Hahn-Banach Separation and the Local-to-Global Gap in the Weil Criterion: Theoretical Notes

Theoretical notes formalizing the precise form of the local-to-global gap in the Hilbert-Pólya programme via Cartan torsion (June 2026). The gap is identified exactly: showing that T_ρ₀(Φ)<0 (Formulation B, Li-Bombieri-Lagarias, Cauchy representation) implies W(Φ★Φ̃)<0 (Formulation A, Weil 1952) requires the existence of a bounded positive operator T relating the two functionals on the cone of admissible functions. Contents:- Precise formulation of the two representations (Weil A and Li-Bombieri-Lagarias B)- The Hahn-Banach separation framework and why it requires an operator T relating F_A and F_B- Connection to Burnol's Sonin spaces and the functional equation symmetry s↔1-s- How Giribet's review of Paper I Appendices A-B unlocks the Hahn-Banach hypotheses- Road map for Paper IV These are working notes, not a preprint of complete results.