Resonance Fields and the Constructive Emergence of BSD Group

This paper presents a constructive, field-theoretic analog of the Birch–Swinnerton–Dyer (BSD) Conjecture using nonlinear twist-compression dynamics and symbolic resonance locking. Rather than imposing algebraic group structure explicitly, the model allows symbolic generators, torsion, rank, and regulator analogs to emerge organically from spatially evolving fields.

We introduce a nonlinear partial differential equation whose stable resonance zones—termed lock zones—encode the symbolic counterparts of elliptic curve invariants. A synthetic LLL-function is constructed from field entropy, and spectral mass gap formation is analyzed through cymatic analogs. Symbolic group closure, rational approximation, and Farey-circle geometry are validated using a suite of reproducible simulations.

The work provides a fully reproducible simulation archive, and while not a proof of the BSD Conjecture, it offers a complementary, physical framework in which the core algebraic structures arise from resonance-based dynamics.

Highlights:

• Emergent symbolic group structure from nonlinear PDEs

• Synthetic Lsym(s)L_{\text{sym}}(s)Lsym(s) constructed via entropy-based resonance

• Triplet closure, regulator analogs, and torsion signatures from field behavior

• Connection to Ford circles, Farey sequences, and rational geometry

• Includes 30+ Python scripts with full reproducibility pipeline

Related software archive:
👉 BSD-Resonance Simulation Suite — https://doi.org/10.5281/zenodo.15380592