The Uniqueness of the Foam Cell: Exhaustive Face Laplacian Check of All Fedorov Parallelohedra

The face adjacency Laplacian is computed for all five Fedorov parallelohedra. The truncated octahedron is proved unique in having prime discriminant (Δ=17), integer eigenvalue products (r₁r₂=16=Δ−1), and perfect square eigenvalue sum (r₁+r₂=9=3²). Pure mathematics — no physics assumptions required. Changes in this version - Criterion (iii) of the §5 Theorem sharpened from "a perfect square" to "= C_A²", tying the test to the framework's colour number rather than an arbitrary perfect square.- New §6 "Scope and Axiomatic Exclusions" added, addressing the Weaire-Phelan structure as outside the theorem's scope (excluded by UFFT's single-cell parallelohedral axiom, not by spectral criterion). Stereohedra and multi-cell tilings also flagged.- §7 Implications softened to reflect the restricted scope.- Abstract amended to mirror §6.- Reference [6] added pointing to D3_Uniqueness_Restriction.md.- Status line updated; "of 63" → "of 72" tail.- Corrects file-attachment error on v1 (wrong file was attached to this concept DOI).