Birch and Swinnerton-Dyer Conjecture: Part I — Research Framework, Computational Certificates, and Open Problems

Abstract

Birch and Swinnerton-Dyer Conjecture: Part I — Research Framework, Computational Certificates, and Open Problems Author: Philippe BeauchampORCID: 0009-0003-7407-394XPublication: Zenodo, 2026Version: 1.0.0 — Part I — Research in Progress Overview This preprint documents Part I of an ongoing independent mathematical research program investigating the Birch and Swinnerton-Dyer (BSD) Conjecture through elliptic-curve arithmetic, computational certificate protocols, regulator and period analysis, integral lattice methods, and a proposed geometric framework inspired by ammonite spiral dynamics. The research record consolidates the current mathematical framework, certificate specifications, theorem statements, computational verification plans, technical audits, and identified open obligations developed during the investigation. It is intended to preserve the progression of the research, provide a structured basis for independent mathematical examination, and establish a documented foundation for subsequent parts. The central objective of the BSD investigation is to examine the relationships among the analytic behavior of an elliptic curve's (L)-function and its arithmetic invariants, including the Mordell–Weil group, regulator, real period, Tamagawa factors, torsion subgroup, and Tate–Shafarevich group. The research also investigates whether selected geometric and dynamical constructions can be connected to these arithmetic objects through rigorous, explicitly verified mathematical correspondences. This release is a research-in-progress preprint. It does not claim a proof of the general Birch and Swinnerton-Dyer Conjecture. The distinction between established mathematics, computational evidence, derived identities, proposed constructions, and unresolved proof obligations is maintained throughout the research record. 1. Research Scope and Mathematical Objectives The research program is organized around the following connected areas: BSD arithmetic framework: examination of the relationship between analytic rank and algebraic rank, together with the arithmetic invariants appearing in the BSD formula. Elliptic-curve data and input binding: identification of the precise curve, original parameters, arithmetic data, and hypotheses required for curve-specific verification. Regulator and integral lattice analysis: investigation of lattice bases, integral relations, saturation, indices, determinant identities, and the conditions under which these calculations represent the arithmetic of a specified elliptic curve. Analytic and computational verification: development of verification protocols for (L)-function calculations, analytic rank, period normalization, numerical integration, and related quantities, subject to exact input data and appropriate error control. Local and global arithmetic invariants: organization of verification requirements associated with torsion, Tamagawa factors, periods, regulators, and the Tate–Shafarevich group. Geometric-to-arithmetic correspondence: investigation of a proposed connection between an Ammonite-inspired geometric model and elliptic-curve arithmetic, with the existence and validity of any such correspondence treated as a separate mathematical obligation. Certificate-based research organization: separation of claims, hypotheses, computational procedures, supporting evidence, and outstanding verification requirements into identifiable research records. These objectives define an ongoing research program rather than a claim that every objective has been achieved. 2. Certificate Architecture — BSD-CERT Series The current archive contains ten principal BSD certificate records, C001–C010. They organize the inputs, arithmetic invariants, computational protocols, and proof obligations required for a curve-specific investigation. C001 — Parameter-Driven Start State and BSD Input BindingStatus: PARTIALLY DERIVED. The project-specific parameter values and BSD arithmetic bridge remain open. C002 — Rational Points, Saturation, Height Pairing, and Regulator ProtocolStatus: OPEN. Curve-specific data have not been recovered. C003 — Hasse–Weil L-Function, Functional Equation, and Analytic RankStatus: OPEN. The protocol is documented; curve-specific values are not certified. C004 — Real Period and Differential NormalizationStatus: OPEN. The curve-specific period remains uncertified. C005 — Tamagawa Factors and Rational TorsionStatus: OPEN. Curve-specific invariants remain uncertified. C006 — Tate–Shafarevich Group and Order EvidenceStatus: OPEN. No curve-specific group order has been certified. C007 — Unified BSD Invariant Ledger and Leading-Term Consistency GateStatus: OPEN. The reference formula is recorded, but the curve-specific invariant ledger is incomplete. C008 — Ammonite Image-to-BSD Certification MapStatus: OPEN. The original construction rule and reproducible map remain unresolved. C009 — Unified Ammonite-Input BSD Certification Audit (SPNIT-Independent)Status: The standard BSD reference framework is recorded; image-driven certification and curve-specific resolution remain open. C010 — Integral-Lattice Transfer, Mordell–Weil Saturation, and Regulator AuditStatus: The method is derived; its curve-specific application remains open. The presence of a certificate record indicates a documented verification task or protocol. It does not, by itself, establish that the associated arithmetic result has been certified. 3. Theorem Register — BSD-T Series The archive contains 25 BSD theorem documents under 22 distinct numbered identifiers. Three identifiers—T016, T017, and T018—each appear on two separate documents in the current archive. The duplicated identifiers are disclosed here rather than silently treated as distinct theorem numbers. BSD-T001 — Leading-Term Formula: Reference Statement and Proof BoundaryStatus: OPEN. Reference formulation only; not a proof. BSD-T002 — Audit of the Candidate Ammonite LimitStatus: The algebraic limit is proved under the recorded assumptions; its arithmetic identification remains open. BSD-T003 — A Fixed Geometric Limit Does Not Determine BSD InvariantsStatus: Conditional proposition proved; the proposed correspondence remains open. BSD-T004 — Conditional Kinematic Identities for the Ammonite SpiralStatus: Derived under stated kinematic assumptions; the BSD connection remains open. BSD-T005 — Dependency Chain for a Curve-Specific Leading-Term AuditStatus: Derived as a proof-organization requirement; curve-specific application remains open. BSD-T006 — Cancellation Limit Has No Curve-Dependent ContentStatus: The elementary identity is proved; BSD identification remains open. BSD-T007 — Conditional Assembly of the BSD Leading-Term IdentityStatus: Conditional formulation; the general conjecture remains open. BSD-T008 — Necessary Conditions for a Geometry-to-Arithmetic BridgeStatus: Conditional methodological proposition; the proposed bridge remains open. BSD-T009 — Unified Analytic–Arithmetic Master Formula and Geometry BoundaryStatus: Formula recorded; the BSD equality remains conjectural and the geometry bridge remains open. BSD-T010 — Root Number Forces Analytic-Rank ParityStatus: Recorded as proved from the functional equation under the standard completed-(L)-function convention. BSD-T011 — Inclusion Index Controls the Height-Regulator DeterminantStatus: Linear-algebra identity proved; curve-specific application remains open. BSD-T012 — Ammonite Geometric Invariants and the Arithmetic-Bridge ObstructionStatus: Geometric identities derived; the geometry-to-arithmetic map remains open. BSD-T013 — Cross-Track Recovery Audit: Hodge/Weil, Navier–Stokes, and ReplicatorStatus: Transfer audit derived; the Ammonite-to-curve arithmetic map remains open. BSD-T014 — Candidate Bridge Admissibility Tests for the Ammonite InputStatus: Admissibility criteria derived; the original construction and arithmetic target remain open. BSD-T015 — Ammonite Measurement Identifiability and Error PropagationStatus: Derived under the logarithmic-spiral model; image-specific parameters and the arithmetic bridge remain open. BSD-T016 — Identifiability Audit of the Candidate Exponential BridgeStatus: Negative identifiability result recorded as proved; arithmetic bridge remains open. BSD-T016 — Similarity Invariance and the Limits of Geometric Curve IdentificationStatus: Conditional mathematical obstruction recorded as proved; arithmetic construction remains open. BSD-T017 — Full Polar-Acceleration Audit of the Constant-Angular-Speed SpiralStatus: Derived under the declared kinematic model. BSD-T017 — Reverse Constraints for a Spiral-to-Confinement TransitionStatus: Derived under the stated model; transition law and BSD bridge remain open. BSD-T018 — Central-Force Reconstruction for an Exact Logarithmic SpiralStatus: Recorded as proved under the stated planar Newtonian assumptions. BSD-T018 — Distance-Parametrized Transition Candidate and IdentifiabilityStatus: Endpoint constraints derived; interpolation remains a candidate, and the original law and BSD bridge remain open. BSD-T019 — Audit of the Proposed Geometric Period IdentificationStatus: Geometric integral proved; identification with an elliptic-curve period remains open. BSD-T020 — Smith Normal Form Does Not Identify the Regulator or Tate–Shafarevich GroupStatus: Algebraic-scope result recorded as proved; proposed BSD identifications remain open. BSD-T021 — Invariance Audit for the Proposed K-01 Spiral ManifoldStatus: The stated non-invariance result is recorded as proved under the displayed Lagrangian. BSD-T022 — Dynamical First-Integral Genus TestStatus: The theorem record is present in the archive. Its detailed conclusion and proof status require confirmation against the complete theorem text before a stronger claim is made. These entries summarize the statuses recorded in the research archive. They are not independent certifications of the underlying mathematic