The Spatial Sterility Theorem and the Arithmetic Lift Conjecture

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. Paper 49 in the open-access research programme "The Geometry of the Critical Line." This paper proves the Spatial Sterility Theorem: the SCT σ-bubble operator in every winding sector m is unitarily equivalent to −d²/dx² + m² on the flat interval [−2, 2] with Dirichlet boundary conditions, achieved by a gauge transformation that eliminates the chiral cross-term (using the algebraic identity 1 − C²D ≡ D) combined with a Liouville coordinate change that absorbs the metric curvature. The spectrum is therefore exactly λ_{m,n} = m² + (nπ/4)² for all m and n. The previously reported finite-element eigenvalue drift is identified as numerical pollution with no analytic content. The reduced spectral zeta is exactly Z_σ(u) = (4/π)^{2u} ζ(2u), the completed form ξ_σ(u) is entire, and the completion-factor decomposition identifies the endpoint values h(i/2) + h(−i/2) unconditionally. The reduced spatial spectral model is therefore sterile: it contains the Weyl volume, the archimedean background, and the completion-factor endpoint terms, but no independent nontrivial-zero data beyond the classical ξ(2u) package. The paper formulates the Arithmetic Lift Conjecture — that the Bost–Connes algebraic quotient acting on the sterile carrier produces the full Weil distribution — as the single remaining open problem, replacing Paper 48's two conjectures. Three theorems are proved unconditionally (Gauge Elimination, Spatial Sterility, Reduced Completion-Factor Endpoint Identification). One conjecture is stated (Arithmetic Lift). The Riemann Hypothesis is not proved.