Separately adopted fixed-background Maxwell and charged probes

On an open region of the oriented, time-oriented Lorentzian spacetime and length-coordinate SI conventions of Smooth matter quantities, current and observer observables, adopt a prescribed-background Maxwell model. The metric $g$ is fixed input. Choose a real $C^2$ electromagnetic two-form $F$ and either adopt the source-free equations $dF=0$ and $d({\star }_{g}F)=0$, or supply a $C^1$ current $j$ and adopt $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, with the exact signs and units of that definition. The source current is prescribed independently of the probe; this postulate does not impose an Einstein equation, matter evolution equation, or field equation for the sources on the unchanged metric.

For a massive charged probe, use the adopted local action and finite-curve conditions of External-field charged-particle action: $m>0$ and $q$ are constant, the path is a $C^2$ future-directed timelike segment parameterized by proper time, and its endpoints are fixed in variations supported in a local potential domain $V$ with a supplied $C^2$ one-form $A$ satisfying $dA=F$. The potential and metric are external data independent of the path. Under the exact hypotheses of Stationary charged action and Lorentz force, action stationarity is equivalent to $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$; that theorem's ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption is inherited only when this equivalent force-law formulation is used. No global potential or trajectory is included.

The probe's own singular field and self-gravity are excluded. No radiation reaction, variable mass or charge, finite-body model, or backreaction estimate is adopted. In particular, calling $g$ or $F$ a test background records this model's omission of their response; it does not claim that the unchanged pair solves the coupled Einstein–Maxwell equations or is close to a coupled solution.

When geometric optics is invoked, it is restricted to the exact $C^4$ real phase, $C^3$ complex amplitudes, four pointwise ansatz constraints, and fixed compact-chart coefficient bounds of Finite WKB Maxwell residual, null rays and polarization quotient. That result supplies finite residual and leading-field estimates, local null rays, and a polarization quotient; it supplies no convergent series, global transport through caustics, exact-solution error estimate, or backreaction theorem.

This is a separately adopted physical model with no empirical premise. The Maxwell and probe assumptions are not deductions from one another, from mathematical axioms, or from source authority.