Quantitative and Uniform $L^2$ Non-Localization on Integrable Polygons

What the math gives to ML

The paper proves a strong anti-localization principle: on integrable polygonal domains, no Laplacian eigenfunction can place arbitrarily little L2 mass inside any fixed positive-measure observation set, even when eigenvalues are degenerate and arbitrary cancellations occur within an eigenspace. The rectangular result is especially transferable because it gives an explicit lower bound depending only on the observation fraction alpha, together with robustness for spectral clusters and quasimodes. A practical neural-network adaptation is to constrain Fourier feature maps shell by shell, using the theorem's lower bound as a calibrated anti-collapse regularizer rather than applying an unstructured spatial smoothness penalty.