Paper #27 established the identity 197/144 = (2·N²gauge − λₓ_₂₆· (F − 1) ) /N²gauge 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·N²gauge − λₓ_₂₆ (F − 1) ) /N²gauge, 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 Ngauge = 12; (II) tr (L₁²) = 2·N²gauge = 288; (III) tr (Aface³) = N²gauge = 144; each identity fails on every other Fedorov parallelohedron. As a corollary, 197/144 = (tr (L₁²) − λₓ_₂₆·β₁) /tr (Aface³), a walk-counting form in integer topological invariants. T68. 3 (rational/transcendental separation): the power sums pₖ = r₁^−k + r₂^−k of the T₁u eigenvalue pair satisfy a rational two-term recurrence pₖ = (9/16) ·p₊−₁ − (1/16) ·p₊−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 — LT, its pseudoinverse, Aface, the boundary operator, face-type and Oₕ 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.
Luke Martin (Mon,) studied this question.