The static and pure electrostatic regimes and their fixed-time reading

Clause (c): in the static regime ${\mathrm{\partial }}_{t}E={\mathrm{\partial }}_{t}B=0$; substituting into the two curl equations of the smooth branch of [F2] leaves $\mathrm{curl}E=0$ and $\mathrm{curl}B={\mu }_{0}J$ at every point and every fixed time, while the two divergence equations are unchanged, giving exactly the displayed four equations; the operators are those of [F4] applied to the fixed-time fields of [F1]. In the pure electrostatic regime $B=0$ and $J=0$, so $\mathrm{div}B=0$ and $\mathrm{curl}B=0$ are identities and the reduction is the pair $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{curl}E=0$.

Clause (d): in the static regime each of $E,B,\rho ,J$ has vanishing partial $t$-derivative at every point of $\mathrm{\Omega }\times I$. For fixed $y$ and two times $t_1<t_2$ in $I$, the mean value theorem [F5] applied to $t\mapsto F(y,t)$ on $[t_1,t_2]$ gives $F(y,t_2)-F(y,t_1)=0$, so $F(y,t)$ depends only on $y$; writing $F_0(y):=F(y,t_0)$ for a fixed $t_0\in I$ we get $F=F_0\circ \pi$ with $\pi (y,t):=y$, and the four reduced equations of clause (c), which are pointwise conditions at each fixed time [F4], are therefore equations between the spatial maps $E_0,B_0,{\rho }_0,J_0$. The same reading is available for the locally integrable time-independent objects of clause (e), for which time independence is an explicit hypothesis rather than a consequence of differentiability.

Clause (e), spatial pairings as time integrals: let the time-independent locally integrable data be read as spatial maps $E_0,B_0,{\rho }_0,J_0$ by the fixed-time reading of step 2.1, and fix $\psi \in C_c^{\mathrm{\infty }}(U)$. For a spatial operator expression $S$ that is linear in the data (for instance $S=\mathrm{div}{E}_0-{\rho }_0\mathrm{/}{\epsilon }_0$), choosing Borel representatives of the coefficients [F7] and moving every distributional spatial derivative onto the test gives the pairing identity $⟨S_U,\psi ⟩={\int }_I⟨S_{\mathrm{\Omega }},\psi (\cdot ,t){⟩}_{\mathrm{\Omega }}dt,$ where $S_U$ denotes the distributional expression on $U$ evaluated with distributional derivatives on $U$ and $S_{\mathrm{\Omega }}$ the same expression on $\mathrm{\Omega }$ evaluated with distributional spatial derivatives; in the displayed instance both sides equal the same absolutely convergent iterated integrals, computed for $S_{\mathrm{\Omega }}$ by moving the spatial derivative onto the test, $-{\int }_I{\int }_{\mathrm{\Omega }}{E}_0\cdot {\mathrm{\nabla }}_y\psi d{\lambda }_{3}dt-\frac{1}{{\epsilon }_0}{\int }_I{\int }_{\mathrm{\Omega }}{\rho }_0\psi d{\lambda }_{3}dt,$ products of locally integrable coefficients with derivatives of the compactly supported test being integrable by Fubini [F7], and for data that are additionally $C^1$ the inner pairing agrees with the classical integral ${\int }_{\mathrm{\Omega }}(\mathrm{div}{E}_0-{\rho }_0\mathrm{/}{\epsilon }_0)\psi (\cdot ,t)d{\lambda }_3$ ([F4]); each $t$-slice $\psi (\cdot ,t)$ is smooth with compact support in $\mathrm{\Omega }$ because the section at $t$ of the compact support of $\psi$ is compact and lies in $\mathrm{\Omega }$ ([F6]). The same computation applies to $\mathrm{div}B$, $\mathrm{curl}E$ and $\mathrm{curl}B-{\mu }_{0}J$.

Clause (e), vanishing of the time derivatives: let $F$ be a time-independent locally integrable function on $U$ (one component of $E$ or $B$, or $\rho$). Choosing a Borel representative of $F$ [F7] and using the product structure and the fixed-time reading of step 3.1, $⟨T_F,{\mathrm{\partial }}_t\psi ⟩={\int }_{U}F(y){\mathrm{\partial }}_t\psi (y,t)d{\lambda }_4={\int }_{\mathrm{\Omega }}F(y)({\int }_I{\mathrm{\partial }}_t\psi (y,t)dt)d{\lambda }_3(y)=0,$ because for each $y$ the function $t\mapsto \psi (y,t)$ is smooth with compact support in $I$, so its integral of derivative vanishes by the fundamental theorem [F5] (the inner integral is a Lebesgue integral of a continuous compactly supported integrand, equal to its Riemann value [F7]); the iterated integral is absolutely convergent since $\psi$ has compact support and $F$ is locally integrable on the bounded region of the support. By [F6] this says ${\mathrm{\partial }}_t{T}_F=0$ in ${\mathcal{D}}^{\mathrm{\prime }}(U)$, so the two time-dependent terms of the weak predicate [F3] vanish identically for time-independent fields.

Clause (e), the equivalence: if the four spatial equations hold in ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$, then for every $t\in I$ the test $\psi (\cdot ,t)$ of step 3.1 gives $⟨S_{\mathrm{\Omega }},\psi (\cdot ,t){⟩}_{\mathrm{\Omega }}=0$, hence the integral of step 3.1 vanishes; together with the vanishing time-derivative pairings of step 4.1 this makes all four weak pairings of [F3] vanish, so the weak equations hold. Conversely, if the four weak pairings of [F3] vanish, choose $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and a normalized temporal bump $\chi \in C_c^{\mathrm{\infty }}(I)$ with ${\int }_I\chi =1$ from [F9] and test with $\psi (y,t)=\eta (y)\chi (t)$, which lies in $C_c^{\mathrm{\infty }}(U)$; by steps 3.1 and 4.1, $0={\int }_I\chi (t)⟨S_{\mathrm{\Omega }},\eta {⟩}_{\mathrm{\Omega }}dt=⟨S_{\mathrm{\Omega }},\eta {⟩}_{\mathrm{\Omega }}$, so each spatial equation holds in ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ since $\eta$ was arbitrary. This is the asserted if-and-only-if, and it uses no product of distributions: all pairings are against a single test function.

Clause (e), stationary point charge: take $I_0:=I$, the constant $C^1$ trajectory $z(t)\equiv x_0$ with $x_0\in \mathrm{\Omega }$, and $q\in \mathbb{R}$. Its graph is $\{x_0\}\times I$, and the graph map is proper into $U=I\times \mathrm{\Omega }$ because the preimage of a compact subset of $U$ is the corresponding compact closed subset of $I$; the source of [F3] is then $⟨{\rho }_{\mathrm{p}\mathrm{t}},\psi ⟩=q{\int }_I\psi (x_0,t)dt$ and $⟨J_{\mathrm{p}\mathrm{t}},\psi ⟩=q{\int }_I{z}^{\mathrm{\prime }}(t)\psi (z(t),t)dt=0$. For each fixed time the spatial reading is the point source pairing a compactly supported smooth spatial test $\eta$ as $\eta \mapsto q\eta (x_0)$, which is exactly the spatial point source of [F8]; each spatial slice $\psi (\cdot ,t)$ belongs to $\mathcal{D}(\mathrm{\Omega })$ by step 3.1, and the spacetime pairing is the time integral of these spatial pairings: $⟨{\rho }_{\mathrm{p}\mathrm{t}},\psi ⟩={\int }_I⟨q{\delta }_{x_0},\psi (\cdot ,t){⟩}_{\mathrm{\Omega }}dt=q{\int }_I\psi (x_0,t)dt$. This integral is finite because $t\mapsto \psi (x_0,t)$ is smooth and compactly supported in $I$.

Clauses (a) and (b) are the definitions of the static and pure electrostatic regimes; clause (c) is step 1.1, clause (d) is step 2.1, clause (e) is steps 3.1-6.1, and clause (f) records that the conditions were imposed in the one frame, that no stability, support, boost or singular-source equation beyond the dictionary of (e), and no empirical claim, is asserted. The verification used only the cited smooth branch, the definitions of the primitives, and the measure/distribution interfaces; no equation of motion and no empirical premise entered.