The Pure Mathematics of the Kelvin Cell: Combinatorial Spectra, O_h Representation Theory, and the Quadratic Ring Q(√17)

This paper is a standalone presentation of the pure-mathematical content underlying the Unified Foam Field Theory. It makes no physical claims. Every theorem stated here is a result of finite linear algebra, group representation theory, or combinatorial topology applied to the truncated octahedron — the unique space-filling polyhedron with full O_h point symmetry. The paper collects in one place the mathematical results that the UFFT physics papers reference repeatedly: (i) the construction of the face Laplacian L on the truncated octahedron's face-adjacency graph; (ii) its spectrum σ(L) = {0, r₁, 4, r₂, 7, 9} with multiplicities (1, 3, 2, 3, 4, 1), where r₁, r₂ are roots of the master equation λ² − 9λ + 16 = 0; (iii) the O_h irreducible decomposition of the 14-dimensional face space; (iv) the ring Q(√17) structure of the eigenvalue algebra; and (v) the uniqueness theorem for the Kelvin cell among Fedorov parallelohedra. The purpose is to allow the mathematical content to be read, verified, and submitted independently of any physics interpretation.