We prove that the largest subgroup of A₄ acting on the cycle space of the FCC lattice is the Klein four-group V₄, with decomposition ker δ¹ ≅ 3·1 ⊕ χₐ ⊕ χᵦ ⊕ χ꜀. The 3-cycles of A₄ are geometrically obstructed by cubic periodicity. The same structure holds for the diamond lattice. We identify a structural parallel between this decomposition and the symmetry of the Apollonian gasket, developed in the companion paper.