Theorem: Classical layer formula

For every smooth classical tensor satisfying $D\gamma =0$,

Its source vanishes outside $0<t<\epsilon$. If that source has spatially compact support, then

The propagated coefficient need not vanish for $t>\epsilon$. It is nonzero precisely when the source vector in (eq:layer) is nonzero.

Proof. Set $D\gamma =0$ in (eq:Xgeneral). The derivative ${\sigma }_{\epsilon }^{\prime }$ is smooth, vanishes off the ramp, and is flat at both endpoints. Finite propagation proves the support inclusion. Since $\square {\mathsf{X}}_1=-{\sigma }_{\epsilon }^{\prime }{a_1}^{0\cdot }$, zero coefficient implies zero source. The converse follows from zero-data uniqueness.