Kleinian Group Structure of the FCC Bond Map and an Apollonian Foam Cosmology

The third paper in a four-paper series connecting FCC lattice cohomology, Apollonian circle packing, and post-quantum cryptography. Papers 1 and 2 proved that the FCC cycle space carries a natural V4 symmetry and that the FCC primitive cell is the symmetric Apollonian seed in R³’¹, with the directed bond map equivariant under both symmetries simultaneously to 10⁻¹⁶. This paper asks what that object is: the group Γ ⊂ SO(3,1) generated by the 24 FCC bond transformations is identified as a Kleinian group acting on hyperbolic three-space H³, whose limit set is conjecturally the Apollonian gasket with Hausdorff dimension 1.3057, and whose spectral bottom is conditionally predicted at λ₀ ≈ 0.906 by Sullivan’s theorem. Beyond this mathematical interpretation, the paper develops a speculative Apollonian foam cosmology in which universes nucleate as V4-symmetric Apollonian seeds, black holes are inter-neighborhood gateways, and the cosmic web fractal dimension is predicted to converge to 1.3057. Mathematical content and cosmological speculation are developed in separate sections and clearly distinguished throughout. Paper 4 (ADLP post-quantum cryptography) is at zenodo.18809541.