A Capacity-Barrier Architecture for the Rank Problem (Conditional BSD Rank Equality via a Sliding Detector and Determinant-Line Locking)

This preprint develops a modular “capacity-barrier” mechanism aimed at the Birch–Swinnerton–Dyer (BSD) rank problem for elliptic curves over the rationals. The core contribution is an analytic/operator-theoretic engine that compares two quantities produced from the completed L-function: a forced local “energy footprint” detected near the central point, and an upper bound coming from an explicit coefficient-budget (“capacity”) associated with a truncated Dirichlet model. The method is intentionally separated into internal analytic modules and explicit arithmetic gates. On the analytic side, the note introduces a sliding detector built from the logarithmic derivative of the completed L-function, and studies its localized exponential energy. A bridge step relates the detector output on a short window to a log-truncated Dirichlet model with a controlled remainder. A Poisson/Plancherel-type capacity estimate then bounds the detector energy in terms of an explicit coefficient budget. Combined with a forcing lower bound (triggered by excess order at the central point), this yields a contradiction unless the excess vanishes. On the arithmetic side, the note isolates normalization and identification issues as explicit assumptions. A determinant-line “locking” phase is used to prevent hidden resolution-dependent rescalings from entering the coefficient budget. The final BSD-rank statement is conditional: the engine proves an internal rank identity between the analytic order of vanishing and a canonical arithmetic readout defined via the locked determinant-line package. Upgrading this identity to the classical BSD rank equality (involving the Mordell–Weil rank) is isolated as a separate Identification Gate assumption. The preprint is written to be readable as a standalone architecture: each module has clear inputs/outputs, and the conditional dependencies are explicitly labeled. The note does not claim an unconditional proof of the full BSD conjecture and does not address the finiteness of the Tate–Shafarevich group or the full leading-term formula. KeywordsBirch–Swinnerton–Dyer conjecture; analytic rank; elliptic curves; L-functions; determinant lines; Selmer complexes; explicit formula; Dirichlet polynomials; operator methods; conditional results NotesThis upload corresponds to the “Short” modular version intended as a conditional preprint.