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 six pairwise Lorentz inner products equal to -1, verified to 10^-14. The bond map satisfies directed V4 equivariance to 8.88x10^-16 across all 144 combinations and defines a flat SO(3,1) connection on T3 with trivial holonomy on all cycles (verified exactly) and non-trivial holonomy on open chains in [0.77, 7.43]. The group Gamma generated by the 36 bond transformations is an infinite discrete subgroup of SO(3,1) with growth rate 36 per word length. A previous conjecture is resolved as a theorem: all 12 orientation-preserving Apollonian generators are in Gamma, each expressible as a product of two Class I generators. Both trace classes {0.3211, 3.1738} are essential to Gamma; neither suffices alone. Computational evidence establishes that the limit set of the full group Gamma is strictly larger than the Apollonian gasket; the correct Apollonian structure resides in the subgroup A contained in Gamma generated by the home-cell bonds, whose limit set is conjecturally the Apollonian gasket with Hausdorff dimension 1.3057. The exact coefficient field for all generators is Q(sqrt(11), sqrt(19)).