A Covariant, Completely Positive Channel from an Exact Sp(3) Relational Construction, Trace-Preserving and Action-Determined Under an Explicit Convention Premise

Abstract

We ask whether a construction’s invariances can be derived from the defined structure of a relational object rather than imposed on it, and whether the physical action such an object induces is determined within the stated convention rather than chosen among admissible placements. Working over ℚ(i) with an exact USp(6) = Sp(3)- covariant construction, we build an effective completely-positive map N = D ∘ E that compresses a thirty-five-dimensional carrier through a six-dimensional memory, and we show its content is gauge-independent — invariant under the memory’s multiplicity basis — and completely positive outright; it is trace-preserving, and so a channel in the strict sense, only under the convention premise stated below. We then separate three objects routinely conflated in constructions of this kind: the stored tensor, the bilinear scoring functional it validates, and the induced channel action. That separation is unconditional, and it dissolves a standing L/L^{T} ambiguity as a category inversion rather than an arithmetic error. Identifying the induced action requires one explicit premise, P, fixing the dual-leg reading of the documented Choi convention; under P, and only under P, the induced action is uniquely the realignment L^{T} — a map that is, in any case, trace-preserving, fully Sp(3)-covariant on the whole opera- 1 tor algebra, and real, with A_physical = [[1/7, 3/14], [4/7, 6/7]]. We keep the plain facts and the premise-relative result distinct throughout, mark the boundary at which a purely structural formalism cannot decide a question formulated as concerning what structure does not represent, and state the remaining frontier as concrete computations.