Exact CFI Parity Formulas for Spin Models

We determine how spin partition functions detect the global parity bit of the original uncolored Cai–Fürer–Immerman graphs. For every connected loopless cubic base, the homogeneous two-spin parity difference has an exact product formula depending on the base only through its order. Its factors classify all positive parameter choices that detect parity, including on pairs indistinguishable by any prescribed dimension of Weisfeiler–Leman refinement. An induced-edge product also gives a single-coefficient difference of independence polynomials. A Fourier transform over edge twists expresses the general spin-model difference as a tensor contraction with edge matrix ∧²A. For Potts models on arbitrary bases of minimum degree two, the response vanishes below degree 3|E(B)|; its coefficient there is a cumulant-weighted sum of flow polynomials over vertex splittings. On cubic bases this reduces to the ordinary flow polynomial, and the entire response has an exact expansion over disjoint cycles. We prove that the base Tutte polynomial determines this response exactly for the positive integer state counts q ≤ 4. A single pair of Tutte-equivalent simple 3-connected cubic bases gives different responses for every q ≥ 5.