair's excluded residues mod 9. The discipline detects each domain's regime and resolves its own law. It does not memorize the previous domain's answer.