RSM Wilson Paper Stack: Dyadic Gaussian Vortex dipoles, Dual separatrices, and a Wilson-loop Grammar for Mirror-paired Carriers

We study a smooth mirror-paired carrier built from a dyadic Gaussian vortex dipole and use it to formulate a Wilson-loop grammar for reciprocal two-lobe structures. The point of the construction is not merely to regularize a classical dipole, but to expose a tractable carrier on which several objects that are often discussed separately become visible at once: a certified divider extracted from an even portrait channel, a binary seam registry, a real separatrix pair in the comoving frame, and a family of closed lobe loops carrying conjugate Wilson-type phases. In this setting the local carrier is not treated as a decorative example. It is used as an explicit realization of the seam Doublet, with a seam-even centered carrier and a seam-odd residue defined on one declared comparison frame. This makes the binary closure law visible on a smooth analytic carrier without changing the abstract Wilson grammar established elsewhere. The paper proceeds in three steps. First, we construct the dyadic Gaussian vortex dipole and fix the mirror-adapted divider and post-seam loop family. Second, we show that in the steady translating frame the carrier supports genuine dual separatrices and dual lobe loops, and that these loops carry opposite circulation flux and hence conjugate Wilson phases under a declared effective connection. Third, we show how the same grammar extends naturally to quasi-two-dimensional plasma carriers, where generalized potential vorticity replaces ordinary fluid vorticity, and we indicate how alternating mirror-paired replication leads toward a Wilson-loop lattice without yet promoting the structure to a higher closure class.