The Dai–Claude (DC) Theorem: A Closed-Form Cumulant Scaling Theorem for the Fisher–Wasserstein Deviation along ϕ^2n Gibbs Families

The Fisher and Wasserstein-2 metrics on a parametric family of probability densities coincide on the Gaussian sublocus and disagree away from it. We organize the diagonal Fisher–Wasserstein deviation along non-Gaussian  Gibbs cumulant directions around a single operator-theoretic identity. Theorem 0 (Spectral Resolvent Identity) identifies, at the Gaussian basepoint, the Fisher–Wasserstein discrepancy as the quadratic form  of the canonical OU resolvent defect operator  (Hermite eigenvalues ) acting on arbitrary mean-zero observable . Theorem 1 (DC Theorem) is the Hermite-basis specialization at , yielding the closed form . Two structural consequences follow: Theorem 2 gives the off-diagonal closed form , and Corollary 3 establishes negative semidefiniteness of the full deviation matrix. A supporting Wasserstein–Hermite lemma provides an independent OU-spectral verification. Finally, Theorem 3 establishes that the diagonal sequence satisfies an order-3 holonomic recurrence, rigorously proven for all  via an explicit Zeilberger creative-telescoping certificate composed with a boundary-absorbing order-1 operator (LCLM construction). Proofs use only Hermite orthogonality, OU spectral decomposition, and the Bény–Osborne Wasserstein gradient flow formulation. The leading-order asymptotic  is established by Stirling–Laplace expansion with numerical verification, and the Borel radius  is rigorously identified from the characteristic polynomial of the holonomic recurrence. Finer sub-leading expansion is the subject of a separate manuscript.