Emergent Three-Dimensional Lattice Saturation and Local Coordination∼ 12 from Local Triadic Closure and Geometric Frustration in Discrete Networks

We present a minimal discrete network model inspired by the Selection-Stitch Model (SSM), where nodes evolve by preferentially closing short-distance triangles. Starting from a random 2D embedding, the system first forms a dense triangulated surface. Local geo- metric frustration triggers independent out-of-plane lifts, allowing 3D voxel formation at a roughly uniform areal rate while 2D frontier growth continues exponentially. In large-scale simulations (2500–22,500 nodes, 400–10,000 steps), local coordination in the 3D bulk satu- rates near 12 (the three-dimensional kissing number) without explicit tuning: median degree = 12.00, maximum degree = 12, 92.6% of nodes ≥12, and average degree = 11.60–11.84. Adding opportunistic triangulation after lifts increases local triangle density from∼2.3 to ∼2.48 per node in dense regions. These results provide a qualitative toy demonstration of how dimensionality, coordination saturation, and defect-like structures can arise from simple local entanglement maximization rules, offering a pathway toward geometric interpretations of cosmic and particle hierarchies as proposed in SSM.