The Reduced Observable and the Scalar Weil Evaluator

Research Note 52 in the "Geometry of the Critical Line" programme. This note fixes the reduced scalar observable produced by the sterile carrier (Paper 49) and assembles the scalar Weil evaluator from its three analytic pieces. The Jacobi reduction K_h(ξ) = (Φ(ξ,a) − 1)·Θ_σ(a), exact via Poisson summation, is identified as the normalized partial trace over the σ-sector at the semigroup level. The sterile Hilbert-space factorization H = H_θ ⊗ H_σ with commuting operators produces a semigroup splitting e^{−aD²} = e^{−aM²} ⊗ e^{−aL_σ}, and the normalized σ-partial trace recovers the angular heat kernel — giving the Jacobi reduction its exact geometric meaning. The log-derivative decomposition of the completed…