Multi-Module Minimal Products: Exact Composition, Closed Profiles, and Gaussian Transfer

We establish an exact composition principle for spherical immersions coupled through coefficient profiles and norm-preserving bilinear maps. Under an explicit mixed-orthogonality condition, the spherical mean-curvature vector splits into orthogonal input and profile contributions. The composition is minimal if and only if every input is minimal and the profile is $W$-minimal for an explicit monomial weight. Using Hsiang–Lawson reduction and Kapouleas–McGrath gluing, we construct closed embedded $W$-minimal profiles, including families with unbounded intermediate Betti numbers. For sufficiently large comparable input dimensions, a Gaussian transfer developed in this paper produces interior profiles from closed embedded Gaussian seeds that are nondegenerate modulo rotations.