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…