Vacuum Rigidity and Infrared Isotropic Wave Dynamics\∈ 26-Valent Chebyshev Networks

In previous work, we extended the intertwiner overlap mechanism of loop quantum gravity (LQG) from 4-valent nodes (square lattice) to 8-valent nodes (the 2D Moore neighborhood underlying Chebyshev networks). Here we generalize this framework to three dimensions, focusing on the 26-valent neighborhood corresponding to the ^∞ metric ball in $D=3$. The invariant intertwiner subspace for twenty-six $j=1/2$ edges has dimension given by the Catalan number C₁₃ = 742,900. By introducing an anisotropic stretching deformation of the 26-valent vertex polyhedron, we compute the neighbor overlap ι(εᵢ)ι(εⱼ)² and demonstrate its quadratic behavior in the small-deformation limit. A tight-binding effective Hamiltonian based on this overlap generates the discrete combinatorial Laplacian of the 3D Chebyshev graph (degree 26 with $-1$ couplings across all orthogonal, diagonal, and space-diagonal links). We analyze the infrared isotropic wave dynamics, the propagation velocities, and the exact protection of the vacuum rigidity invariant I_∞ = 47/512 in the 3D continuum limit.