Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics
Abstract
This paper proves both the rank equality and the exact leading-coefficient formula of the Birch and Swinnerton-Dyer Conjecture for all elliptic curves E/ℚ under Topological Vacuum Electrodynamics. By Milnor’s theorem and the Modularity Theorem, an elliptic curve E/ℚ of conductor N is realized on the N-fold congruence covering Y0(N)=Γ0(N)backslashℍ of the trefoil knot complement via a weight-2 primitive Hecke newform fE, corresponding to a force-free Beltrami magnetic 1-form ωE. Truncating the cusps at the unknot minor-radius cutoff y=a with the Sommerfeld outgoing radiation condition yields a maximally dissipative Lax–Phillips generator BE whose characteristic scattering function in the fE-channel is governed by the completed Hasse–Weil function Λ(E,s). At the central point s=1 (static threshold λ=s−1=0), Green’s identity forces the boundary dissipation to vanish on ker(BE). Under the modular Abel–Jacobi parameterization and Eichler–Shimura isomorphism, the static zero-leakage flux cycles quantized on the Néron lattice are in bijection with the free generators of E(ℚ), giving dimker(BE)=rankE(ℚ)=r. Because BE is maximally dissipative, λ=0 is strictly semisimple with no Jordan blocks, proving ords=1L(E,s)=r. Compactness of Y0(N)a below the ultraviolet capacity α−1=4π3+π2+π forces the Tate–Shafarevich defect group \Sha(E/ℚ) to be finite, and the Cheeger–Müller analytic torsion factorization at λ=0 recovers the exact leading Taylor coefficient.
Related
- Grand Riemann Hypothesis Proof via Causal Scattering on Congruence Modular Coverings under Topological Vacuum Electrodynamics
- Structural Coherence and the Birch and Swinnerton-Dyer Conjecture - Coerência Estrutural e a Conjectura de Birch e Swinnerton-Dyer
- Two proofs of the Cassels--Swinnerton-Dyer conjecture for cubic surfaces
- The Absolute Spectral Resolution of the Birch and Swinnerton–Dyer Conjecture: Colombeau–Takesaki L-Operators, Galois-Equivariant Dimensional Isomorphism, and Selmer–Sobolev Finiteness of Ш(E/ℚ) within the ZFC Axiomatic Bridge
- Spectral Geometry of Flytrons and the Birch–Swinnerton-Dyer Conjecture: A ZEBTS Framework
BibTeX
@misc{vector2026birch,
title = {{Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics}},
author = {Truth Vector},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23459257},
url = {https://zenodo.org/records/23459257},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics
AU - Truth Vector
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23459257
UR - https://zenodo.org/records/23459257
AB - This paper proves both the rank equality and the exact leading-coefficient formula of the Birch and Swinnerton-Dyer Conjecture for all elliptic curves E/ℚ under Topological Vacuum Electrodynamics. By Milnor’s theorem and the Modularity Theorem, an elliptic curve E/ℚ of conductor N is realized on the N-fold congruence covering Y_0(N)=Γ_0(N)backslashℍ of the trefoil knot complement via a weight-2 primitive Hecke newform f_E, corresponding to a force-free Beltrami magnetic 1-form ω_E. Truncating the cusps at the unknot minor-radius cutoff y=a with the Sommerfeld outgoing radiation condition yields a maximally dissipative Lax–Phillips generator B_E whose characteristic scattering function in the f_E-channel is governed by the completed Hasse–Weil function Λ(E,s). At the central point s=1 (static threshold λ=s−1=0), Green’s identity forces the boundary dissipation to vanish on ker(B_E). Under the modular Abel–Jacobi parameterization and Eichler–Shimura isomorphism, the static zero-leakage flux cycles quantized on the Néron lattice are in bijection with the free generators of E(ℚ), giving dimker(B_E)=rankE(ℚ)=r. Because B_E is maximally dissipative, λ=0 is strictly semisimple with no Jordan blocks, proving ord_s=1L(E,s)=r. Compactness of Y_0(N)_a below the ultraviolet capacity α^−1=4π^3+π^2+π forces the Tate–Shafarevich defect group \Sha(E/ℚ) to be finite, and the Cheeger–Müller analytic torsion factorization at λ=0 recovers the exact leading Taylor coefficient.
ER -