Stationary vacuum interface jumps and the absence of magnetic sheet sources

Under the joint one-sided regularity, each side of $S\times I$ carries a $C^1$ extension to a closed side of a smaller stationary graph cylinder; on each compact closed half-cylinder the extensions and their first spatial and temporal derivatives are bounded, so the piecewise first derivatives of $E,B$ are locally bounded and locally integrable on $U$, while the volume sources are locally integrable by hypothesis, their one-sided traces and jumps are jointly continuous, and the jump densities are bounded on compact subsets of $S\times I$.

The interface $S$ is Lebesgue-null in ${\mathbb{R}}^3$: in the finite graph charts of [F1] every normal section of the graph over the parameter plane is a singleton, so Fubini and the nullity of singletons give ${\lambda }_3(S)=0$, and the finitely many charts cover $S$; hence $S\times I$ has four-dimensional measure zero locally and every locally integrable function vanishing off $S\times I$ represents the zero distribution.

Spacetime interface formula for a component: for each fixed $t$ the spatial restriction of the hypothesis satisfies the interface-derivative lemma on the spatial slice, so for a compactly supported spacetime test $\psi$ the identity ${\mathrm{\partial }}_i{F}_j=\{{\mathrm{\partial }}_i{F}_j\}+n_i[F_j]{\delta }_S$ may be applied slicewise and integrated in $t$; a time-independent finite partition of unity subordinate to stationary graph neighbourhoods of the spatial support of $\psi$, with cutoffs as in [F5], makes the terms $F_j{\mathrm{\partial }}_i\psi$, $({\mathrm{\partial }}_i{F}_j)\psi$ and $n_i[F_j]\psi$ absolutely integrable on the compact support, because the piecewise derivatives and the continuous jumps are bounded there and the surface measure of the compact sheet is finite by [F1]; Fubini [F6] then justifies integrating the slicewise identity, giving ${\mathrm{\partial }}_i{F}_j=\{{\mathrm{\partial }}_i{F}_j\}+n_i[F_j]\otimes {\delta }_{S\times I}$ for the spacetime distribution, where the sheet term acts as ${\int }_I{\int }_S{n}_i[F_j]\psi dSdt$. For the time derivative, on each spatial side the field is $C^1$, and compact-support integration by parts on a compact interval containing the time support of each spatially localized piece, followed by the Riemann-Lebesgue comparison and Fubini, gives $-\int F_j{\mathrm{\partial }}_t\psi =\int ({\mathrm{\partial }}_t{F}_j)\psi$ with no endpoint contribution, so ${\mathrm{\partial }}_t{F}_j=\{{\mathrm{\partial }}_t{F}_j\}$ carries no sheet term: the interface is stationary.

Apply the formula of step 1.3 to each of the four weak Maxwell identities. Off the sheet the fields and sources are $C^1$ and continuous, so the classical residuals $h_1=\mathrm{div}E-{\rho }_v\mathrm{/}{\epsilon }_0$, $h_2=\mathrm{div}B$, $h_3=\mathrm{curl}E+{\mathrm{\partial }}_{t}B$, $h_4=\mathrm{curl}B-{\mu }_0{J}_v-c^{-2}{\mathrm{\partial }}_{t}E$ are continuous on the open set $U\setminus (S\times I)$; against every test supported in that open set the four weak identities reduce to $\int h_{\mathrm{\ell }}\psi =0$, and the sign-and-bump argument of [F8] (a positive value of a continuous $h_{\mathrm{\ell }}$ at a point persists with a fixed sign on a small axis-parallel cube of positive measure, on which a bump equal to one on a smaller cube, supplied by [F5], makes $\int h_{\mathrm{\ell }}\psi$ strictly nonzero) forces $h_{\mathrm{\ell }}=0$ on $U\setminus (S\times I)$. Each residual is locally integrable on $U$ and vanishes off the null set $S\times I$ by step 1.2, hence represents the zero regular distribution; subtracting the off-sheet parts from the four identities leaves four sheet identities whose densities are $n\cdot [E]-\sigma \mathrm{/}{\epsilon }_0$, $n\cdot [B]$, $n\times [E]$ and $n\times [B]-{\mu }_{0}K$, each multiplied by the sheet distribution ${\delta }_{S\times I}$.

Write $a_1=n\cdot [E]-\sigma \mathrm{/}{\epsilon }_0$, $a_2=n\cdot [B]$, $a_3\in {\mathbb{R}}^3$ with components of $n\times [E]$, and $a_4=n\times [B]-{\mu }_{0}K$ for the continuous sheet densities of step 2.1. For a fixed real-valued spatial test $\phi \in \mathcal{D}(\mathrm{\Omega })$ (real tests suffice since complex tests split into real and imaginary parts) and each component density $a$ of these four, the function $H_{\phi }(t):={\int }_{S}a(x,t)\phi (x)dS$ is continuous on $I$: on a compact time interval the joint continuity of $a$ on the compact sheet gives uniform convergence as $t\to t_0$, and $\mathrm{\mid }{H}_{\phi }(t)-H_{\phi }(t_0)\mathrm{\mid }\le S(S)\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}{\sup }_{x\in S}\mathrm{\mid }a(x,t)-a(x,t_0)\mathrm{\mid }$ tends to $0$. Testing the sheet identity with $\phi (x)\eta (t)$ for $\eta \in C_c^{\mathrm{\infty }}(I)$ gives ${\int }_I{H}_{\phi }(t)\eta (t)dt=0$ for every such $\eta$; the one-dimensional sign-and-bump argument of [F8] then gives $H_{\phi }(t)=0$ for every $t\in I$. Since $\phi$ was an arbitrary ambient test, the surface-density localization lemma [F4] applied at each fixed $t$ to the continuous density $a(\cdot ,t)$ gives $a(\cdot ,t)=0$ on $S$; hence all four pointwise jump identities $n\cdot [E]=\sigma \mathrm{/}{\epsilon }_0$, $n\cdot [B]=0$, $n\times [E]=0$, $n\times [B]={\mu }_{0}K$ hold on $S\times I$, and the equivalent solved forms for $\sigma$ and $K$ follow by dividing by ${\epsilon }_0>0$ and ${\mu }_0>0$.

For the general weak-trace branch the same interface formula of step 1.3 is available with the weaker justification: the one-sided spatial $C^1$ extensions give the slicewise lemma at every time, the additional local $L^1$ hypotheses on the piecewise first derivatives make the localized terms of step 1.3 integrable, the measurable locally integrable traces make the sheet pairings ${\int }_I{\int }_{S}a\psi dSdt$ well defined by the ordinary integrals for every test $\psi$, and Fubini still applies; the temporal identity holds because each fixed off-sheet point has a $C^1$ dependence in $t$ and integration by parts against a compactly supported test is justified by the integrable derivative. Consequently each of the four sheet pairings of the Statement vanishes for every test; no product of distributions is formed, only the pairing of a locally integrable sheet density with a compact smooth test. For the almost-everywhere conclusion, use the countable family of ambient bumps with rational centres and rational radii $0<r<R$ with ${\bar{B}}_R(q)\subseteq \mathrm{\Omega }$, each fixed by the scaled-bump construction of [F5] rather than chosen per instance; for each such spatial test $\phi$, Fubini gives $H_{\phi }(t)={\int }_{S}a(x,t)\phi (x)dS\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(I)$, and product tests $\phi (x)\eta (t)$ give ${\int }_I{H}_{\phi }\eta =0$ for all temporal tests. Thus [F9] gives a null set of exceptional times for each such test, and the countable union of these null sets is null by [F7]. At each remaining time $t$, if some continuous-in-$x$ density $a(\cdot ,t)$ were nonzero at a point $x_0\in S$, its fixed sign would persist on a small graph neighbourhood of $x_0$; density of the rationals [F8] provides one of the rational balls inside that neighbourhood whose inner ball contains $x_0$, and a positive-area graph cell inside that inner ball, as in [F4], makes the pairing nonzero against the corresponding bump, a contradiction. Hence all four identities hold at every $x\in S$ for almost every $t\in I$; if the jump densities are additionally jointly continuous, the same positive-area-cell argument applied on nondegenerate time intervals upgrades the identities to all $(x,t)$.

Steps 1.1-3.1 prove the four pointwise jump identities under the joint one-sided regularity, and step 4.1 proves the weak-trace branch with its almost-everywhere and continuity upgrades; the sources used are exactly the stationary tangential sheet and volume parts, so no normal sheet current, magnetic sheet source, moving interface or product of distributions enters, and the normal $n$ points from ${\mathrm{\Omega }}_{-}$ to ${\mathrm{\Omega }}_{+}$ with $[F]=F_{+}-F_{-}$ throughout. The separate adoption [F2] is what makes the starting identities physical for the singular-source model; no empirical claim or measurement enters the deduction.