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2025
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| Online-Zugang: | https://doi.org/10.5281/zenodo.17171003 |
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| _version_ | 1866902101065465856 |
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| author | Platek, Nir |
| author_facet | Platek, Nir |
| contents | <p>This work develops a finite-scale robustness framework that yields a complete, <strong>fully rigorous solution to the Birch–Swinnerton–Dyer conjecture for elliptic curves over ℚ</strong>. The method establishes finite-window positivity of the explicit GL(2) kernel through a deterministic barrier argument and closes the formula via a nonnegative compensated functional and Lipschitz stability at the central line.</p> <p><strong>Four Pillars</strong></p> <ul> <li> <p><strong>(P1) Gaussian flow on Paley–Wiener slices.</strong> A frequency-side Gaussian multiplier preserves Paley–Wiener geometry and produces a monotone Sobolev barrier that concentrates spectral mass near the central window.</p> </li> <li> <p><strong>(P2) Coarea–barrier transport.</strong> The flowed prime-shell functional has a slice-uniform Riesz-gradient denominator, yielding deterministic suppression of prime contributions at positive flow without Sard or transversality assumptions.</p> </li> <li> <p><strong>(P3) Archimedean center and Tauberian window.</strong> An explicit archimedean integral provides a strictly positive central value; combined with local Lipschitz control and a small-<strong>Δ</strong> Tauberian window, this stabilizes the explicit formula on a finite frequency interval.</p> </li> <li> <p><strong>(P4) Sharp de-mollification absorption.</strong> A quantified first-moment inequality separates archimedean and near-window prime contributions, absorbing the error into the Sobolev barrier and enforcing nonnegativity of the compensated functional.</p> </li> </ul> <p><strong>Technical Highlights</strong></p> <ul> <li> <p>Explicit archimedean constant via the trigamma kernel with rigorous analytic enclosures.</p> </li> <li> <p>Slice-uniform coarea Jacobian <em>c</em><sub>∇</sub>(<em>s</em>) > 0 ensuring constant denominator bounds.</p> </li> <li> <p>Gaussian shell-sum envelopes with exponential decay.</p> </li> <li> <p>Beurling–Selberg extremals and Rankin–Selberg <em>L</em><sup>2</sup>-energy as an independent audit route with explicit thresholds (<em>N</em><sub>0</sub>, δ*).</p> </li> <li> <p>Comprehensive constants ledger, feasibility schedule, and Sage/Pari scripts for verification.</p> </li> </ul> <p><strong>Result</strong></p> <p><strong>These components yield a rigorous resolution of the Clay Millennium Problem</strong> on the Birch–Swinnerton–Dyer conjecture: for every elliptic curve <em>E</em>/<strong>ℚ</strong>, the analytic rank equals the Mordell–Weil rank and the full leading-coefficient formula holds with Sha(<em>E</em>) finite.</p> <p><strong>Outlook</strong></p> <p>The robustness principle—finite-scale positivity stabilized by Gaussian flow, coarea–barrier transport, Tauberian evaluation, and sharp de-mollification—extends beyond elliptic curves to modular abelian varieties, symmetric-power and Rankin–Selberg <em>L</em>-functions, and computable BSD certificates for curve families.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17171003 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | Birch–Swinnerton–Dyer: A Robustness Framework via Gaussian Flow, Coarea–Barrier Transport, Paley–Wiener Damping, and a Small–Δ Tauberian Window Platek, Nir <p>This work develops a finite-scale robustness framework that yields a complete, <strong>fully rigorous solution to the Birch–Swinnerton–Dyer conjecture for elliptic curves over ℚ</strong>. The method establishes finite-window positivity of the explicit GL(2) kernel through a deterministic barrier argument and closes the formula via a nonnegative compensated functional and Lipschitz stability at the central line.</p> <p><strong>Four Pillars</strong></p> <ul> <li> <p><strong>(P1) Gaussian flow on Paley–Wiener slices.</strong> A frequency-side Gaussian multiplier preserves Paley–Wiener geometry and produces a monotone Sobolev barrier that concentrates spectral mass near the central window.</p> </li> <li> <p><strong>(P2) Coarea–barrier transport.</strong> The flowed prime-shell functional has a slice-uniform Riesz-gradient denominator, yielding deterministic suppression of prime contributions at positive flow without Sard or transversality assumptions.</p> </li> <li> <p><strong>(P3) Archimedean center and Tauberian window.</strong> An explicit archimedean integral provides a strictly positive central value; combined with local Lipschitz control and a small-<strong>Δ</strong> Tauberian window, this stabilizes the explicit formula on a finite frequency interval.</p> </li> <li> <p><strong>(P4) Sharp de-mollification absorption.</strong> A quantified first-moment inequality separates archimedean and near-window prime contributions, absorbing the error into the Sobolev barrier and enforcing nonnegativity of the compensated functional.</p> </li> </ul> <p><strong>Technical Highlights</strong></p> <ul> <li> <p>Explicit archimedean constant via the trigamma kernel with rigorous analytic enclosures.</p> </li> <li> <p>Slice-uniform coarea Jacobian <em>c</em><sub>∇</sub>(<em>s</em>) > 0 ensuring constant denominator bounds.</p> </li> <li> <p>Gaussian shell-sum envelopes with exponential decay.</p> </li> <li> <p>Beurling–Selberg extremals and Rankin–Selberg <em>L</em><sup>2</sup>-energy as an independent audit route with explicit thresholds (<em>N</em><sub>0</sub>, δ*).</p> </li> <li> <p>Comprehensive constants ledger, feasibility schedule, and Sage/Pari scripts for verification.</p> </li> </ul> <p><strong>Result</strong></p> <p><strong>These components yield a rigorous resolution of the Clay Millennium Problem</strong> on the Birch–Swinnerton–Dyer conjecture: for every elliptic curve <em>E</em>/<strong>ℚ</strong>, the analytic rank equals the Mordell–Weil rank and the full leading-coefficient formula holds with Sha(<em>E</em>) finite.</p> <p><strong>Outlook</strong></p> <p>The robustness principle—finite-scale positivity stabilized by Gaussian flow, coarea–barrier transport, Tauberian evaluation, and sharp de-mollification—extends beyond elliptic curves to modular abelian varieties, symmetric-power and Rankin–Selberg <em>L</em>-functions, and computable BSD certificates for curve families.</p> |
| title | Birch–Swinnerton–Dyer: A Robustness Framework via Gaussian Flow, Coarea–Barrier Transport, Paley–Wiener Damping, and a Small–Δ Tauberian Window |
| url | https://doi.org/10.5281/zenodo.17171003 |