Propagation of the Gauss constraints, persistent defects and the full-Maxwell corollary

In the smooth branch, mixed partials commute and the divergence of a curl vanishes, so ${\mathrm{\partial }}_t{D}_B=\mathrm{div}({\mathrm{\partial }}_{t}B)=-\mathrm{div}(\mathrm{curl}E)=0$ pointwise, and ${\mathrm{\partial }}_t{D}_E=\mathrm{div}({\mathrm{\partial }}_{t}E)-{\epsilon }_0^{-1}{\mathrm{\partial }}_t\rho =c^2\mathrm{div}(\mathrm{curl}B)-c^2{\mu }_0\mathrm{div}J-{\epsilon }_0^{-1}{\mathrm{\partial }}_t\rho =-{\epsilon }_0^{-1}(\mathrm{div}J+{\mathrm{\partial }}_t\rho )=0$ pointwise, using $c^2{\mu }_0={\epsilon }_0^{-1}$ and the separately assumed continuity in the last equality. For each fixed $x$ the two functions $t\mapsto D_B(x,t)$ and $t\mapsto D_E(x,t)$ are differentiable on the interval $I$ with zero derivative, hence constant; therefore zero constraints at any $t_0$ persist at every time and a nonzero value at a point persists unchanged.

In the distributional branch the identical componentwise cancellation holds: for every test, the distributional divergence of the distributional curl of $B$ pairs to ${\sum }_{i,j,k}{\epsilon }_{ijk}⟨{\mathrm{\partial }}_i{\mathrm{\partial }}_j{B}_k,\psi ⟩$, with the three-index coefficients defined in Charge continuity as a compatibility condition of the vacuum Maxwell equations, Proof 1.2, which the commuting derivatives and the antisymmetry of ${\epsilon }_{ijk}$ present as its own negative, hence zero; applying the two evolution identities and the independent continuity identity, all derivatives being distributional, gives ${\mathrm{\partial }}_t{D}_B={\mathrm{\partial }}_t{D}_E=0$ in ${\mathcal{D}}^{\mathrm{\prime }}(U)$, without forming any product of distributions, since $D_B,D_E$ themselves are distributions only through their derivatives acting on tests.

For the continuous-representative statement assume ${\mathrm{\partial }}_{t}h=0$ in ${\mathcal{D}}^{\mathrm{\prime }}(U)$ for a continuous $h$, and let $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. The function $F(t):={\int }_{\mathrm{\Omega }}h(x,t)\phi (x)d{\lambda }_3(x)$ is continuous on $I$: on a compact time interval the integrands have the fixed compact spatial support of $\phi$ and a uniform bound coming from continuity of $h$ on the compact set $\mathrm{supp}\phi \times$ that interval, so dominated convergence applies. For $\eta \in C_c^{\mathrm{\infty }}(I)$ with ${\int }_I\eta =0$, extend $\eta$ by zero and put $H(t):={\int }_{-\mathrm{\infty }}^t\eta (v)dv$; the FTC makes $H$ smooth with $H^{\mathrm{\prime }}=\eta$, and, choosing $a<b$ in $I$ with $\mathrm{supp}\eta \subset (a,b)$, $H$ vanishes outside $[a,b]$: it is zero before $a$ and equals the zero total integral after $b$. Thus $\mathrm{supp}H\subseteq [a,b]\subset I$, so $\mathrm{\Psi }(x,t):=\phi (x)H(t)$ is a compact smooth test with ${\mathrm{\partial }}_t\mathrm{\Psi }=\phi \eta$; applying the derivative equation to this primitive test gives $0=⟨{\mathrm{\partial }}_{t}h,\mathrm{\Psi }⟩=-⟨h,{\mathrm{\partial }}_t\mathrm{\Psi }⟩=-{\int }_{I}F(t)\eta (t)dt$, the multiple integral being identified with the iterated one by the Borel product structure and Fubini and its Riemann-Lebesgue value by the comparison lemma. Choosing a nonnegative compact bump $\beta$ of integral one and writing an arbitrary temporal test as $\eta ={\eta }_0+({\int }_I\eta )\beta$ with ${\int }_I{\eta }_0=0$ gives ${\int }_I(F-C){\eta }_0=0$ for all ${\eta }_0$ with zero integral and $C:={\int }_{I}F\beta$; since also ${\int }_I(F-C)\beta ={\int }_{I}F\beta -C{\int }_I\beta =0$, every $\eta \in C_c^{\mathrm{\infty }}(I)$ satisfies ${\int }_I(F-C)\eta =0$, and the sign-and-bump argument (a nonzero value of the continuous $F-C$ at a point persists with one sign on a subinterval of positive measure, and a nonnegative bump equal to one on a smaller subinterval has strictly nonzero pairing by monotonicity and the positive finite measure of that subinterval) then forces $F-C=0$, that is, $F=C$ constant on $I$. For complex-valued functions this sign-and-bump argument is applied separately to their real and imaginary parts with real tests; it also applies componentwise to the spatial difference below. Since $\phi$ is arbitrary, $h(\cdot ,t_1)-h(\cdot ,t_2)$ is a continuous function on $\mathrm{\Omega }$ whose pairing against every spatial test vanishes, so the same sign-and-bump argument in ${\mathbb{R}}^3$ gives $h(\cdot ,t_1)=h(\cdot ,t_2)$; hence $h$ is independent of $t$.

In clause (c), a solution of the full postulate with the regularity of (a) satisfies the two evolution equations and, by the smooth branch of Charge continuity as a compatibility condition of the vacuum Maxwell equations, source continuity; its Gauss defects vanish identically by the two Gauss equations, and step 1.1 applies to its evolution and continuity hypotheses with the conclusion that the already vanishing defects persist. The original locally integrable weak branch has the four identities in ${\mathcal{D}}^{\mathrm{\prime }}$, hence zero defects and the statement of step 1.2. For the admitted point and stationary-sheet models, the separate adoption supplies the four identities and the distributional branch of Charge continuity as a compatibility condition of the vacuum Maxwell equations supplies the compatibility identity, so the defects vanish and step 1.2 applies. Conversely, if in (a) or in the continuous-representative case of (b) the defects vanish at one time, then by constancy they vanish at all times, which is exactly the pair of Gauss equations; adding them to the two evolution equations and the independently assumed continuity yields all four Maxwell identities on $U$, pointwise in (a) and distributionally in (b).

Steps 1.1-2.1 prove the smooth and distributional propagation statements, the persistence and nonzero-persistence readings for the smooth branch, the time independence of a continuous representative of a defect under Countable Choice, and the full-Maxwell corollaries and their converse; the nonzero-defect persistence is exactly the statement that evolution plus continuity cannot repair an initial Gauss defect, and no existence, matter dynamics or empirical claim enters the argument.