Packing unit squares in a square of side just below an integer: a two-strip construction with O(√K) deficit

Let s(n) be the side of the smallest square containing n non-overlapping unit squares. We show that for every integer K ≥ 16, s(n) < K whenever n ≤ K² − 8√K − 19; equivalently, the wasted area is W(s) ≤ 8√K + 19 for s in [K − 1/K, K). The best known general bound W(s) = O(s3/5) gives only a deficit of order K3/5 in this regime. The proof is an explicit construction: two strips of squares tilted by an angle of order K−1/2, one on the bottom wall and one on the right wall, with an axis-parallel grid elsewhere. Exact rational certificates of 683 improved upper bounds obtained with optimised versions of this construction are deposited at https://doi.org/10.5281/zenodo.23624433.