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| Formatua: | Recurso digital |
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Zenodo
2026
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| Sarrera elektronikoa: | https://doi.org/10.5281/zenodo.19084943 |
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Etiketa erantsi
Etiketarik gabe, Izan zaitez lehena erregistro honi etiketa jartzen!
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| _version_ | 1866901937152065536 |
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| author | Keeble, Clifford |
| author_facet | Keeble, Clifford |
| contents | <p>Paper 140 validated the quintic compiler at rank 1, verifying the five-factor grammar on the curve 37a1 (ε = −1, gate open). This paper completes the rank 0 channel by running the compiler on 11a1: the elliptic curve y² + y = x³ − x² − 10x − 20, which is the modular curve X₀(11) and the curve of smallest conductor in the Cremona database. The parity gate is closed (ε = +1): the archimedean ψ-mode (ε_∞ = −1) is cancelled by the split multiplicative ψ-mode at 11 (w₁₁ = −1), giving ε = (−1)(−1) = +1. No zero is forced at s = 1, and L(E, 1) ≠ 0 confirms rank 0. The five-factor grammar produces L(E, 1) = Ω · c₁₁ · |Sha| / |E(ℚ)_tors|² = 1.269209 × 5 × 1 / 25 = Ω/5. The rational BSD quotient is 1/5 — the quintic filter's simplest output. The number 5 pervades the curve: conductor 11 = L₅ (the fifth Lucas number, equal to φ⁵ + ψ⁵), torsion order 5, Tamagawa number 5, Kodaira type I₅ (five Néron components), and isogenies of degree 5 and 25 within the isogeny class. The compiler's first curve is tuned entirely to the quintic. With both rank 0 and rank 1 channels validated, the compiler passes every test case where BSD is a theorem.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19084943 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Boot Sequence: BSD at Rank 0 on the Compiler's First Curve Keeble, Clifford Birch and Swinnerton-Dyer conjecture, elliptic curves, golden ratio, quintic filter, root number, parity gate, modular curve, Tamagawa number, L-function, icosahedral symmetry, Poincaré homology sphere, Bootstrap Universe Programme <p>Paper 140 validated the quintic compiler at rank 1, verifying the five-factor grammar on the curve 37a1 (ε = −1, gate open). This paper completes the rank 0 channel by running the compiler on 11a1: the elliptic curve y² + y = x³ − x² − 10x − 20, which is the modular curve X₀(11) and the curve of smallest conductor in the Cremona database. The parity gate is closed (ε = +1): the archimedean ψ-mode (ε_∞ = −1) is cancelled by the split multiplicative ψ-mode at 11 (w₁₁ = −1), giving ε = (−1)(−1) = +1. No zero is forced at s = 1, and L(E, 1) ≠ 0 confirms rank 0. The five-factor grammar produces L(E, 1) = Ω · c₁₁ · |Sha| / |E(ℚ)_tors|² = 1.269209 × 5 × 1 / 25 = Ω/5. The rational BSD quotient is 1/5 — the quintic filter's simplest output. The number 5 pervades the curve: conductor 11 = L₅ (the fifth Lucas number, equal to φ⁵ + ψ⁵), torsion order 5, Tamagawa number 5, Kodaira type I₅ (five Néron components), and isogenies of degree 5 and 25 within the isogeny class. The compiler's first curve is tuned entirely to the quintic. With both rank 0 and rank 1 channels validated, the compiler passes every test case where BSD is a theorem.</p> |
| title | The Boot Sequence: BSD at Rank 0 on the Compiler's First Curve |
| topic | Birch and Swinnerton-Dyer conjecture, elliptic curves, golden ratio, quintic filter, root number, parity gate, modular curve, Tamagawa number, L-function, icosahedral symmetry, Poincaré homology sphere, Bootstrap Universe Programme |
| url | https://doi.org/10.5281/zenodo.19084943 |