The Laplacian Spectrum of the Truncated Octahedron Face Adjacency Graph

We compute the exact spectrum of the graph Laplacian on the face adjacency graph of the truncated octahedron (Kelvin cell). The 14 faces (6 squares, 8 hexagons) form a graph where two faces are adjacent if they share an edge. The Laplacian eigenvalues are: 0 (×1), (9−√17)/2 (×3), 4 (×2), (9+√17)/2 (×3), 7 (×4), 9 (×1). The characteristic polynomial factors completely: p(λ) = λ(λ²−9λ+16)³(λ−4)²(λ−7)⁴(λ−9). All eigenvalues are algebraic numbers over Q(√17). Using the full octahedral symmetry group O_h (order 48), each eigenspace is identified with an irreducible representation: A1g, T1u, Eg, T1u, A1g⊕T2g, A2u. All results verified by trace identities, numerical computation, and character-theoretic decomposition. The truncated octahedron face graph arises naturally in BCC lattice tiling, where the spectral gap (9−√17)/2 ≈ 2.438 governs diffusion and equilibration through face-to-face coupling.UFFT Paper #5. Foundational spectrum paper — all subsequent UFFT derivations depend on these eigenvalues. Part of the Unified Foam Field Theory (B + V = D). Zero free parameters. GitHub: https://github.com/WebEnvy/UnifiedFoamFieldTheory