OG-II-08 — M — Scalar Matter on an Aperiodic Carrier: Positive Inertia and Normalization Obstructions

We construct a scalar rigid-core matter sector on an aperiodic D₆/H₃ carrier with positive internal rest energy and a massive continuum limit. A quadratic-plus-cofactor energy has a degree-controlled lower bound. When the target is an isometric copy of the physical shell and κ₂=κ₄ in that metric convention, an explicit identity map saturates the bound, with an exact ninety-tetrahedron volume certificate. For the distinct regular-tetrahedron nerve metric, an edge-star and cone-angle argument excludes saturation by any nonzero-degree finite piecewise-linear map. A gauge-covariant center field uses lumped mass, positive simplex stiffness and an exact cochain extension of the edge connection. Variational midpoint preserves total Noether charge and, in a static background, positive energy on arbitrary positive clock steps. For smooth prescribed changing backgrounds we prove mesh-uniform stability and compact-time strong convergence of well-prepared finite-energy data. A Ritz comparison gives O(h) spatial error for smooth solutions, with O(h+δmax²) total error in a static background and O(h+δmax) for the stated changing-background scheme. The dispersion relation gives positive structural inertia with effective mass E₀/c². Total positive-frequency charge can be normalized, but exact identification of every local source with a positive number observable is obstructed; narrow spectral bands admit a sharp quantitative approximation. Independent matter and gauge normalization freedoms remain. The result is a fully specified admitted scalar completion, with metric-dependent saturation and controlled dynamics, rather than a numerical mass or charge prediction from geometry alone. Variational homogeneity preserves energy ratios under a common scale change; concavity and comparison estimates bound their dependence on relative coefficients.