Near-Closure and Neutral Discrete Substrates: A Structural Selection of the Rhombic Dodecahedral Tiling

We study neutral discrete substrates in R3 using only structural assumptions: discreteness, single-scale uniformity, and coherence under infinite extension. Our guiding motif is maximal coherent local stacking of equal-scale domains: in three dimensions the maximal “kissing” coordination is 12, and any substrate that saturates this bound at every site realizes the largest possible local coordination compatible with equal-scale exclusion. We introduce a concrete near-closure functional that separates (i) deviation from continuum isotropy (measured by low-order spherical-harmonic content of the neighbor-direction measure) from (ii) extension inconsistency (defect density under repeated patch extension). Within an axiom class that enforces periodic global extension, maximal coordination 12, cubic neutrality (no distinguished axis at the symmetry level), and a monohedral Voronoi partition, we prove a rigidity theorem: up to Euclidean similarity, the only admissible substrate is the face-centered cubic (FCC) Delone set. Equivalently, the unique admissible Voronoi (Wigner–Seitz) cell is the rhombic dodecahedron, yielding the rhombic dodecahedral tiling of R3. Simple cubic and body-centered cubic are excluded by non-maximal coordination, and hexagonal close packing is excluded by axial bias under the cubic-neutrality axiom. No dynamics, energetics, or empirical claims are made; the result is purely structural.