Proposition: Transition defect and plateau qualification

For the displayed prescription,

There is a compact $\xi$ in the transition region for which $W_{\sigma }[K\xi ]\ne \xi$.

If $\mathrm{supp}\xi$ lies strictly inside a region where $\sigma =1$, the displayed defect vanishes and the two fields agree.

Proof. Direct differentiation gives

Substitute into (eq:ramp-equation) and subtract ${\square }_{\eta }{\xi }^{\mu }$.

Choose $t_c\in (0,ϵ)$ with $0<\sigma (t_c)<1$. Let a compact spatial bump $\psi$ equal one near ${\mathbf{x}}_c$, and choose a compact time bump $a$ in the transition region with

Set ${\xi }^0=a\psi$, ${\xi }^i=0$. Near the detecting point,

The right side of (eq:ramp-defect) there is

The difference field cannot vanish identically.

For plateau-supported $\xi$, the right side is zero and the difference has zero data. Smooth uniqueness is local: the wave energy on an inward-moving ball has nonpositive boundary flux, and starts at zero. Enlarging the ball treats any chosen point. Thus the homogeneous difference vanishes without an assumption of finite total energy.