Multi-Sector Phase Atlases in the VGM Program: Branch-Preserving Comparison and an Exact Coupled-Vector Benchmark

This paper develops a branch-preserving framework for comparing phase and critical-set atlases across explicitly specified sectors of the VGM research program. Sector solution spaces are pulled back to a common parameter base without identifying distinct states that project to the same controls. The framework separates critical-set intersections, coincidence of loci, physical equivalence, and the dynamics of an interacting system. It gives precise conditions for regular pullbacks and transverse intersections, distinguishes uncertainty overlap from certified matching, and states the additional obligations required for global coverage and topology verification. An exact two-vector benchmark makes the construction concrete. On a fixed flat periodic background, two massive vector fields interact through a specified quadratic mass-mixing term and are driven by prescribed conserved currents. A uniform positive-definiteness bound establishes a unique static response, an empty internal static critical set, and positive squared frequencies for all six physical polarizations throughout the declared control window. Exact response and energy data, symbolic checks, and reproducible figures accompany the release. A Schur-complement calculation shows why the critical set of an interacting system cannot generally be obtained by intersecting the critical sets of its uncoupled sectors. The results apply to the declared leading-order benchmark and comparison framework. They do not establish microscopic OFT matching, a universal multi-sector phase law, coupled gravitational dynamics, or observational confirmation.