Local approximation widths for elliptic equations with measurable coefficients

We prove that the local energy approximation widths of scalar, symmetric, uniformly elliptic equations in dimensions two and three decay as $\exp(-c n^{1/(d-1)})$, with no regularity assumption on the coefficients beyond measurability and boundedness. The result applies to the normalized cutoff operator on nested cubes in AIM Problem 517. Its proof compares the derivative of a harmonic energy Gram matrix with tangential boundary energy, and the Gram matrix itself with the energy of an ordinary harmonic extension. Spherical harmonic counting then bounds the determinant and the restriction singular values. A finite ball covering transfers the estimate to the prescribed cutoff. Exact propagation through radial layers supplies reproducible numerical checks of the determinant estimate. The constants may depend on coefficient contrast.