arXiv:2611.12996v1 Announce Type: new Abstract: We study geometric dependence of the dimensions of planar spline spaces $S_{2r+1}^r$ through families with a single moving interior vertex and determine their dimensions throughout the admissible parameter domains. For $r=1$, we extend the Wu--Deng one-parameter classification to the interior of the central quadrilateral. The dimension increases by one at exactly two isolated points, one due to a local direction-count change and the other to a global compatibility defect. For $r=2$, subdividing one triangle of the Yuan--Stillman configuration gives a twelve-triangle family with fixed local direction counts; its dimension increases by one precisely on an open line segment. For every $r\ge3$, we construct a family with $2r+12$ triangles and fixed local direction counts, proving geometric dependence at degree $2r+1$. The moving vertex lies in an open triangle, and the dimension exceeds Schumaker's lower bound by one exactly on its symmetry axis. The lower collinear vertices may have arbitrary strictly ordered heights. The low-order proofs use exact cofactor elimination. For higher orders, weighted interpolation and block elimination bound the global defect by one, and explicit extension and obstruction arguments determine the exceptional set.
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