Gauge reachability: Lorenz gauge on the all-space slab and Coulomb gauge on a bounded smooth domain, with residual freedom

Branch (a). The pair $(\phi ,A)$ satisfies the hypotheses of [F5] verbatim: it is of class $C^{\mathrm{\infty }}$ in the extension sense on the slab and $G$ is its Lorenz divergence, with $E,B$ its field pair in the vocabulary of [F1] and [F2]. By F5 there is a gauge function $\chi$ of class $C^{\mathrm{\infty }}$ in the extension sense with $\chi (\cdot ,t_0)=0$, ${\mathrm{\partial }}_t\chi (\cdot ,t_0)=0$ and $W\chi =G$ on ${\mathbb{R}}^3\times (t_0,T)$; by F5 the transformed pair has the same $E$ and $B$, and by F5 it is in Lorenz gauge because $G(T_{\chi }(\phi ,A))=G-W\chi =0$ there; by F5 two such gauge functions differ by a solution of $W\chi =0$. The countable-choice assumption is carried by [F5]. This is exactly assertion (a).

Branch (b): data class and an open time interval. Let $N$ be the actual neighbourhood of $\bar{\mathrm{\Omega }}\times \bar{I}$ on which $(\phi ,A)$ is $C^{\mathrm{\infty }}$, and let $N_{\mathrm{\partial }}$ be a neighbourhood of $\mathrm{\partial }\mathrm{\Omega }\times \bar{I}$ on which the chosen boundary family is smooth in boundary charts. Since $\bar{\mathrm{\Omega }}$ is compact (Bounded C^k domains and boundary charts), the set of times $t$ for which a product neighbourhood of $\bar{\mathrm{\Omega }}\times \{t\}$ is contained in $N$ is open in $\mathbb{R}$, and it contains $\bar{I}$; let $J$ be its connected component containing $I$, and likewise choose $J_{\mathrm{\partial }}$ for $N_{\mathrm{\partial }}$ and replace $J$ by $J\cap J_{\mathrm{\partial }}$. Then $J\supseteq \bar{I}$ is an open interval, and the potentials, the bulk datum $F:={\epsilon }_0\mathrm{div}A$ and the chosen boundary family are smooth on $\bar{\mathrm{\Omega }}\times J$ and in its boundary atlas times $J$, in the sense required by [F6]; the derivative bounds below use only $\bar{\mathrm{\Omega }}\times K$ with compact $K\subset J$, never compactness of all of $\bar{I}$. If $I$ is empty then $\mathrm{\Omega }\times I$ is empty and the conclusions of (b) are vacuous; the construction below therefore assumes $I$ nonempty.

Branch (c): the difference of two representations. Assume first that $I$ has at least two elements, and put $\delta \phi :=\tilde{\phi }-\phi$ and $\delta A:=\tilde{A}-A$, which are jointly $C^2$ on $\mathrm{\Omega }\times I$ in the convention of [F8]. Equality of the field pairs gives, by the componentwise linearity of $\mathrm{\nabla }$, ${\mathrm{\partial }}_t$ and $\mathrm{curl}$ [F9], $\mathrm{curl}\delta A=B(\tilde{\phi },\tilde{A})-B(\phi ,A)=0$ and $\mathrm{\nabla }\delta \phi +{\mathrm{\partial }}_t\delta A=-E(\tilde{\phi },\tilde{A})+E(\phi ,A)=0$ on $\mathrm{\Omega }\times I$. Both pairs being in Lorenz gauge means $G(\phi ,A)=G(\tilde{\phi },\tilde{A})=0$ on $\mathrm{\Omega }\times I$.

Branch (b), Dirichlet case. Apply [F6] with $ϵ={\epsilon }_0>0$ and bulk datum $F={\epsilon }_0\mathrm{div}A$ on the open interval $J$, and with the Dirichlet family $g_D$; the data are smooth in the required sense by step 1.2. This gives a unique solution $\psi$ on $\mathrm{\Omega }\times J$, the restriction of one $C^{\mathrm{\infty }}$ ambient function on a neighbourhood of $\bar{\mathrm{\Omega }}\times J$, with $-{\epsilon }_0\mathrm{\Delta }\psi =F$ on $\mathrm{\Omega }\times J$ and $\psi =g_D$ on $\mathrm{\partial }\mathrm{\Omega }\times J$. On $\mathrm{\Omega }\times I\subseteq \mathrm{\Omega }\times J$ we get $\mathrm{\Delta }\psi =-\mathrm{div}A$ (since ${\epsilon }_0>0$), hence $\mathrm{div}(A+\mathrm{\nabla }\psi )=\mathrm{div}A+\mathrm{div}\mathrm{\nabla }\psi =\mathrm{div}A+\mathrm{\Delta }\psi =0$ by the definitions [F1]; on $\mathrm{\partial }\mathrm{\Omega }\times I$, $\psi =g_D$. Therefore $T_{\psi }(\phi ,A)$ is in Coulomb gauge on $\mathrm{\Omega }\times I$ and has the same field pair as $(\phi ,A)$: apply [F3] to the smooth ambient pairs on the open product $\mathrm{\Omega }\times J$ and restrict to $\mathrm{\Omega }\times I$, including a singleton $I$.

Branch (b), Neumann case. Let ${\mathrm{\Omega }}_j$ be a connected component of $\mathrm{\Omega }$ and define, for $t\in J$, $c_j(t):={\int }_{{\mathrm{\Omega }}_j}F(\cdot ,t)dx+{\int }_{\mathrm{\partial }{\mathrm{\Omega }}_j}\kappa (\cdot ,t)dS$. The bulk integral is a parameter integral of the smooth family $F$ over the fixed compact set ${\bar{\mathrm{\Omega }}}_j$, and the boundary integral is a finite sum of chart parameter integrals of the smooth family $\kappa$ over the fixed boundary atlas, so $c_j$ is $C^{\mathrm{\infty }}$ on $J$ by repeated applications of [F11] on compact time subintervals. Here $\mathrm{\mid }{\mathrm{\Omega }}_j\mathrm{\mid }$ below is its finite positive Lebesgue volume: boundedness gives finiteness, and the nonempty open component contains a nondegenerate cube of positive volume by [F11]; there are finitely many components by [F6], and distinct component closures cannot meet: a sufficiently small connected one-sided graph subpatch near a common boundary point would intersect both and put them in the same component. Thus componentwise constants define smooth functions up to the closed spatial domain; the compatibility hypothesis gives $c_j(t)=0$ for $t\in I$. Put $\widehat{F}:=F-c_j(t)\mathrm{/}\mathrm{\mid }{\mathrm{\Omega }}_j\mathrm{\mid }$ on each ${\mathrm{\Omega }}_j$. Then $\widehat{F}$ is $C^{\mathrm{\infty }}$ on $\bar{\mathrm{\Omega }}\times J$, $\widehat{F}=F$ on $\mathrm{\Omega }\times I$, and for every $t\in J$ and component ${\mathrm{\Omega }}_j$, ${\int }_{{\mathrm{\Omega }}_j}\widehat{F}(\cdot ,t)+{\int }_{\mathrm{\partial }{\mathrm{\Omega }}_j}\kappa (\cdot ,t)={\int }_{{\mathrm{\Omega }}_j}F(\cdot ,t)-c_j(t)+{\int }_{\mathrm{\partial }{\mathrm{\Omega }}_j}\kappa (\cdot ,t)=0$, so the compatibility required by [F6] holds throughout $J$. Apply [F6] with the zero-mean normalization on each component and the datum $(\widehat{F},\kappa )$: this gives a unique normalized solution $\psi$ on $\mathrm{\Omega }\times J$, the restriction of one smooth ambient function, with $-{\epsilon }_0\mathrm{\Delta }\psi =\widehat{F}$ on $\mathrm{\Omega }\times J$ and ${\epsilon }_0{\mathrm{\partial }}_{\nu }\psi =\kappa$ on $\mathrm{\partial }\mathrm{\Omega }\times J$. On $\mathrm{\Omega }\times I$ we have $\mathrm{\Delta }\psi =-\mathrm{div}A$ as before, hence $\mathrm{div}(A+\mathrm{\nabla }\psi )=0$; on $\mathrm{\partial }\mathrm{\Omega }\times I$, ${\epsilon }_0{\mathrm{\partial }}_{\nu }\psi =\kappa$. Hence $T_{\psi }(\phi ,A)$ is in Coulomb gauge on $\mathrm{\Omega }\times I$ with the same field pair, applying [F3] on $\mathrm{\Omega }\times J$ before restriction, also for singleton $I$.

Branch (c): the radial scalar witness. By step 1.3 the field $\delta A$ is jointly $C^2$ on the open star-shaped $\mathrm{\Omega }$ times the interval $I$ and satisfies $\mathrm{curl}\delta A(\cdot ,t)=0$ for every $t$; the scalar clause of [F8] applied with $C=\delta A$ gives the jointly $C^2$ function ${\chi }_0:=-{\varphi }_{\delta A}$, that is ${\chi }_0(x,t)={\int }_0^1⟨\delta A(a+s(x-a),t),x-a⟩ds$, with $\mathrm{\nabla }{\chi }_0(\cdot ,t)=\delta A(\cdot ,t)$ for every $t\in I$, and ${\chi }_0(a,t)=0$ for every $t\in I$. The construction is the radial one of that lemma, so ${\chi }_0$ is the restriction of a $C^2$ function on an open radial-admissibility neighbourhood of $\mathrm{\Omega }\times I$.

Branch (b): uniqueness and residual freedom. Let ${\psi }^{\mathrm{\prime }}$ be any other generator of the statement's class on $\mathrm{\Omega }\times I$: it is $C^{\mathrm{\infty }}$ near $\bar{\mathrm{\Omega }}\times \bar{I}$, and on $\mathrm{\Omega }\times I$ it satisfies $\mathrm{\Delta }{\psi }^{\mathrm{\prime }}=-\mathrm{div}A$ as well as the same boundary data in the relevant case. Then $\chi :={\psi }^{\mathrm{\prime }}-\psi$ is $C^{\mathrm{\infty }}$ near $\bar{\mathrm{\Omega }}\times \bar{I}$ and $\mathrm{\Delta }\chi (\cdot ,t)=0$ on $\mathrm{\Omega }$ for every $t\in I$, i.e. $\chi$ is spatially harmonic at every $t$; this proves the general residual sentence. Fix $t\in I$ and apply Green's identity [F7] on $\mathrm{\Omega }$ with $u=v=\chi (\cdot ,t)$: ${\int }_{\mathrm{\Omega }}(\chi \mathrm{\Delta }\chi +\mathrm{\mid }\mathrm{\nabla }\chi {\mathrm{\mid }}^2)dx={\int }_{\mathrm{\partial }\mathrm{\Omega }}\chi {\mathrm{\partial }}_{\nu }\chi dS$, whose right-hand side vanishes in the Dirichlet case because $\chi =0$ on $\mathrm{\partial }\mathrm{\Omega }$ (both generators have the value $g_D$) and in the Neumann case because ${\mathrm{\partial }}_{\nu }\chi =0$ (both have conormal datum $\kappa \mathrm{/}{\epsilon }_0$); the first term vanishes because $\mathrm{\Delta }\chi =0$, so ${\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }\chi (\cdot ,t){\mathrm{\mid }}^2=0$ and hence $\mathrm{\nabla }\chi (\cdot ,t)=0$ on $\mathrm{\Omega }$ (the integrand is continuous and nonnegative). Therefore $\chi (\cdot ,t)$ is constant on each connected component of $\mathrm{\Omega }$; in the Dirichlet case the constant is the boundary value of $\chi$, which is zero, so $\chi (\cdot ,t)=0$ and ${\psi }^{\mathrm{\prime }}=\psi$ on $\mathrm{\Omega }\times I$; in the Neumann case the constants are the only freedom, and imposing ${\int }_{{\mathrm{\Omega }}_j}{\psi }^{\mathrm{\prime }}(\cdot ,t)={\int }_{{\mathrm{\Omega }}_j}\psi (\cdot ,t)=0$ makes each of them zero. This proves uniqueness of generator values with the stated data and normalization, and the componentwise constant freedom without normalization. If $I$ has at least two elements, equality of the generator restrictions implies equality of their spatial and relative time derivatives, so the transformed pairs agree in the Dirichlet or normalized Neumann case. If $I=\{t_0\}$, equality of generator values need not give equality of ambient time derivatives: with zero original potentials and zero Dirichlet data, the ambient generators $0$ and $t-t_0$ both solve the required spatial problem at $t_0$, but their transforms there are $(0,0)$ and $(-1,0)$, respectively. Thus no singleton transformed-pair uniqueness follows. The Green identity carries its own ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis, already assumed in branch (b).

Branch (c): the auxiliary scalar has zero gradient. By [F9] the mixed partials of the jointly $C^2$ function ${\chi }_0$ commute, so ${\mathrm{\partial }}_t\mathrm{\nabla }{\chi }_0=\mathrm{\nabla }{\mathrm{\partial }}_t{\chi }_0$ componentwise; by step 2.3 and the identity $\mathrm{\nabla }\delta \phi +{\mathrm{\partial }}_t\delta A=0$ of step 1.3, $\mathrm{\nabla }(\delta \phi +{\mathrm{\partial }}_t{\chi }_0)=\mathrm{\nabla }\delta \phi +{\mathrm{\partial }}_t\mathrm{\nabla }{\chi }_0=\mathrm{\nabla }\delta \phi +{\mathrm{\partial }}_t\delta A=0$ on $\mathrm{\Omega }\times I$.

Branch (c): the auxiliary scalar is position independent. Fix $t\in I$. The function $x\mapsto g(x,t):=\delta \phi (x,t)+{\mathrm{\partial }}_t{\chi }_0(x,t)$ is $C^1$ on $\mathrm{\Omega }$ with $\mathrm{\nabla }g(\cdot ,t)=0$ by step 3.2, and $\mathrm{\Omega }$ is star-shaped with centre $a$, so for $x\in \mathrm{\Omega }$ the radial path $s\mapsto g(a+s(x-a),t)$ is defined on $[0,1]$ and, by the chain rule [F9], has derivative $⟨\mathrm{\nabla }g(a+s(x-a),t),x-a⟩=0$ at every interior point; the zero-derivative corollary [F9] applied on $[0,1]$ makes it constant, so $g(x,t)=g(a,t)$ for every $x\in \mathrm{\Omega }$. In particular the common value depends on $t$ only.

Branch (c): construction of the gauge generator. Assume first that $I$ is open, and put $h(t):=g(a,t)=\delta \phi (a,t)+{\mathrm{\partial }}_t{\chi }_0(a,t)$ for $t\in I$; since ${\chi }_0(a,t)=0$ for every $t$, step 2.3 gives $h(t)=\delta \phi (a,t)$, a $C^1$ function of $t$ on the open interval $I$. By [F9] the function $\eta (t):=-{\int }_{t_{\ast }}^{t}h(u)du$, with a fixed $t_{\ast }\in I$, is a primitive of $-h$ on $I$, hence $C^2$ because $h$ is $C^1$, and $\eta$ is independent of $x$. Put $\chi :={\chi }_0+\eta$ on $\mathrm{\Omega }\times I$, which is jointly $C^2$ there. Then $\mathrm{\nabla }\chi =\mathrm{\nabla }{\chi }_0=\delta A$ by step 2.3, and ${\mathrm{\partial }}_t\chi ={\mathrm{\partial }}_t{\chi }_0-h$; since $h(t)=g(x,t)=\delta \phi (x,t)+{\mathrm{\partial }}_t{\chi }_0(x,t)$ for every $x$ by step 4.1, this reads ${\mathrm{\partial }}_t\chi =-\delta \phi$. Hence $T_{\chi }(\phi ,A)=(\phi -{\mathrm{\partial }}_t\chi ,A+\mathrm{\nabla }\chi )=(\phi +\delta \phi ,A+\delta A)=(\tilde{\phi },\tilde{A})$ on $\mathrm{\Omega }\times I$. For a non-open interval $I$ the same construction is performed with the ambient $C^2$ extensions supplied by the restriction convention of [F8]: ${\chi }_0$ is already a restriction from an open radial-admissibility set, the common value admits the ambient $C^1$ representative $t\mapsto \tilde{\phi }(a,t)-\phi (a,t)+{\mathrm{\partial }}_t{\chi }_0(a,t)$ on an open time interval around $I$, its primitive is added to ${\chi }_0$ inside that neighbourhood, and the resulting function is a $C^2$ restriction whose values on $\mathrm{\Omega }\times I$ satisfy the two displayed identities by the same pointwise computation.

Branch (c): the generator has vanishing wave expression. The pair $(\phi ,A)$ is jointly $C^2$ and the generator $\chi$ of step 5.1 is jointly $C^2$, so the gauge-shift identity [F4] applies and gives $G(T_{\chi }(\phi ,A))=G(\phi ,A)-W\chi$ on $\mathrm{\Omega }\times I$. By step 5.1 $T_{\chi }(\phi ,A)=(\tilde{\phi },\tilde{A})$, and both pairs are in Lorenz gauge by step 1.3, so $0=0-W\chi$, that is $W\chi =0$ on $\mathrm{\Omega }\times I$.

Branch (c): degenerate and empty intervals. If $I=\mathrm{\varnothing }$, then $\mathrm{\Omega }\times I=\mathrm{\varnothing }$ and the assertion is vacuous: there are no points at which an identity is required. If $I=\{t_0\}$ is a singleton, then by the convention stated in (c) the ambient $C^2$ extensions and the same-field and Lorenz conditions on an ambient open time neighbourhood $J\ni t_0$ are part of the data; apply the construction of step 1.3 and steps 5.1 and 6.1 with $I$ replaced by $J$ (a nondegenerate open interval) and restrict the resulting generator, its transform identity and $W\chi =0$ to $\mathrm{\Omega }\times \{t_0\}$. This covers every degenerate case and completes (c).

Scope and conclusion. Step 2.1 and step 2.2 construct the required Coulomb generators in the two declared boundary families, and step 3.1 proves the uniqueness clauses and the residual classification, so (b) is complete up to the auxiliary non-uniqueness of ambient continuations recorded in [F6]. Step 1.1 is (a). Step 1.3 and steps 4.1 through 7.1 construct, for two Lorenz $C^2$ representations of one field pair on a star-shaped domain, a $C^2$ gauge generator with $W\chi =0$, completing (c). Nothing in the branches asserts a gauge function meeting arbitrary prescribed boundary or initial restrictions: in (a) the gauge Cauchy data are zero and the spatial domain is all of ${\mathbb{R}}^3$; in (b) the boundary family and its compatibility are those stated; and the instantaneous character of the Coulomb potential is a property of the chosen representation, since by [F2] only the field pair $(E,B)$ is physical content. The Axiom of Countable Choice is used in (a) through [F5] and in (b) through [F6] and [F7]; the full Axiom of Choice is used in (b) through [F6] and its Sobolev suppliers. No empirical claim is made anywhere.