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 =…