Remark: Agreement on compact sources is not the missing hypothesis

Agreement of two linear extensions on ${\mathcal{D}}_{\mathrm{int}}$ alone does not determine their values on $\mathcal{H}$. For example, for $b\in {C_c}^{\infty }((0,T))$ and an interior time $t_{\ast }$, the map

agrees on ${\mathcal{D}}_{\mathrm{int}}$ and retains zero initial data. In general it is not a right inverse of ${\square }_{+}$.

By contrast, two extensions that both return smooth solutions of the same wave equation with the same data must agree on their common source domain, by Lemma lem:C. The equation, smoothness, and initial data are the decisive additional requirements.