A singular repulsive power tail is stable

From the coordinate formula in [F1], $\mathrm{\parallel }x{\mathrm{\parallel }}_2=\sqrt{{\sum }_{r=1}^d{x}_r^2}$ is continuous: each coordinate square is continuous by $u^2-v^2=(u-v)(u+v)$, and for $u,v\ge 0$, $\mathrm{\mid }\sqrt{u}-\sqrt{v}\mathrm{\mid }\le \sqrt{\mathrm{\mid }u-v\mathrm{\mid }}$, since, if $u\ge v$, $(\sqrt{u}-\sqrt{v}{)}^2\le (\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})=u-v$, with the other case symmetric. Thus the square-root composition is continuous, including at $x=0$. On ${\mathbb{R}}^d\setminus \{0\}$, the function $x\mapsto \mathrm{\parallel }x{\mathrm{\parallel }}_2^{-d-ϵ}$ is therefore continuous by [F2]. The complement of $\{0\}$ is Borel and $\{0\}$ is closed, so the piecewise function is Borel; it is even because $\mathrm{\parallel }-x{\mathrm{\parallel }}_2=\mathrm{\parallel }x{\mathrm{\parallel }}_2$. For $\mathrm{\parallel }x{\mathrm{\parallel }}_2\ge 1$ it satisfies the tail bound with $C=1$ and $R=1$.

Every pair term is nonnegative or $+\mathrm{\infty }$. Thus every finite pair sum satisfies $U_N\ge 0$, so the stability constant is $B=0$. This proves the claimed model-class membership.