Defect-Bound Modes of the Naive Dirac Operator on the FCC Lattice: Spectral Multiplet Structure at Tetrahedral and Octahedral Voids

The face-centered cubic (FCC) lattice has two classes of interstitial void: tetrahedral, with coordination 4, and octahedral, with coordination 6. A trapped extra node at either void is a localized point defect coupled to its surrounding FCC sites through nearest-neighbor bonds. We compute the bound-mode multiplet of the naive Dirac operator at each defect class and decompose it under the residual point-group symmetry. At a tetrahedral defect the multiplet has dimension 4 and decomposes under T_d ≅ S_4 as A_1 ⊕ T_1. At an octahedral defect the multiplet has dimension 6 and decomposes under O_h as A_{1g} ⊕ E_g ⊕ T_{1u}. The 3-dimensional T_1 piece of the tetrahedral multiplet has the dimension and group-theoretic properties needed to carry an internal SU(3) color triplet; the matter paper of Kulkarni [Phys. Open 27, 100423 (2026)] builds such an action on the three perfect matchings of K_4, the bonded graph of the tetrahedral defect's surrounding sites. The 3-dimensional T_{1u} piece of the octahedral multiplet is the spatial-vector representation of O_h, not an internal triplet. The bonded graph of the octahedron's six surrounding sites is the complete tripartite K_{2,2,2} rather than K_6, because the three antipodal vertex pairs sit at second-nearest-neighbor distance in the FCC lattice and are not physical bonds; K_{2,2,2} has 8 perfect matchings and 30 skew-edge pairs, neither giving the 3-color structure needed for the matter-paper construction. The same obstruction is reached on independent combinatorial grounds in a companion dark-matter paper, which identifies octahedral defects with a dark-sector matter species; the spectral analysis here gives independent representation-theoretic evidence for that identification. Tetrahedral defects admit the matter-paper color algebra; octahedral defects do not. As supporting background we give a complete analytic classification of the free-field FCC zero-mode spectrum from an exact factorization of the kinetic vector field, with three topological classes (a singlet at Γ, a quartet at L-points, and three boundary nodal loops) summing to zero index by Nielsen-Ninomiya. The defect mechanism is unlike the minimally doubled and supersymmetric-lattice constructions, which act on the bulk Dirac operator; the analysis here leaves the bulk operator unchanged. The local result depends on the cuboctahedral K = 12 first-shell structure of FCC, which is not present in HCP (anticuboctahedral D_{3h}, with a preferred c-axis at rank 4 in the bulk), BCC (K = 8 with broken K_4 at tetrahedral voids), or hypercubic lattices. Among the two 3D close-packed lattices only FCC carries the full O_h first-shell symmetry needed for an isotropic substrate.