A Defect–Convexity Filtration for Regular-Simplex-Faceted Polytopes: Exact spectra, gaps, re-entry, and quantitative realization thresholds

Abstract

We propose a two-parameter framework for organizing convex polytopes whose facets are regular simplices or controlled perturbations thereof. The framework separates metric regularity of the edge set from strict convexity of the gluing. For a convex simplicial polytope P, we introduce a scale-invariant edge-regularity defect together with a strict-convexity margin defined by the minimum exterior dihedral angle over its ridges. These quantities induce a realization spectrum Σ_D(ε, γ), consisting of the facet counts realizable in dimension D under prescribed regularity tolerance ε and convexity margin γ. Classical classifications provide exact benchmark spectra. In dimension three, convex polyhedra with equilateral triangular facets realize the facet counts {4, 6, 8, 10, 12, 14, 16, 20}; in dimension four, convex polytopes with regular tetrahedral facets realize {5, 8, 16, 40, 600}. The three-dimensional benchmark displays a distinctive structure: a seven-term consecutive arithmetic run, followed by a gap at 18 and a terminal re-entry at 20. We formalize notions of spectral gaps and re-entry and introduce quantitative realization thresholds that measure how much regularity must be relaxed, while maintaining a prescribed degree of strict convexity, before a previously absent facet count becomes realizable. This formulation turns discrete non-existence phenomena into quantitative questions in realization geometry and provides a natural interface with rigidity theory, realization spaces, and edge-length perturbation methods. The benchmark classifications themselves are prior mathematical results. The contribution of this work is the proposed defect–convexity filtration, the associated spectrum Σ_D(ε, γ), and a quantitative research program for studying gaps, re-entry, robustness, and dimension-dependent realization thresholds in regular-simplex-faceted polytopes.