THE DETERMINANT MEDIATION TRANSFER THEOREM

Abstract

This article develops the Determinant Mediation Transfer Theorem (DMT-T), a general framework for comparing multiple determinant-type realizations through a common carrier while preserving provenance, certification, and obstruction data. The theory is formulated in an ambient Picard groupoid and separates three logically distinct levels of obstruction: intrinsic holonomy in (H^1), the declared-port mediation defect in (C^1), and the residual absolute defect in (C^0/\mathcal S_{\mathrm{cert}}). The main Transfer Extension Theorem proves that direct comparison maps with trivial cycle holonomy extend uniquely to coherent comparisons within each connected component. Certified reference ports then convert the resulting path-dependent construction into a local two-endpoint formula. The article also establishes an Engine Computation Theorem, a bridge-port classification for disconnected comparison systems, a Relative Concentration Theorem, a conservation and descent principle, and a No-Self-Certification Theorem. The latter shows that the formalism can internally refute incompatible comparison systems, but can establish successful mediation only relative to externally declared admissible and certified data. An ordered extension develops two-sided bounds, valuation and divisor-level control, and a conditional squeeze theorem under order-convex certification. A worked example illustrates intrinsic holonomy failure, incorrect port choices, detectable and invisible residual defects, benchmark limitations, and cross-component bridge freedom. The final section explains how the Common Determinant Object (CDO) architecture developed in connection with the author’s Birch and Swinnerton-Dyer (BSD) program is recovered as a concrete instance of DMT-T. In that setting, DMT-T formalizes the transfer, certification, provenance, and obstruction bookkeeping underlying comparisons among analytic, Selmer-theoretic, regulator, local, and finite determinant realizations. The present article establishes the abstract determinant-mediation framework; it should not by itself be read as a proof of the BSD conjecture. Keywords: determinant functors; Picard groupoids; determinant lines; comparison morphisms; cohomological obstruction; holonomy; certified mediation; Iwasawa theory; arithmetic geometry; algebraic (K)-theory.