= Norm and spectral bounds for subdivision regularity
{title2=$\max\rho(A_{\epsilon_p}\cdots A_{\epsilon_1})^{1/p}\le\widehat\rho\le\max\|A_{\epsilon_p}\cdots A_{\epsilon_1}\|^{1/p}$}
Finite local <subdivision matrices> for the scaled difference scheme track <derivative> variations in nested parameter intervals. A product-norm bound below one proves uniform contraction for every interval and therefore gives a <lower bound> on <derivative> <continuity>. A product <eigenvalue> of modulus at least one is an obstruction when its mode is reachable from the data and visible in the limit: generic data excite it, so the corresponding <derivative> differences fail to vanish. Harmless <polynomial> or unobservable modes must be removed before applying such an <upper bound>. Borderline smoothness needs direct analysis, not rounding a numerical exponent. Stability and convergence of the underlying scheme must be checked rather than inferred from mask width.
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