Forty-four unit squares need a square of side seven.

Let s(n) be the side of the smallest square into which n unit squares with pairwise disjoint interiors can be packed, rotations allowed. We prove s(44) = 7. The proof is a weighted closed cover of [0, 7]2: an explicit measure made of 15,525 weighted points and 3,912 uniformly weighted pieces of the interior grid lines, of total mass 2199755099/50000000 < 44, such that every closed unit square in [0, 7]2 has mass at least one. The covering property is verified by two independently written exact rational-arithmetic checkers, one of them both with and without symmetry reduction. The certificate has only 0.4% of slack, and finding it required a new separation step for low-mass “valleys” formed by nearly axis-parallel squares resting on the walls. We describe that step because the same obstruction should appear in other cases with a thin margin.