Local potential bases on Freudenthal meshes

Local bases for finite-element potential spaces provide the algebraic decompositions needed for vertex-patch corrections. On a consistently oriented Freudenthal mesh of the cube, we prove that continuous vector polynomials of degree $k+1$ with continuous curl have a basis supported on single vertex stars for every $k\ge5$ and every mesh size. This establishes the local-support assertion in the second conjecture of Farrell, Mitchell and Scott. The proof uses exact integer certificates together with analytical arguments covering all grids and degrees. At degree six, bounded windows transfer a fixed normal form and two-part local repairs. At higher degrees, clipped barycentric gaps reduce the construction to finitely many types, while affine inequalities verify their validity over unbounded parameter regions. We also obtain an exact dimension formula and provide reproducible basis-construction code. The conclusion is algebraic locality; a uniform bound on decomposition norms is a separate question.