The Critical-Line Amplitude p^{−r/2} as a KMS State at β = 1/2

Research Note 21 in the "Geometry of the Critical Line" programme. RN20 identified the primitive-orbit factor log p as a geometric primitive-length factor in the orbital integral on the flat 2-torus. The remaining arithmetic inputs for the Weil prime-side kernel are the critical-line amplitude p^{−r/2} and the dimensional reduction t^{−1} → t^{−1/2}. This note realises on the SCT critical sector the abstract KMS amplitude mechanism introduced in RN10. The algebra generated by the prime translations T_n and the Hecke correspondences μ_p satisfies the commutation relation μ_p T_n = T_{np} μ_p, with μ_p*μ_p = 1. Under the RN10/Bost–Connes KMS framework at inverse temperature β = 1/2, the expectation of the range projection μ_p^r μ_p^{*r} evaluates to p^{−r/2}. At β = 1/2 < 1, the Bost–Connes system is in its unique scaling-invariant KMS phase; range projections evaluate directly as p^{−rβ} without the partition-function normalisation ζ(β)^{−1} that appears in the Gibbs trace regime β > 1. Physically, p^{−r/2} is the thermal probability that the winding sector index m is divisible by p^r at the critical temperature. Combined with RN20's log p, this isolates the dimensional reduction as the sole remaining arithmetic input. The assembly is formal and conditional; no arithmetic theorem or completed dictionary claim is made. The dimensional reduction is treated in RN22. Part of a 46-paper open-access programme on the geometry of the Riemann zeta function's critical line, anchored by the SCT 5-Manifold and the cover equation Φ + e^{iπ − 1/Φ} = 0.