Lemma: Existence and uniqueness of the displayed prescription

Equations (eq:W1)–(eq:W2) with (eq:harmonic-data) have unique smooth coefficient fields. Uniqueness does not require spatial decay of the compared smooth solutions.

Proof. For a smooth spatially compact forcing $F$, the solution of ${\square }_{\eta }u=F$ with zero coordinate-time data is

The quotient has its continuous value $t-s$ at zero frequency. Spatial Fourier transformation verifies the equation and data. Smoothness follows by differentiating the Fourier integral, or by the standard local wave estimates. Apply this construction first to (eq:W1), and then to (eq:W2). The second forcing is spatially compact because $A_1,A_2$ are.

For uniqueness, let $u$ be a smooth homogeneous solution with zero data. On a shrinking ball of radius $R-(t-t_0)$, its wave energy

has nonpositive derivative. The boundary flux is

Since $E(t_0)=0$, the solution vanishes on the corresponding interior cone. Taking $R$ large enough treats any chosen point. The argument is local and imposes no condition at spatial infinity. Apply it successively to the differences of the first and second coefficient solutions.