Bounded mesh ratio gives an h^{-2s} exact condition-number bound

Given: Countable Choice, fixed parameters, a bounded domain, a finite distinct site set with $h_X\le 1$, and the stated mesh-ratio bound.

For $N\ge 2$ and separation radius $0<q_X\le 1$, ${\kappa }_2(K_X)\le C_{0}N{q}_X^{-2\nu }$; for $N=1$ the condition number is $1$. (Separated Matérn sites have a positive Gram lower bound)

The domain lies in a fixed ball and the disjoint balls of radius $q_X$ about the sites have volume proportional to $q_X^d$; finite additivity gives the corresponding packing bound. (Radial kernels, polynomial augmentation, and site geometry, Euclidean balls have positive finite Lebesgue measure, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume, A linear map $T$ of ${\mathbb{R}}^n$ sends Lebesgue measurable sets to Lebesgue measurable sets, with ${\lambda }_n(T[E])=\mathrm{\mid }\det T\mathrm{\mid }{\lambda }_n(E)$ when $T$ is invertible and $T[E]$ Lebesgue null when it is not)

The mesh ratio and separation radius satisfy ${\rho }_X=h_X\mathrm{/}{q}_X$, and the condition number is finite for each invertible Gram matrix. (Radial kernels, polynomial augmentation, and site geometry, The condition number kappa_p(A) = ||A||_p ||A^{-1}||_p of a nonsingular linear system)

Countable Choice is inherited from the Matérn separation estimate [F1] and the Lebesgue measure and ball-packing suppliers used in [F2]. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

Proof technique: packing followed by the separation estimate.