Combinatorial Properties of Tetrahedral Complexes: Growth Invariants, Face-Bonding Rules, and Fine/Coarse-Graining Duality

We establish a set of combinatorial and geometric results for simplicial complexes built from tetrahedra. The central result is the INV-H invariant: any local growth process that attaches a new tetrahedron via a single boundary face (adding a new vertex) satisfies S = 2N + 2 exactly, where S is the number of free boundary faces and N the total number of tetrahedra. This implies the boundary-to-volume ratio σ = S/N → 2 and prevents holographic scaling (S ∼ N^{2/3}) by purely local growth. We further classify boundary-face increments ΔS ∈ {−4,−2, 0, +2, +4} according to the number of shared faces in each attachment step, and show that only multi-face bonding (sharing 2 or 3 faces simultaneously) can achieve sub-linear growth of S. We prove that the tetrahedron is the unique minimal 3-simplex: all convex 3-polytopes decompose into tetrahedra but not vice versa, establishing the tetrahedron as the natural IR cutoff of any fine-graining procedure. Finally, we derive the holographic scaling relation S_∂ ∼ N^{2/3} tet analytically from the isoperimetric geometry of the sphere, and discuss its role as the target relation for any coarse-graining scheme on tetrahedral complexes.