Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates

We prove positive definiteness and quantitative condition-number bounds for continuous Sobolev Gram matrices of deterministic shallow networks with Gaussian and tanh-difference activations. The prescribed feature families include the optimal Sobolev approximation constructions of Part I. On every fixed nonempty bounded Lipschitz domain in ${\mathbb{R}}^d$, $d\ge 2$, and for every fixed integer $m\ge 0$, the raw and column-normalized Gram matrices in the full $H^m$ inner product have spectral condition numbers bounded by $\exp (C{N}^{2\mathrm{/}d}\mathrm{/}\log (N+2))$ for sufficiently large $N$, where $N$ is the feature count and $C>0$ is independent of $N$. The upper bounds hold for midpoint and Gauss-Legendre offsets. For midpoint offsets and a suitable angular sequence, matching lower bounds on fixed ellipsoids establish sharpness of this exponent over the stated domain class for raw Gaussian matrices and both raw and normalized tanh-difference matrices. The proof combines homogeneous differential operators that isolate direction groups with one-dimensional coefficient-recovery estimates. Two-dimensional $L^2$ experiments on a square illustrate condition-number growth and normalization effects.