A concrete FlatTorus3 instance on \(\mathbb {T}^3 = (\mathbb {R}/\mathbb {Z})^3\), realized as \(\mathrm{Fin}\, 3 \to \mathrm{AddCircle}\, 1\). All 23 fields proved: IBP via \(1\)D FTC + Fubini + periodicity, harmonic \(\Rightarrow \) constant via the energy method, Laplacian maximum principle via the second derivative test, etc.

A concrete VelocityDecayConditions instance for the Coulomb kernel \(\Psi (r) = r^{-3}\) with Schwartz-class \(f\) and nonzero \(E\), \(B\). All 17 integrability/Fubini/IBP conditions verified.