Paper J: Vacancy–Resonant Octahedral Scaffold (VROS)

This work introduces the Vacancy-Resonant Octahedral Scaffold (VROS) as a minimal, symmetry-driven mechanism for isotropic onset in the holosphere program. Starting from three axioms — causal propagation on boundary links, massless (standing-wave) stabilization, and discrete Oh symmetry — we study spherical shells with vacancies placed on complete Oh orbits. We prove that exactly six axial vacancies at plus or minus x, y, and z form the unique minimal configuration that eliminates the T1u vector components while preserving an A1g breathing ground mode. Variational arguments and interlacing bounds establish stability and show that adding non-axial vacancies forces leakage into higher multipoles, degrading coherence and quality factor. Numerical simulations with cotangent Laplacians on icosahedral refinements solve the generalized eigenproblem L v = lambda M v, measure T1u leakage with a mass-weighted projector, and report median and IQR statistics across random seeds. The results confirm two falsifiable signatures: (1) leakage scales approximately as h squared under refinement, and (2) a small, symmetric vacancy set (six, sometimes twelve) maximizes isotropy and quality factor, whereas larger or random sets over-constrain and inject higher order content. We also compare Dirichlet anchors to Robin (impedance) anchors, showing that moderate Robin impedance reduces leakage relative to Dirichlet while overly large impedance can over-constrain. Together, the theory and numerics establish VROS as a parameter-free, symmetry-selected scaffold that provides a credible first rung of the growth ladder from dipole rails to full octahedral shells.