The Reduced Observable and the Scalar Weil Evaluator

Scoped supersession notice (22 September 2026). This record belongs to the Papers 49–52 and RN50–RN52 source arc consolidated in Restricted Weil Forms: Constraints, Negative Index, and Certified Sign, by Pavel V. Kramarenko-Byrd (DOI: 10.5281/zenodo.22885925). For the restricted-Weil-form material treated in that article, the new article supersedes the earlier programme presentation and is the current consolidated reference. Supersession is limited to that scope; other results, geometric claims, arithmetic-lift constructions, and open problems in this record are not thereby replaced or resolved. The original description below and deposited files are retained for provenance. 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 spectral zeta ξ_σ produces all three Weil terms as residue classes of a single contour integral: the prime sum via the free-energy derivative F'_θ(s) = −d/ds log ζ(2s) (simultaneously performing logarithmic differentiation and Möbius cancellation to produce the von Mangoldt series), the endpoint terms via the completion poles at u = 0 and u = 1/2, and the archimedean term via the digamma kernel. The scalar Weil evaluator is a closed theorem: the sterile carrier spectrum determines the complete Weil distribution through purely analytic operations. No crossed-product algebra is used. The later positivity papers take this scalar evaluator as their starting point. This note does not prove Weil positivity, does not prove the Riemann Hypothesis, and does not construct a Bost–Connes internal realisation.