BSD Grid v1.4_r4: A Grid-Theory Module Toward BSD via Symmetric-Power Contraction

# Overview This record uploads **BSD Grid v1.4_r4 (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 \(π\) over \(Q\) in a balanced admissible regime, and for low degrees \(2 ≤ m ≤ 4\), the paper uses a calibrated contraction interface of the form\[Eglob(Sym^mπ) ≤ λ_m^2\,Eglob(π), 0<λ_m<1,\]equivalently 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 \(Ceq(PB)\) 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 - `BSD_Grid_v1.4_r4.