Spectral Geometry of Flytrons and the Birch–Swinnerton-Dyer Conjecture: A ZEBTS Framework
Abstract
We present the ZEBTS (Zeta–Elliptic–BSD–Spectral) framework, which reformulates the Birch–Swinnerton–Dyer (BSD) conjecture in the language of spectral geometry and operator theory. The central object of the construction is the flytron operator DE,χD_{E,\chi}DE,χ acting on a truncated adelic Hilbert space; its properties are governed by a minimal set of axioms (F1–F6) that encode the arithmetic and geometric structure of an elliptic curve E/QE/\mathbb{Q}E/Q in spectral terms. Under these axioms, we establish a Spectral Reduction Theorem that transforms the classical BSD leading-coefficient formula into an explicit identity involving the regularized determinant of DE,χD_{E,\chi}DE,χ.
The theoretical reduction is complemented by a fully reproducible numerical protocol with rigorous error control. We implement the method for the elliptic curve E=37a1E = 37a1E=37a1 (rank 111) and demonstrate convergence of the spectral estimates to the classical arithmetic invariants: the real period Ω37\Omega_{37}Ω37, the regulator R37R_{37}R37, Tamagawa numbers cpc_pcp, and the special value L′(E37,1)L'(E_{37},1)L′(E37,1). Convergence diagnostics include the truncation residual ΔN\Delta_NΔN, monotonic decay of the relative error, stabilization of the zero-mode count, and residual analysis over the interval s∈[0.9,1.1]s \in [0.9,1.1]s∈[0.9,1.1]. At the highest resolution (Level 3), the spectral estimate for L′(E37,1)L'(E_{37},1)L′(E37,1) achieves an error below 0.7%0.7\%0.7% and is cross-validated against modular-symbol computations, classical BSD invariants, and Néron-model data.
Our approach introduces several novel elements: (i) an axiomatic spectral model for BSD where each axiom directly controls a specific aspect of numerical stability and convergence; (ii) an explicit spectral identity equivalent to the BSD formula, with fully traceable normalization factors; (iii) a systematic local-to-global dictionary mapping spectral densities ρp\rho_pρp to arithmetic invariants cpc_pcp; and (iv) a reproducible computational pipeline with automated verification checks, unit tests, and regression suites. The results for 37a1 provide concrete evidence that the ZEBTS framework yields a viable and rigorous pathway for studying the BSD conjecture via spectral methods. The protocols and data are designed for full reproducibility and are aligned with the standards expected by peer-reviewed journals in mathematical physics and number theory.
Related
- 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
- 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
- Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics
- ZEBTS: Discrete Geometry, Flytrons, and Emergent Spectra
BibTeX
@misc{mukha2026spectral,
title = {{Spectral Geometry of Flytrons and the Birch–Swinnerton-Dyer Conjecture: A ZEBTS Framework}},
author = {Anatolii Mukha},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23618007},
url = {https://zenodo.org/records/23618007},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Geometry of Flytrons and the Birch–Swinnerton-Dyer Conjecture: A ZEBTS Framework
AU - Anatolii Mukha
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23618007
UR - https://zenodo.org/records/23618007
AB - We present the ZEBTS (Zeta–Elliptic–BSD–Spectral) framework, which reformulates the Birch–Swinnerton–Dyer (BSD) conjecture in the language of spectral geometry and operator theory. The central object of the construction is the flytron operator DE,χD_{E,\chi}DE,χ acting on a truncated adelic Hilbert space; its properties are governed by a minimal set of axioms (F1–F6) that encode the arithmetic and geometric structure of an elliptic curve E/QE/\mathbb{Q}E/Q in spectral terms. Under these axioms, we establish a Spectral Reduction Theorem that transforms the classical BSD leading-coefficient formula into an explicit identity involving the regularized determinant of DE,χD_{E,\chi}DE,χ. The theoretical reduction is complemented by a fully reproducible numerical protocol with rigorous error control. We implement the method for the elliptic curve E=37a1E = 37a1E=37a1 (rank 111) and demonstrate convergence of the spectral estimates to the classical arithmetic invariants: the real period Ω37\Omega_{37}Ω37, the regulator R37R_{37}R37, Tamagawa numbers cpc_pcp, and the special value L′(E37,1)L'(E_{37},1)L′(E37,1). Convergence diagnostics include the truncation residual ΔN\Delta_NΔN, monotonic decay of the relative error, stabilization of the zero-mode count, and residual analysis over the interval s∈[0.9,1.1]s \in [0.9,1.1]s∈[0.9,1.1]. At the highest resolution (Level 3), the spectral estimate for L′(E37,1)L'(E_{37},1)L′(E37,1) achieves an error below 0.7%0.7\%0.7% and is cross-validated against modular-symbol computations, classical BSD invariants, and Néron-model data. Our approach introduces several novel elements: (i) an axiomatic spectral model for BSD where each axiom directly controls a specific aspect of numerical stability and convergence; (ii) an explicit spectral identity equivalent to the BSD formula, with fully traceable normalization factors; (iii) a systematic local-to-global dictionary mapping spectral densities ρp\rho_pρp to arithmetic invariants cpc_pcp; and (iv) a reproducible computational pipeline with automated verification checks, unit tests, and regression suites. The results for 37a1 provide concrete evidence that the ZEBTS framework yields a viable and rigorous pathway for studying the BSD conjecture via spectral methods. The protocols and data are designed for full reproducibility and are aligned with the standards expected by peer-reviewed journals in mathematical physics and number theory.
ER -