A Flat SO(3,1) Connection on the FCC Lattice and Its Relation to Apollonian Circle Packing

We establish a V₄-equivariant directed bond map from the FCC lattice to SO(3,1), realizing each bond as a composition of Lorentz inversions. The FCC primitive cell is the symmetric Apollonian seed in R³·¹, with all pairwise Lorentz inner products equal to -1. The bond map defines a flat SO(3,1) connection on T³ and satisfies directed equivariance to 10⁻¹⁶. The generated group Γ is an infinite discrete subgroup of SO(3,1) with orbit dimension trending toward the Apollonian value D = 1.3057.