Proof: Verification

Proof. The time-derivative jump is $-\mathrm{id}$, so the leading $-{{\partial }_t}^2$ produces the correctly normalized delta. Ordinary matrix ODE uniqueness supplies the mode functions. On compact time intervals, the usual wave-equation energy estimate bounds the Fourier-mode solutions and their fixed derivatives by polynomial momentum weights. Thus the formulas define distributions on compact tests. Their time support, inverse equations, and Green uniqueness identify them with Theorem thm:greens.