Logarithmic Density, Metric Correction, and the Chiral Spectral Obstruction

Paper 40 in the "Geometry of the Critical Line" programme. This paper proves three culminating results of the Symmetric Complex Transcendental (SCT) geometry programme: Density theorems: the raw winding-sector eigenvalue counting function satisfies Nᵣaw (E) ~ E² log E / (4ω), with a conditional post-quotient match to the Riemann counting N (T) ~ T log T / (2π) that uniquely determines the metric constant k = π/8. The chiral spectral obstruction (Theorem 6. 5): the corrected Frobenius boundary analysis gives indicial roots with Re (r₁) ≈ 3/2 and Re (r₂) ≈ −1/2. The Friedrichs form-domain condition (finite kinetic and potential energy, inherited from the global Laplace–Beltrami operator) excludes the singular r₂-branch. For the admissible r₁-branch, all boundary terms vanish as O (ε²), and the Wronskian identity yields 0 = ∫ (A′−2B) |ψ|² > 0 — a contradiction. No real eigenvalues exist in any nonzero winding sector of the Friedrichs domain. The spectral gap grows linearly with the winding number: |Im (E²) | ≥ mπ/4 for k = π/8. This is the SCT specialisation of the general theorem proved in RN11 (DOI: 10. 5281/zenodo. 19230352). A conditional critical-line theorem (Theorem 10. 3): assuming only the SCT–Connes dictionary (that the SCT critical sector is unitarily equivalent to the archimedean component of the Connes–Marcolli adele class space), the spectral exclusion forces all nontrivial zeros of the Riemann zeta function onto the critical line. The Riemann Hypothesis remains open. The paper states exactly where geometry ends and arithmetic must enter.