What question did this study set out to answer?

The aim is to establish a framework to control defect accumulation in Birch–Swinnerton–Dyer comparisons using operator-theoretic methods.

February 2, 2026Open Access

Unit Rigidity in the BSD Determinant Line via a Capacity Barrier

Key Points

The aim is to establish a framework to control defect accumulation in Birch–Swinnerton–Dyer comparisons using operator-theoretic methods.
Developed a spectral determinant line from finite-level strict perfect complexes over p-adic integers.
Introduced a determinant-line transport mechanism to align an arithmetic reference element.
Applied a budget dominance principle to enforce uniform ceiling transitions along finite-level towers.
Examined the capacity-barrier interface to analyze p-corank in the context of obstructions.
Established canonical local unit control at all primes except for a finite exceptional set.
Produced zero p-corank for the p-primary obstruction, indicating structural finiteness for the Shafarevich–Tate group.
Decoupled internal proofs from interface assumptions to enhance audit-friendliness.

Abstract

This preprint develops an operator-theoretic and determinant-line framework for controlling p-adic “defect accumulation” in Birch–Swinnerton–Dyer type comparisons. The core construction defines a spectral determinant line internally from finite-level, strict perfect complexes over the p-adic integers and equips it with a canonical integral structure. A determinant-line transport mechanism then places an arithmetic reference element in the same one-dimensional line, so that the global mismatch is encoded by a single rational scalar. The main methodological contribution is a “budget dominance” (uniform ceiling) principle: unimodular reduction transitions along the finite-level tower prevent the creation of new p-power denominators. This yields canonical local unit control at all primes outside a finite, explicitly described exceptional set, reducing all remaining normalization ambiguity to a finite audit boundary. An archimedean orientation/positivity calibration then fixes the remaining sign ambiguity, producing a “locking” conclusion for the completed comparison scalar. In addition to the locking spine, the manuscript isolates a standalone Path B mechanism. Under a capacity-barrier interface (implemented via a projected test family and a first-hit rigidity window), the framework forces zero p-corank for the p-primary obstruction: no nontrivial p-divisible component can persist without generating an unbounded p-adic drift, which contradicts the uniform ceiling. With an integral bridge identifying the operator complex with the classical Selmer complex in the perfect derived category, this zero p-corank conclusion upgrades to a structural finiteness criterion for the p-primary Shafarevich–Tate group. The paper is written to be audit-friendly: it separates what is proved internally (determinant-line transport, unimodular tower control, unit-rigidity locking) from explicit interface assumptions (capacity-barrier window rigidity and the integral Selmer bridge). A small normalization-only audit on sample curves is included to validate consistency of conventions; it is not used to infer arithmetic finiteness.

Unit Rigidity in the BSD Determinant Line via a Capacity Barrier

Key Points

Abstract

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