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Let V _h ^k be the continuous vector Lagrange space of degree k, with homogeneous Dirichlet trace, on the uniform three-dimensional Freudenthal triangulation, and let Qk h = div V k h . Zhang proved a mesh-uniform right inverse of the divergence for k ≥ 6. In his edge stage, however, the actual edge trace functions already have degree k ≥ 4; degree six is used only to repair elementwise divergence means. The missing point is to turn that observation into a genuinely local and order-independent lifting. We give a separate protected raw edge-star lemma for k = 4, 5. The complete geometry consists of seven classes and thirty-seven oriented/boundary configurations; an independent census verifies all 117 boundary-decorated source envelopes. For every configuration we specify an exact rational local linear map. Its source trace space, target reproduction, and protection of all non-target edges are certified by integer matrix identities. Exact Bernstein mass and stiffness calculations give the uniform reference-patch bound C _ref < 385. The protected maps have dependency depth zero, permit a 189-colour assembly, and have overlap at most 19. An exactly verified continuous piecewise-quartic two-cube macro-patch operator then repairs the element means without changing any edge trace. For k = 4, 5, combining this construction with Zhang’s mean, vertex, and face stages and the low-degree vanishing of the final face-zero, cell-mean-zero residual yields mesh-independent inf–sup stability. Zhang’s theorem supplies the cases k ≥ 6, giving the result for every fixed k ≥ 4.