Local quadratic isoperimetric stability of the truncated octahedron among parallelohedra

We prove that the Archimedean truncated octahedron is a strict local minimizer of the isoperimetric quotient among all parallelohedra in ${\mathbb{R}}^3$. After volume normalization and minimization over rotations, the deficit controls the squared Hausdorff distance from the Archimedean cell. This extends the corresponding result from the five-dimensional lattice-Voronoi subclass to the full ten-dimensional local parameter space. A six-zone parametrization and Schur’s lemma reduce the Hessian to two positive definite $2\times 2$ matrices.