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

Version 3.0 (6 October 2026): the printed formula C_A = F_hx/F − 1 does not evaluate to 3 and is withdrawn; C_A = 3 is dim(T2g), the colour number, and the sum identity r1 + r2 = 9 = C_A² is unchanged. References to unpublished internal review notes have been removed. The independent confirmation that the face graph is the tetrakis hexahedral graph, whose tabulated characteristic polynomial has discriminant 17, is recorded in the UFFT Core Framework, Part X §10. Corrected following the external referee audit of 6 October 2026. 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).