We compute the complete eigenvalue spectrum of the face adjacency Laplacian for each of the five Fedorov parallelohedra — the convex polytopes that tile Euclidean three-space by translation. We show that the truncated octahedron is the unique member of this family whose face Laplacian has irrational eigenvalues, and that these eigenvalues satisfy a quadratic with prime discriminant. Several characterising properties of this spectrum are proved, including a product-discriminant relation and a sum-square relation that fail for every other Fedorov cell. The results are elementary (requiring only integer arithmetic and one square root) and fully reproducible by direct matrix diagonalisation. Standalone mathematics preprint — deliberately written to be readable and verifiable without any UFFT context. Every numerical claim is reproduced by the attached verifyₛpectraᵥ2. py, which diagonalises the face Laplacian of all five Fedorov parallelohedra from first principles (requires only NumPy). Runs in under a second. The companion. docx is a formatted version of the same content for ease of reading. Author also publishes the Unified Foam Field Theory series (Papers #1–#67 on Zenodo) ; this standalone paper is cited independently as pure combinatorics and is therefore not assigned a number in the public UFFT sequence.
Luke Martin (Fri,) studied this question.