A Gaussian Obstruction to Uniform Mesh Bounds for Differential Operators

We classify the nonzero finite-order differential operators with real polynomial coefficients that preserve hyperbolicity and retain a positive uniform fraction of additive zero mesh on the whole Laguerre-Polya class: they are exactly the nonzero scalar multiples of the identity. For every other such operator and every prescribed positive input mesh, the infimum of the output mesh is zero, even for inputs with exactly three simple zeros. A strengthening Gaussian factor yields a local Hermite cluster; for an arbitrary fixed operator at a nonsingular center we compute its common drift and next common dilation, without assuming global hyperbolicity preservation. The inherited cubic-Gaussian construction and classical antecedents are explicitly credited. This is a complete scoped theorem, not a full resolution of the broad AIM-ANALYSIS-0146 zero-spacing question. The manuscript is AI-assisted, self-audited and unrefereed, with no absolute-priority or independent-review claim. Exact standard-library Python code and identical normal/optimized outputs cover 450 operator/parameter cases and 9,505 checks; the written proofs establish the general statements.