Killing exchange current and gauge-compensated charged first integral

Let $\mathrm{\Omega }$ be an oriented open subset of the time-oriented Lorentzian domain of Smooth formal matter, EM tensors and observer quantities. Let $F$ and $j$ be smooth, let $T_m^{ab}$ be smooth and symmetric, and suppose

$dF=0,{\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b,{\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c\mathrm{.}$

Let $T_{\mathrm{E}\mathrm{M}}$ be the tensor of Smooth formal matter, EM tensors and observer quantities and assume the exact conditional identity of Curved electromagnetic stress divergence, ${\mathrm{\nabla }}_a{T}_{\mathrm{E}\mathrm{M}}^{ab}=-F^b{}_c{j}^c$, under its stated smoothness and source-sign hypotheses. For a smooth Killing field $\xi$ on $\mathrm{\Omega }$, meaning ${\mathcal{L}}_{\xi }g=0$, define

$J_{\xi }^a=(T_m^{ab}+T_{\mathrm{E}\mathrm{M}}^{ab}){\xi }_b\mathrm{.}$

Then ${\mathrm{\nabla }}_a{J}_{\xi }^a=0$. If $C$ is a finite smooth singular $4$-chain in $\mathrm{\Omega }$ with $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$ for smooth singular $3$-chains, and ${\mathcal{Q}}_{\xi }(\mathrm{\Sigma })={\int }_{\mathrm{\Sigma }}{\iota }_{J_{\xi }}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, then

${\mathcal{Q}}_{\xi }({\mathrm{\Sigma }}_2)-{\mathcal{Q}}_{\xi }({\mathrm{\Sigma }}_1)=-{\mathcal{Q}}_{\xi }(B)\mathrm{.}$

In particular the two slice fluxes are equal when the side flux ${\mathcal{Q}}_{\xi }(B)$ vanishes.

There is also a local test-particle identity. Let $V\subseteq \mathrm{\Omega }$ be open, let $A$ be a supplied smooth one-form on $V$ with $F=dA$, and let $\chi$ be a smooth scalar with ${\mathcal{L}}_{\xi }A=d\chi$. If $I\subseteq \mathbb{R}$ is any interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and a $C^2$ curve $z:I\to V$ in the sense of clause (a) of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold has velocity $U=dz\mathrm{/}d\tau$ (one-sided at included endpoints), $g(U,U)=-c^2$, and satisfies $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$ with the C${}^2$-curve covariant acceleration of Covariant acceleration along a $C^2$ curve, for constants $m>0,q\in \mathbb{R}$, where $F^{\mathrm{♯}}(U{)}^a=F^a{}_b{U}^b$, then

$C=mg(U,\xi )+q(A(\xi )-\chi )$

is constant along the curve. This is a conditional mathematical identity for a supplied field and an existing curve; it constructs no particle trajectory, global potential, Einstein solution, or point-source self-field.

• The current $J_{\xi }$ is a Killing flux built from stress, not electric charge. The finite-chain formula includes a side flux and does not assert an asymptotic or universal global energy.
• If the symmetry parameter has length units in length coordinates, $\xi$ is dimensionless, $A(\xi )$ and $\chi$ have units $\mathrm{T}\mathrm{m}$, and the test-particle constant has momentum units; multiplying the stationary time-translation constant by $-c$ gives energy units.
• The statement uses no Einstein equation, no matter action, no Maxwell source construction, and no existence theorem for the curve.
• The draft lem-emg-maxwell-stress-divergence carrier has a complete conditional proof at its exact interface: smooth $F,j$, ${\mu }_0>0$, $dF=0$, and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ imply ${\mathrm{\nabla }}_a{T}_{\mathrm{E}\mathrm{M}}^{ab}=-F^{bc}{j}_c$. Its current Step 1 record is ready; the dated owner final-dispositions record resolves the earlier authoring-readiness hold at the current source binding. The declared action-variation, current-forms, and seven typed mathematical inputs are also current and ready at their inspected interfaces. This establishes authoring readiness only; no independent proof approval or publication is inferred. The stress-current deduction remains conditional on the stated model and supplier identity, while the local test-particle clause is proved separately.