The Square P₂ Boundary and Bilinear Semiprime Capacity: Cube-Root RootRough, Least-Factor Rows, Cofactor Switching and the Buchstab log 2 Margin

At the cube-root cutoff z³ ≥ (n+1)² − 1 every rough survivor is prime or semiprime, giving the exact identity P(n) = S₃(n) − E₂(n) (least-factor rows, cofactor switching); 'sublinear fit-depth' is demoted to an optional route that is 'much stronger than the original Legendre statement'; a standard linear sieve gives lower bound 0 at s = 3/2 (a no-go); the audit for 100 ≤ n ≤ 3000 finds primes are only 60.16% of S₃, so 'prime majority' is not a usable theorem.