Spectral Determinants and the BSD Conjecture

This manuscript reorganizes a determinant-line formulation of the Birch and Swinnerton–Dyer (BSD) comparison into a modular proof pipeline with explicit input/output contracts. The goal is not to bypass the Tate–Shafarevich group, but to place it in the correct canonical container — the Selmer/Bloch–Kato determinant line equipped with its integral lattices — and to separate “normalization/rigidity closure” from any arithmetic finiteness upgrades. Core architecture The argument is presented in two layers: 1) Facade layer (communication view): Spectral side: produces a canonical determinant-line germ at the special point using modular-symbol and period data. Arithmetic side: the canonical Selmer/Bloch–Kato determinant line target (not “Mordell–Weil without Tate–Shafarevich”). Locking mechanism: transports the spectral germ into the arithmetic container and isolates the residual mismatch as a single scalar parameter. 2) Mechanism layer (five-module pipeline): Local Arithmetic Interface: fixes local conventions and removes all non-selected-prime valuation freedom from the comparison scalar. Spectral Matching Engine: constructs the spectral determinant-line germ. Determinant-Line Transport: performs the transport and defines a single global mismatch scalar. Unit-Rigidity Closure: collapses the mismatch scalar to a sign and fixes the sign canonically via archimedean calibration, yielding a locked identity. Arithmetic Finiteness Upgrade (optional): upgrades the locked determinant-line identity to lattice-index and classical BSD-factor statements only when an external arithmetic finiteness/control package is supplied (for example, standard rank 0/1 regimes or suitable Iwasawa/Euler-system inputs). Main unconditional contribution Under the explicitly stated unit-rigidity admissibility contract, and assuming an integral transport condition at one “locking prime” (a single lattice-integrality input sufficient to eliminate any remaining prime-power scaling), the manuscript proves a unit-rigidity locking theorem: after local normalizations, the transported spectral trivialization matches the arithmetic reference trivialization with no residual unit ambiguity. As a result, any remaining discrepancy cannot be attributed to normalization choices and must live purely at the integral/lattice index level inside the Selmer determinant-line container. What is not claimed The manuscript does not claim new unconditional finiteness results for the Tate–Shafarevich group. Any conversion from “determinant-line defect” to “cardinality of Tate–Shafarevich” is isolated in the optional upgrade module and requires imported arithmetic input. This separation is deliberate and referee-safe: the paper pinpoints exactly where Tate–Shafarevich-type information can enter (as an integral index) and keeps it logically separate from the unit-locking mechanism. Intended audience and use The work is intended for readers familiar with determinant functors, Selmer complexes, and modern BSD/ETNC-style frameworks, and it is structured to support modular verification: each module can be checked independently, and the dependency map records precisely where each hypothesis is used.