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 Tₙ and the Hecke correspondences μₚ satisfies the commutation relation μₚ Tₙ = T₍₏ μₚ, with μₚ*μₚ = 1. Under the RN10/Bost–Connes KMS framework at inverse temperature β = 1/2, the expectation of the range projection μₚʳ μₚ^*r evaluates to p^−r/2. At β = 1/2 1. Physically, p^−r/2 is the thermal probability that the winding sector index m is divisible by pʳ 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.