Controlled test-wave rays, polarization and Maxwell residual

Use the fixed-background Maxwell test model of Separately adopted fixed-background Maxwell and charged probes on a smooth oriented, time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$ and its adopted length-coordinate SI conventions. Let $U\subseteq M$ be open. Let $\theta \in C^4(U,\mathbb{R})$ be dimensionless, set $k=d\theta$, and assume $k$ is nowhere zero and $g^{-1}(k,k)=0$. Let $a_0,a_1\in C^3(U;T^{\ast }M{\otimes }_{\mathbb{R}}\mathbb{C})$, with covariant derivatives and contractions extended complex-linearly. Use $[k]={\mathrm{m}}^{-1}$, $[a_0]=[a_1]=\mathrm{T}\mathrm{m}$, and dimensionless $ϵ$, as in Finite WKB Maxwell residual, null rays and polarization quotient. Define

$(\mathrm{\square }a{)}_b:=g^{ac}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_c{a}_b,(Pa{)}_b:=(\mathrm{\square }a{)}_b-{\mathrm{Ric}}_b{}^c{a}_c,(La{)}_b:=2k^c{\mathrm{\nabla }}_c{a}_b+({\mathrm{\nabla }}_c{k}^c)a_b\mathrm{.}$

Assume the four pointwise constraints $g^{-1}(k,a_0)=0$, $L{a}_0=0$, $iL{a}_1+P{a}_0=0$, and $i{k}^a(a_1{)}_a+{\mathrm{\nabla }}^a(a_0{)}_a=0$. For $0<ϵ\le 1$, form the complex ansatz $A_{ϵ}=ϵe^{i\theta \mathrm{/}ϵ}(a_0+ϵa_1)$ and its real potential ${\widehat{A}}_{ϵ}:=\mathrm{Re}{A}_{ϵ}$ with real field ${\widehat{F}}_{ϵ}:=d{\widehat{A}}_{ϵ}$. Use the fixed-chart component max norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty },K}$ of Finite WKB Maxwell residual, null rays and polarization quotient, which is zero on an empty compact set. On a nonempty compact coordinate set $K$, assume finite constants $K_0,D_0,C_0,B_0,B_1$ satisfy $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty },K}\le K_0$, $\mathrm{\parallel }{d}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le D_0$, $\mathrm{\parallel }P{a}_1-\mathrm{\nabla }{d}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le C_0$, $\mathrm{\parallel }d{a}_0+ik\wedge a_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le B_0$, and $\mathrm{\parallel }d{a}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le B_1$. The real field obeys $d{\widehat{F}}_{ϵ}=0$ and

$\mathrm{\parallel }{\mathrm{\nabla }}^a({\widehat{F}}_{ϵ}{)}_{ab}{\mathrm{\parallel }}_{\mathrm{\infty },K}\le ϵK_0{D}_0+{ϵ}^2{C}_0,\mathrm{\parallel }{\widehat{F}}_{ϵ}-\mathrm{Re}(e^{i\theta \mathrm{/}ϵ}ik\wedge a_0){\mathrm{\parallel }}_{\mathrm{\infty },K}\le ϵB_0+{ϵ}^2{B}_1\mathrm{.}$

Each local integral curve of $k^{\mathrm{♯}}$ is an affinely parameterized null geodesic. Along it $a_0$ is transported according to the supplied scalar transport amplitude, and the leading physical polarization is the class in $k^{\perp }\mathrm{/}{\mathrm{span}}_{\mathbb{C}}\{k\}$, a two-dimensional complex space with positive induced Hermitian form. These are exact finite-order deductions under the displayed ansatz constraints. The residual is a component norm in the chosen chart, not an observer-independent tensor norm. The conclusion asserts no convergent WKB series, existence of amplitudes across caustics, exact Maxwell solution for a nonzero residual, solution-distance estimate, or coupled Einstein–Maxwell family.