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| Format: | Recurso digital |
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Zenodo
2026
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| Hasła przedmiotowe: | |
| Dostęp online: | https://doi.org/10.5281/zenodo.18780549 |
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- <p>Papers 9 and 10 in <em>The Geometry of the Critical Line</em> programme.</p> <p>This record contains two further papers in the programme, completing the analytic and geometric bridge between the cover equation and the SCT 5-manifold.</p> <p><strong>Paper 9: The Analytic Vortex and the Lambert Spectrum</strong> 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.</p> <p><strong>Paper 10: The Phase-Locking Bridge</strong> 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.</p> <p>Both papers maintain the strict category boundaries emphasised throughout the programme: analytic vortex (complex plane) vs. geometric measure (manifold). Open problems are stated explicitly.</p> <p>Status: these papers extend the framework; they do not claim a proof of the Riemann Hypothesis.</p>