# Overview This record uploads **BSD Grid v1. 4ᵣ4 (PDF-only) **, a companion / module paper in the **Grid Theory** program. It organizes a BSD-oriented upgrade pipeline by treating analytic and arithmetic components as explicitly separated modules: - **Main hub: ** *Grid Theory v2. 4* (DOI: 10. 5281/zenodo. 18709740) - **Symmetric-power contraction module: ** *Sym-Power v2. 3* (DOI: 10. 5281/zenodo. 18710856) - **This record: ** BSD-oriented module that links the contraction interface to BSD-style rank statements through standard arithmetic bridges. This is intentionally **not** a claim of a complete unconditional proof of BSD. Instead, it is a **program-level closure interface** with explicit “proved vs imported vs assumed” status labeling. # Status legend (read this first) - ** (P) ** proved in this paper (within a fixed parameter box PB and stated inputs) - ** (I) ** imported input (standard external theorems; cited and used as black boxes) - ** (H) ** standing hypothesis / engine-level assumption (recorded as an acceptance-test interface) - ** (V) ** protocol / acceptance-test item (checkable template rather than a proved theorem) # Theory summary ## 1) Analytic module (symmetric-power contraction) For holomorphic \ (GL (2) \) newforms \ (\) over \ (Q\) in a balanced admissible regime, and for low degrees \ (2 m 4\), the paper uses a calibrated contraction interface of the form₆₋₎₁ (Symᵐ) ₘ²\, E₆₋₎₁ (), 0<ₘ<1, expressed in terms of windowed mean-square deviation statistics. The low-degree range \ (2 m 4\) is aligned with the known functoriality window. A key reviewer-facing point: in this unconditional low-degree regime, the calibration Gram matrix dimension is fixed, so the norm-equivalence constant \ (C₄ₐ (PB) \) is treated as a modest PB-dependent constant (no large-\ (m\) conditioning growth in this range). ## 2) BSD-oriented upgrade (program interface) The paper packages an upgrade path from analytic control to BSD-type rank conclusions in a strictly modular way: - **Analytic-to-threshold step: ** recorded as explicit acceptance-test inputs (H/V), not claimed as new Euler-system mathematics. - **Arithmetic bridge (imported): ** standard Gross–Zagier / Kolyvagin consequences (I) are used as black boxes to convert analytic rank \ (0/1\) information into arithmetic rank statements and finiteness of \ (\) in the relevant cases. Thus, the novelty is primarily the **engine architecture**: a parameter-box disciplined interface that makes explicit what must be checked (acceptance tests) and what is imported (arithmetic bridges), rather than claiming new deep arithmetic theorems. # Relations to other records (DOIs) - Grid Theory v2. 4 (main hub): DOI 10. 5281/zenodo. 18709740- Sym-Power v2. 3 (contraction module): DOI 10. 5281/zenodo. 18710856 # Contents - `BSDGridᵥ1. 4ᵣ4. pdf` (PDF-only) # Suggested keywords Birch and Swinnerton-Dyer; elliptic curves; L-functions; analytic rank; arithmetic rank; Gross–Zagier; Kolyvagin; Heegner points; symmetric powers; functoriality; Sato–Tate; explicit formula; Grid Theory; acceptance tests; program closure
Byoungwoo Lee (Fri,) studied this question.