= Characteristic domain from data on a half-line
For <Noncharacteristic Cauchy data> parameterized by $t\geq0$, the <method of characteristics> produces a surface $\Phi(s,t)=(x,y,z)$. The part uniquely fixed as a classical graph through the data must retain both the allowed footpoint range and a nonsingular branch of the projected map. On a branch where inversion gives $(s(x,y),t(x,y))$, require $t(x,y)\geq0$, together with the branch condition and a nonzero <Jacobian determinant>. The strict inequality $t>0$ describes its open interior; the endpoint characteristic $t=0$ supplies a boundary trace. A <fold caustic> can supply another boundary.
An algebraic formula obtained by eliminating $s,t$ can extend beyond this <domain of dependence>. Its values there need not be selected by the original data, because their characteristic footpoints can lie on the unprescribed half of the data curve. Tracking the parameter range is therefore part of the uniqueness argument, separately from selecting a real square-root branch.
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