| _version_ | 1866901878174908416 |
|---|---|
| author | Kramarenko-Byrd, Pavel |
| author_facet | Kramarenko-Byrd, Pavel |
| contents | <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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18780549 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Geometry of the Critical Line: Further Developments (Papers 9–10) Kramarenko-Byrd, Pavel Riemann zeta function SCT 5-manifold Lambert W function exponential integral Picard's theorem phase transition analytic vortex transcendental equation special functions Lambert spectrum essential singularity cover equation exponential integral Ei inverse problem Pure mathematics Complex analysis <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> |
| title | The Geometry of the Critical Line: Further Developments (Papers 9–10) |
| topic | Riemann zeta function SCT 5-manifold Lambert W function exponential integral Picard's theorem phase transition analytic vortex transcendental equation special functions Lambert spectrum essential singularity cover equation exponential integral Ei inverse problem Pure mathematics Complex analysis |
| url | https://doi.org/10.5281/zenodo.18780549 |