Square-Anchored Obstruction Residue Field, Atomic Reduction and the Exact Sieve Identity

Splits the obstruction into atomic primes with forbidden phase −n² mod a, proves the survivor–prime equivalence, CRT collapse and the exact anchored inclusion–exclusion L(n) = 2n·φ(Q_n)/Q_n + Δ(n) (Legendre iff L(n) > 0 for all n ≥ 2); shows the generic Jacobsthal shortcut is too strong (j(p_49#) = 742 > 2n = 454) and that no fixed non-constant integer polynomial is prime for all large n; finite check n ≤ 3000 (minimum prime count 2, at n = 1, 2, 3, 5).