The statistical mechanics of sieve-theoretic primes: the critical spectrum, the log-power dichotomy, and two families that are never unique

For six sieve-theoretic prime families the critical exponents equal the counting exponents, and the open/closed character of the symmetry-broken phase is the power of the logarithm, hence the number of simultaneous prime conditions. Confinement to a single residue class obstructs only when the class lies in the kernel of the character: Landau and Friedlander-Iwaniec primes occupy the kernel class 1 mod 4 and their KMS state is never unique at any temperature, while twin and Sophie Germain primes occupy the generating class 2 mod 3, so the detecting sets are cofinite and no congruence obstructs them at all, consistently with Theorem 5.5(5) and Lemma 5.6 of the foundational paper. All class occupancies and counts are verified by full enumeration to 200000.