Scott–Vogelius Stability on Three-Dimensional Freudenthal Meshes for k ≥ 4: A Quartic Macroelement Proof

We prove mesh-size-independent inf-sup stability of the Scott–Vogelius pair on the standard three-dimensional Freudenthal meshes for every velocity degree k ≥ 4. Zhang proved the corresponding result for k ≥ 6, while Farrell, Mitchell, and Scott subsequently formulated stability for k ≥ 4 as a conjecture supported by numerical evidence. The degree restriction in Zhang’s constructive argument occurs in the edge-matching step: the edge liftings themselves extend to k ≥ 4, whereas degree-six local bubbles are used to restore elementwise mean divergence without changing the divergence traces already prescribed on the mesh edges. We replace these degree-six bubbles by a continuous piecewise-quartic macroelement correction supported on a 2 × 2 × 1 block of Freudenthal cubes. The central finite-dimensional statement is certified exactly over the rational numbers. For the edge-trace operator E and the operator M collecting the 24 elementwise divergence integrals, we obtain rank_Q(E) = 270 and rank_Q([E; M]) = 293. It follows that the restriction of M to ker(E) has rank 293 − 270 = 23 = 24 − 1, which is exactly the dimension of the compatibility hyperplane imposed by the global divergence theorem. Thus every compatible collection of elementwise mean-divergence defects admits a continuous piecewise-quartic correction whose divergence vanishes identically on all tetrahedral edges. A scaling argument and uniformly bounded macroelement overlap convert this exact local surjectivity into a mesh-independent H¹-stable correction. Substituting the quartic macroelement correction into Zhang’s edge-matching construction extends the required right-inverse estimate to k = 4. Since the remaining stages of the construction require only k ≥ 4, this proves uniform Scott–Vogelius inf-sup stability on the three-dimensional Freudenthal meshes for every k ≥ 4. The computer-assisted component uses exact rational arithmetic only. No floating-point tolerance or numerical rank decision is involved. A reproducible verification script reconstructing the macroelement matrices directly from the reference mesh is included with this archival record.