Track A4: Gaussian mass energies and stability

The kernel $1/\text{abs}x$ is locally integrable in three dimensions, with scaling degree one. Two extensions with scaling degree at most one differ by a point-supported distribution, whose nonzero terms have scaling degree at least three. Hence its extension is unique in that class. VII-H's four-dimensional Wick-square contact freedom is not imported CheungVIIH.

Euclidean translations and rotations preserve the energy. A general coordinate change must transform both mass measures and the distance kernel; retaining the Euclidean kernel in arbitrary new coordinates changes the prescription.

Finally, if a constraint root $z(\theta )$ satisfies

the mean-value theorem gives $\text{abs}z(\theta )-z({\theta }^{\prime })\le L\text{abs}\theta -{\theta }^{\prime }/c$. Without nondegeneracy, $z=\sqrt{\theta }$ from $z^2-\theta =0$ is a counterexample to Lipschitz boundary movement.