arXiv:2609.06364v2 Announce Type: replace Abstract: We study vertex evaluation on the full space of continuously differentiable piecewise cubic polynomials on a fixed planar triangulation. A complete dual description turns failure of interpolation into balanced edge weights with nonzero vertex loading. It leads to exact counterexamples with 17, 25, 31 and 41 vertices, each with one relation and the classical lower-bound dimension. The 31- and 41-vertex examples have interior graphs that are trees. We give a geometric construction criterion for a sector-completion family and a certified implicit construction outside that family. With all boundary vertices fixed, the nearby bad configurations form a four-dimensional smooth locus in the 17-vertex case, two intersecting components of dimensions six and five in the 25-vertex case, and isolated transverse points in the two tree examples. A common Schur-complement estimate explains their different interpolation costs. We first realize arbitrary defect by decomposable chains of the 41-vertex example. We then construct obstructions from individually interpolatory blocks, prove that the capacity of one complete edge is exactly five, and build a fixed-square family with arbitrarily large strongly indecomposable defect. The latter excludes two triangle-disjoint connected normal bad submeshes even when those submeshes may have holes.