Real Spectral Exclusion for Singular Chiral Sturm–Liouville Operators

Research Note 11 in the "Geometry of the Critical Line" programme. We prove that a class of singular non-self-adjoint Sturm–Liouville operators with chiral (first-order) coupling admits no real eigenvalues in the Friedrichs form domain. The operator family is H^ (m) = −d²/dx² + Vₘ (x) + imA (x) d/dx + imB (x) on a bounded interval with confining singularities at both endpoints, where m is a nonzero integer parameter. The proof proceeds in three steps: (1) a Frobenius analysis classifies local solutions into a regular branch (|ψ| = O (ε^3/2) ) and a singular branch (|ψ| = O (ε^−1/2) ) ; (2) the Friedrichs form-domain condition (finite kinetic and potential energy) excludes the singular branch; (3) a Wronskian identity, with all boundary terms vanishing for the regular branch, yields 0 = ∫ (A′−2B) |ψ|² > 0, a contradiction. The positivity condition A′−2B > 0 holds for any cross-coupling g with g′ > 0. The spectral gap grows linearly with |m|. No perturbation theory is used. The result is applied to the Symmetric Complex Transcendental (SCT) 5-manifold family, where it provides the spectral exclusion theorem used in the conditional Riemann Hypothesis reduction of Paper 40 (DOI: 10. 5281/zenodo. 19230354).