Cell-Integer Identities of the Truncated Octahedron and the Single-Cell Obstruction to 197/144

Paper #27 established the identity 197/144 = (2·N²_gauge − λ_{T2g}·(F − 1))/N²_gauge for the rational part of the two-loop QED anomalous magnetic moment C₂, via the chain foam → QED → QED's two-loop rational → identity holds. That chain imports the QED calculation as an intermediary: the identity is proved, but 197/144 is not independently computed from the face graph. This paper does not close that gap — a direct two-loop lattice computation remains open — but it closes a family of surrounding questions at theorem level. T68.1 (reconciliation identity): the two cell-integer rewritings of 197/144, (F² + 1)/(E − V)² and (2·N²_gauge − λ_{T2g}(F − 1))/N²_gauge, agree if and only if F ∈ {1, 14}; the truncated octahedron is the unique Fedorov parallelohedron at which the naive rewriting coincides with the structural rewriting. T68.2 (three cell-integer identities): on the truncated octahedron, (I) the hex–hex polytope edge count equals N_gauge = 12; (II) tr(L₁²) = 2·N²_gauge = 288; (III) tr(A_face³) = N²_gauge = 144; each identity fails on every other Fedorov parallelohedron. As a corollary, 197/144 = (tr(L₁²) − λ_{T2g}·β₁)/tr(A_face³), a walk-counting form in integer topological invariants. T68.3 (rational/transcendental separation): the power sums p_k = r₁⁻ᵏ + r₂⁻ᵏ of the T₁u eigenvalue pair satisfy a rational two-term recurrence p_k = (9/16)·pk−1 − (1/16)·pk−2 with rational initial values, so every trace of a polynomial in the face-graph resolvent is rational; combined with lattice-PT Poisson-summation orthogonality, no rational single-cell operator-trace leaks into the transcendental two-loop sector. T68.4 (single-cell obstruction): an exhaustive search over natural trace-polynomial combinations of the 14-dim face-graph operators — L_T, its pseudoinverse, A_face, the boundary operator, face-type and O_h isotypic projectors, Hashimoto non-backtracking matrix, graph resolvent at rational arguments, Ihara zeta values, heat kernel at rational times — produces no expression evaluating to 197/144; the minimum L¹ distance from any bounded single-cell rational observable to 197/144 is 1/144. Consequence: the full Tier-2 derivation of 197/144 genuinely requires the BCC multi-cell lattice Feynman rules, not only the single-cell face graph. Paper #27's identity remains the canonical statement at Tier 3 until the multi-cell Lemmas 3.1–3.3 close; Paper #68 fixes the Tier-1 surrounding architecture.