The Geometry of the Critical Line: Further Developments (Papers 9–10)

Papers 9 and 10 in The Geometry of the Critical Line programme. This record contains two further papers in the programme, completing the analytic and geometric bridge between the cover equation and the SCT 5-manifold. Paper 9: The Analytic Vortex and the Lambert Spectrum promotes the cover equation Φ + exp(iπ − 1/Φ) = 0 from a discrete root condition to a continuous complex function C(z) = z − exp(−1/z). Its antiderivative contains the Exponential Integral Ei(−1/z), introducing an essential singularity at z = 0. Picard's Great Theorem formalises the "Left Modality" (phase collapse) as a theorem-level analytic obstruction. The scaled constraint Cα(z) = z − exp(−α/z) yields a countable Lambert spectrum zₖ(α) = −α/Wₖ(−α), all roots simple and structurally stable except at the phase transition α = 1/e. Paper 10: The Phase-Locking Bridge defines a concrete bridge map f(α) = Im(Φ₀(α)) with Φ₀(α) = −α/W₀(−α) and solves numerically the inverse problem f(α) = 1/√2. This yields a unique calibration constant α⋆ ≈ 0.9989095719923, linking the analytic scale of the cover constraint (Paper 9) to the metric coupling coefficient k = 1/√2 of the SCT 5-manifold (Paper 4). The correspondence is presented as an explicit inverse problem, not a derivation from first principles. Both papers maintain the strict category boundaries emphasised throughout the programme: analytic vortex (complex plane) vs. geometric measure (manifold). Open problems are stated explicitly. Status: these papers extend the framework; they do not claim a proof of the Riemann Hypothesis.