This paper demonstrates that the discrete, persistent geometries generated by the Sc-Rubs recursive scalar-field engine correspond to eigen-structures of a Laplacian-biharmonic operator with threshold and curvature regulation. Octahedral, spherical, cubic, and intermediate morphologies arise as stability plateaux across parameter sweeps, and correlate with changes in the principal eigenvalue ordering. This establishes Sc-Rubs as a generative engine for reproducible eigen-structures in a classical field framework. Supplementary materials include solver parameters, mesh files, and visual outputs for independent replication.