Apollonian–Icosahedral Unification: Fifteen Copies of the Apollonian Group Inside O(5,1)

The FCC Apollonian group Γ ⊂ O(3,1), generated by inversions through four mutually tangent circles, embeds into the icosahedral Apollonian group Γ₅ ⊂ O(5,1), generated by inversions through six mutually tangent hyperspheres. Any four of the six icosahedral seed vectors satisfy the mutual tangency condition ⟨cᵢ, cⱼ⟩ = −1, and the twelve inversions they generate are elements of Γ₅ (verified to 4.90×10⁻¹⁴). There are C(6,4) = 15 such four-element subsets, each generating a copy of Γ inside Γ₅. The icosahedral group I ≅ A₅ acts transitively on the 15 copies by permuting the six seed vectors, and Γ₅ is generated by their union. The 15 copies biject with the 15 unordered pairs from {1,…,6} — each copy is determined by the two seed vectors not included. This resolves the final open question in the companion series: the FCC and icosahedral Apollonian groups are not parallel constructions but nested ones, related by the inclusion of four tangent circles into six tangent hyperspheres. The embedding is controlled by the Descartes quadratic form (N−1)I_N − J_N of Paper 10, with the Lorentz signature living in the full six-dimensional space. Twelfth and final paper in a series; companion papers at DOI 10.5281/zenodo.18793518, 10.5281/zenodo.18828073, 10.5281/zenodo.19009461, 10.5281/zenodo.19027249, 10.5281/zenodo.19027545.