The free twin-prime Bost-Connes system is unconditionally supercritical: resolving the phase dichotomy via exact partial sums

For the free (Toeplitz-Cuntz) variant of a localized Bost-Connes system the Gibbs series is geometric in the fugacity L_S(beta), so the phase dichotomy at the classical critical point is the single comparison L_S(1) vs 1. For the twin primes the Brun-identity route is inconclusive because the familiar value of Brun's constant is a Hardy-Littlewood extrapolation, not a theorem. The dichotomy is resolved unconditionally by the instrument the problem admits: partial sums of a positive series are rigorous lower bounds, and a deterministic segmented sieve to 1.1 x 10^10 (29,889,853 twin pairs) gives sum 1/p = 1.00195935 > 1. Hence the free twin-prime system is in Regime (S), beta_c^free > 1: on an explicit interval above beta = 1 the classical system is in its closed broken phase while the free system admits no KMS state at all. The precise beta_c^free is recorded as tail-conditional. Two methodological remarks close the paper: occupancy of residue classes does not imply divergence of per-character detecting series (Chen's theorem in arithmetic progressions remains the named gap for Chen-family uniqueness), and average correlation from Rankin-Selberg poles does not touch the pointwise parity barrier.