The Minimal Simplex Principle: Tetrahedral Coordination as a Combinatorial Necessity in R³

Abstract

Several independent approaches to quantum gravity employ tetrahedral or 4-valent building blocks — loop quantum gravity uses 4-valent spin-network nodes, causaldynamical triangulations sum over 4-simplices — but in each case the tetrahedron is introduced as a convenient discretization, not derived as necessary. This paperproves that no derivation is needed, because none is possible: in Euclidean R³, the minimal number of points required to enclose non-zero volume is exactly four(Lemma 1), and hence the minimal coordination number of any discrete structure supporting volume is z=4 (Theorem 1, the Minimal Simplex Principle). This is astatement about the definition of volume itself, not a physical postulate.