General and conditional formulas

Strictly, the closed support of ${\sigma }_{\epsilon }^{\prime }$ can meet both $0$ and $\epsilon$, even though the function vanishes at those endpoints. The support statement is about a smooth layer source, not a hypersurface distribution.

A nonzero tensor in the layer can have ${a_1}^{0\mu }=0$ and hence zero dressing. For example, a constant purely spatial trace-free tensor has this property. Thus “nonzero layer data” must refer to the projected source in (eq:layer), not to arbitrary nonzero metric components.