Metric dependence, flux sectors and constitutive geometry

In coordinates, each pullback coefficient is the finite sum of products described in [F2]. Every form coefficient composed with $\sigma$ is continuous by [F1] and [F3], and each derivative of $\sigma$ is continuous by [F1]. Finite sums of continuous real functions are continuous by the triangle inequality. For a product $fg$ at a point $u_0$, first restrict to a neighbourhood where $\mathrm{\mid }g(u)-g(u_0)\mathrm{\mid }<1$, so $\mathrm{\mid }g(u)\mathrm{\mid }\le \mathrm{\mid }g(u_0)\mathrm{\mid }+1$, then use $\mathrm{\mid }f(u)g(u)-f(u_0)g(u_0)\mathrm{\mid }\le \mathrm{\mid }f(u)-f(u_0)\mathrm{\mid }(\mathrm{\mid }g(u_0)\mathrm{\mid }+1)+\mathrm{\mid }f(u_0)\mathrm{\mid }\mathrm{\mid }g(u)-g(u_0)\mathrm{\mid }$. Continuity of $f$ and $g$ makes the right side arbitrarily small. Induction proves continuity for every finite product, hence $g_{\sigma }$ is continuous on ${\mathrm{\Delta }}^k$. If two smooth extensions of $\sigma$ are chosen, they agree as functions on the relative interior of ${\mathrm{\Delta }}^k$, which is open in its affine span; hence their tangential derivatives and pullback coefficients agree there. For any boundary point $u$, the points $u_s=(1-s)u+sb$, where $b=(1\mathrm{/}(k+1),\dots ,1\mathrm{/}(k+1))$ and $0<s\le 1$, have all coordinates positive and coordinate sum below $1$, so they lie in the relative interior and converge to $u$. Continuity of both coefficient functions makes them equal at $u$ too. Thus $g_{\sigma }$ is independent of the extension. No immersion is needed; when $D{\sigma }_u$ has rank below $k$, alternation makes the pullback coefficient zero at $u$.

The set ${\mathrm{\Delta }}^k$ in [F4] is closed and bounded, hence compact by [F5]. Its boundary consists of its coordinate faces $u_i=0$ and its upper face ${\sum }_i{u}_i=1$. If $k=1$, these are the two endpoints $0,1$, covered by two intervals of arbitrarily small total length, so the boundary has content zero by [F7]. If $k\ge 2$, each coordinate face is the graph of the constant zero function over its closed bounded coordinate projection, and the upper face is the graph of $u_k=1-{\sum }_{i<k}{u}_i$ over the closed bounded simplex $\{u_i\ge 0,{\sum }_{i<k}{u}_i\le 1\}$. These bases are compact by [F5]. The upper-face function is continuous because $\mathrm{\mid }{\sum }_{i<k}(u_i-v_i)\mathrm{\mid }\le (k-1)\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$, and the lower-face functions are constant; [F6] therefore makes each face content zero. Given $\epsilon >0$, cover each of the finitely many faces by a finite cube family with total volume less than $\epsilon \mathrm{/}(k+1)$; their union is a finite cover of the boundary with total volume less than $\epsilon$ by [F7]. Thus $\mathrm{\partial }{\mathrm{\Delta }}^k$ has content zero, and [F7] makes the bounded set ${\mathrm{\Delta }}^k$ Jordan measurable.

By [F9], the continuous coefficient $g_{\sigma }$ is Riemann integrable on the compact Jordan simplex from step 1.2, so each simplex integral in the Definition is finite.

A finite chain has finitely many nonzero coefficients by [F10]. Its integral is therefore a finite sum of finite real numbers; the zero chain has the empty sum, equal to $0$ by [F10]. If a chain is presented with repeated copies of the same simplex, [F10] and linearity combine their coefficients without changing the result. The positive affine parameter orientation is fixed by [F8]. The Definition covers $C^1$ coefficient forms and the continuous coefficients of their first exterior derivatives by [F12]. For smooth $\omega$, the coefficient and affine parameter integral are exactly those in [F11], so the two integral conventions agree.

Each displayed spatial form is an alternating covector field because the coordinate 1-forms are a basis and wedge products are alternating. Since $dt=d{x}^0\mathrm{/}c$, the time-space part of $F$ has $F_{0i}=-E_i\mathrm{/}c$, and its spatial part has $F_{ij}={ϵ}_{ijk}{B}_k$; hence it matches the existing electromagnetic two-form components. The definitions of $D,H,{\rho }_f,J_f$ give exactly the coefficient fields used in $G$ and $\mathcal{J}$.

On an oriented spatial 2-surface, the coefficients of $\mathcal{B}$ and $\mathcal{D}$ have units T and C m${}^{-2}$, respectively, so their integrals have units T m${}^2$ = Wb and C. The time-space terms have units (s)(V m${}^{-1}$)(m) = V s = Wb for $dt\wedge \mathcal{E}$, and (s)(A m${}^{-1}$)(m) = A s = C for $dt\wedge \mathcal{H}$. Likewise ${\rho }_{f}vo{l}_3$ and $dt\wedge {\mathcal{J}}_{\mathrm{s}\mathrm{p}}$ each integrate with units C.

A smooth singular simplex has compact Jordan parameter domain ${\mathrm{\Delta }}^2$ and its pullback coefficient is continuous, so [F8] gives a finite Riemann integral. The same local convention gives finite integrals for continuous spacetime 2-forms and 3-forms on their respective compact parameter simplices. For a smooth form this agrees with the usual simplex integral of [F8]. For $C^1$ coefficients as in [F9], the coordinate formula for $d$ agrees with the standard one on smooth coefficients by [F6]. The definitions assign $F$ and $G$ independently from the field and excitation data; [F5] is not applied as a constitutive law here.

Each $D_i=\{u_j\ge 0:j\mathrm{\ne }i,{\sum }_{j\mathrm{\ne }i}{u}_j\le 1\}$ is the standard two-simplex up to a coordinate permutation, and $T$ is the standard three-simplex, so [F9] gives that all are compact Jordan. The coefficient functions $f_i$ and ${\mathrm{\partial }}_i{f}_i$ are continuous on them by the $C^1$ hypothesis [F8], so their integrals exist by [F9]. Fix $i$ and fix the other two coordinates in $D_i$. The $u_i$-section of $T$ is $[0,1-{\sum }_{j\mathrm{\ne }i}{u}_j]$. If $1-{\sum }_{j\mathrm{\ne }i}{u}_j>0$, [F3] makes its integral of ${\mathrm{\partial }}_i{f}_i$ the upper endpoint value of $f_i$ minus the lower endpoint value. If the upper endpoint is $0$, the section is a singleton: its bounded integrand has zero integral (cover the point by intervals of arbitrarily small length), and the endpoint difference is also zero. Thus the same identity holds for every section. Integrating this identity over $D_i$ by [F4], then summing over $i$, writes ${\int }_{T}d\eta$ as the sum of all upper endpoint terms minus all lower endpoint terms.

On the three lower faces, in increasing order of their remaining coordinates, the induced boundary integrals are $-\int f_{1}d{u}^2\wedge d{u}^3$ on $u_1=0$, $-\int f_{2}d{u}^1\wedge d{u}^3$ on $u_2=0$, and $-\int f_{3}d{u}^1\wedge d{u}^2$ on $u_3=0$. The first and third signs are the boundary signs in [F2]; the middle face has positive signed face but the restriction of $\eta$ is already $-f_{2}d{u}^1\wedge d{u}^3$. Thus the lower endpoint terms from step 1.1 are exactly the three lower-face contributions to ${\int }_{\mathrm{\partial }T}\eta$.

On the upper face, set $u_1=1-u_2-u_3$. Substitution gives $\eta {\mathrm{\mid }}_{u_1+u_2+u_3=1}=(f_1+f_2+f_3)d{u}^2\wedge d{u}^3$. Its outward normal is proportional to $(1,1,1)$, while the ordered tangent vectors $(-1,1,0),(-1,0,1)$ have cross product $(1,1,1)$, so $(u_2,u_3)$ gives the positive face orientation. The ordered affine face parameters differ from $(u_2,u_3)$ by an affine map with determinant of absolute value $1$; [F5] therefore identifies the upper endpoint sum in step 1.1 with the upper-face integral. Adding the four face contributions using [F6] proves ${\int }_{T}d\eta ={\int }_{\mathrm{\partial }T}\eta$.

Expand in the fixed chart. The spatial part of $dF$ is $(\mathrm{div}B)vo{l}_3$ and its $dt$ part is $dt\wedge ({\mathrm{\partial }}_t\mathcal{B}+d_{\mathrm{\Omega }}\mathcal{E})$, where $d_{\mathrm{\Omega }}\mathcal{E}=({\mathrm{\partial }}_2{E}_3-{\mathrm{\partial }}_3{E}_2)d{x}^2\wedge d{x}^3+({\mathrm{\partial }}_3{E}_1-{\mathrm{\partial }}_1{E}_3)d{x}^3\wedge d{x}^1+({\mathrm{\partial }}_1{E}_2-{\mathrm{\partial }}_2{E}_1)d{x}^1\wedge d{x}^2$. Thus $d_{\mathrm{\Omega }}\mathcal{E}$ has the curl components in the ordered spatial 2-form basis. Likewise, the spatial part of $dG$ is $(\mathrm{div}D)vo{l}_3$ and its $dt$ part is $dt\wedge ({\mathrm{\partial }}_t\mathcal{D}-d_{\mathrm{\Omega }}\mathcal{H})$, where $d_{\mathrm{\Omega }}\mathcal{H}$ is given by the same curl formula with $H$ in place of $E$; $\mathcal{J}={\rho }_{f}vo{l}_3-dt\wedge {\mathcal{J}}_{\mathrm{s}\mathrm{p}}$. These coefficient identities follow by applying the $C^1$ component formula of [F1] to each displayed wedge term. Equality of the independent spatial and $dt$ coefficients therefore gives both directions of the claimed equivalence.

In vacuum $P=M=0$, so $D={ϵ}_{0}E$ and $H={\mu }_0^{-1}B$ by [A2]. Applying the basis formula [F3] to the six terms of $F$ gives ${\star }_{\eta }F=cdt\wedge (B_{1}d{x}^1+B_{2}d{x}^2+B_{3}d{x}^3)+c^{-1}(E_{1}d{x}^2\wedge d{x}^3+E_{2}d{x}^3\wedge d{x}^1+E_{3}d{x}^1\wedge d{x}^2)$. Using ${ϵ}_0{\mu }_0{c}^2=1$, this equals ${\mu }_{0}cG$. The Lorentzian sign check ${\star }_{\eta }^2=-\mathrm{id}$ on 2-forms is [F4], so the closure has the stated signature convention.

From [A1] and [A2], $\mathrm{div}D={ϵ}_0\mathrm{div}E+\mathrm{div}P=\rho +\mathrm{div}P=\rho -{\rho }_b={\rho }_f$. Also, $\mathrm{curl}H={\mu }_0^{-1}\mathrm{curl}B-\mathrm{curl}M=J+({\mu }_0{c}^2{)}^{-1}{\mathrm{\partial }}_{t}E-\mathrm{curl}M=J+{ϵ}_0{\mathrm{\partial }}_{t}E-\mathrm{curl}M$. Since $J_f=J-{\mathrm{\partial }}_{t}P-\mathrm{curl}M$ and ${\mathrm{\partial }}_{t}D={ϵ}_0{\mathrm{\partial }}_{t}E+{\mathrm{\partial }}_{t}P$, this is $\mathrm{curl}H=J_f+{\mathrm{\partial }}_{t}D$. The two remaining component equations, $\mathrm{div}B=0$ and $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$, are unchanged from [A1]. Thus the component equations hold, and step 1.1 gives $dF=0$ and $dG=\mathcal{J}$.

Under the additional $C^2$ hypothesis, $\mathcal{J}=dG$ is $C^1$ and $d\mathcal{J}$ is defined by [F1]. In repeated-index notation, $d(dG)=\frac{1}{2}{\mathrm{\partial }}_{\kappa }{\mathrm{\partial }}_{\lambda }{G}_{\mu \nu }d{x}^{\kappa }\wedge d{x}^{\lambda }\wedge d{x}^{\mu }\wedge d{x}^{\nu }$. Terms with $\kappa =\lambda$ vanish by alternation; for each $\kappa \mathrm{\ne }\lambda$, exchanging the derivative indices leaves the coefficient unchanged by [F6] and reverses the wedge sign. Thus all terms cancel and $d\mathcal{J}=d(dG)=0$. From the definition $\mathcal{J}={\rho }_{f}vo{l}_3-dt\wedge {\mathcal{J}}_{\mathrm{s}\mathrm{p}}$ in [F1], its coefficient along $dt\wedge vo{l}_3$ is ${\mathrm{\partial }}_t{\rho }_f+\mathrm{div}{J}_f$; setting this coefficient to zero gives the stated continuity equation.

Take $t_0<t_1$ in $I$ and let $e_0,e_1,e_2$ be the ordered vertices of ${\mathrm{\Delta }}^2$. The product prism has the oriented triangulation $P={\tau }_0-{\tau }_1+{\tau }_2$, with ${\tau }_0=[(t_0,e_0),(t_1,e_0),(t_1,e_1),(t_1,e_2)]$, ${\tau }_1=[(t_0,e_0),(t_0,e_1),(t_1,e_1),(t_1,e_2)]$, and ${\tau }_2=[(t_0,e_0),(t_0,e_1),(t_0,e_2),(t_1,e_2)]$. In coordinates $(t,u^1,u^2)$, their ordered vertex determinants are $t_1-t_0,-(t_1-t_0),t_1-t_0$, so the coefficients orient all three simplices consistently. Applying the signed-face formula [F7] cancels their internal faces and gives $\mathrm{\partial }P={\iota }_{t_1\mathrm{\#}}{\mathrm{\Delta }}^2-{\iota }_{t_0\mathrm{\#}}{\mathrm{\Delta }}^2-P(\mathrm{\partial }{\mathrm{\Delta }}^2)$, where ${\iota }_t(u)=(t,u)$ and $P(\mathrm{\partial }{\mathrm{\Delta }}^2)=P([e_1,e_2])-P([e_0,e_2])+P([e_0,e_1])$, with $P([e_a,e_b])=[(t_0,e_a),(t_1,e_a),(t_1,e_b)]-[(t_0,e_a),(t_0,e_b),(t_1,e_b)]$. For each singular simplex $\sigma$ in $z$, compose these parameter maps with $q_{\sigma }(t,u)=(t,\sigma (u))$. The top and bottom chains become the time-slice copies of $z$; after summing with the real coefficients of $z$, side terms cancel because $\mathrm{\partial }z=0$. Each composite $q=q_{\sigma }\circ {\tau }_i$ is a smooth singular 3-simplex into $U$: the affine parameter maps and the smooth-simplex extensions of $\sigma$ give an extension to a neighbourhood, which may be restricted so its image lies in the open set $U$. Hence $q^{\ast }F$ is a $C^1$ two-form on a neighbourhood of its standard parameter simplex. Write $F=\frac{1}{2}{F}_{\mu \nu }(y)d{y}^{\mu }\wedge d{y}^{\nu }$ in laboratory coordinates. By [F10], the pullback coefficients are ${\eta }_{ij}=F_{\mu \nu }(q){\mathrm{\partial }}_i{q}^{\mu }{\mathrm{\partial }}_j{q}^{\nu }$, so $(d\eta {)}_{kij}={\mathrm{\partial }}_k{\eta }_{ij}+{\mathrm{\partial }}_i{\eta }_{jk}+{\mathrm{\partial }}_j{\eta }_{ki}$. By [F9], the terms differentiating $F_{\mu \nu }$ are $({\mathrm{\partial }}_{\lambda }{F}_{\mu \nu })(q)({\mathrm{\partial }}_k{q}^{\lambda }{\mathrm{\partial }}_i{q}^{\mu }{\mathrm{\partial }}_j{q}^{\nu }+{\mathrm{\partial }}_i{q}^{\lambda }{\mathrm{\partial }}_j{q}^{\mu }{\mathrm{\partial }}_k{q}^{\nu }+{\mathrm{\partial }}_j{q}^{\lambda }{\mathrm{\partial }}_k{q}^{\mu }{\mathrm{\partial }}_i{q}^{\nu })$, which is exactly $(dF{)}_{\lambda \mu \nu }(q){\mathrm{\partial }}_k{q}^{\lambda }{\mathrm{\partial }}_i{q}^{\mu }{\mathrm{\partial }}_j{q}^{\nu }$. The remaining terms are $F_{\mu \nu }(q)({\mathrm{\partial }}_{ki}{q}^{\mu }{\mathrm{\partial }}_j{q}^{\nu }+{\mathrm{\partial }}_i{q}^{\mu }{\mathrm{\partial }}_{kj}{q}^{\nu }+{\mathrm{\partial }}_{ij}{q}^{\mu }{\mathrm{\partial }}_k{q}^{\nu }+{\mathrm{\partial }}_j{q}^{\mu }{\mathrm{\partial }}_{ik}{q}^{\nu }+{\mathrm{\partial }}_{jk}{q}^{\mu }{\mathrm{\partial }}_i{q}^{\nu }+{\mathrm{\partial }}_k{q}^{\mu }{\mathrm{\partial }}_{ji}{q}^{\nu })$. The first and fourth cancel after interchanging $\mu ,\nu$ and using ${\mathrm{\partial }}_{ki}q={\mathrm{\partial }}_{ik}q$; the second and fifth cancel by ${\mathrm{\partial }}_{kj}q={\mathrm{\partial }}_{jk}q$; the third and sixth cancel by ${\mathrm{\partial }}_{ij}q={\mathrm{\partial }}_{ji}q$. Each cancellation uses $F_{\mu \nu }=-F_{\nu \mu }$ and [F6]. Thus $d(q^{\ast }F)=q^{\ast }(dF)=0$ by step 2.1. The $C^1$ Stokes identity [F8] applies to each pullback on the standard parameter 3-simplex, so the boundary integrals sum to zero. The top and bottom terms are ${\mathrm{\Phi }}_z(t_1)$ and $-{\mathrm{\Phi }}_z(t_0)$ because the time-slice pullback of $F=\mathcal{B}-dt\wedge \mathcal{E}$ is $\mathcal{B}$ by [F1]. Finite-chain linearity in [F1] gives ${\mathrm{\Phi }}_z(t_1)-{\mathrm{\Phi }}_z(t_0)=0$. Hence the cycle flux is constant.

By linearity of $d$ and [F1], $d{F}_{\theta }=\cos (\theta )dF+\sin (\theta )d{\star }_{\eta }F=0$. By [F2] and [F3], ${\star }_{\eta }{F}_{\theta }=\cos (\theta ){\star }_{\eta }F-\sin (\theta )F$, so $d{\star }_{\eta }{F}_{\theta }=\cos (\theta )d{\star }_{\eta }F-\sin (\theta )dF=0$. Thus the rotated form solves both source-free vacuum equations.

In a vacuum region with nonzero free-current form $\mathcal{J}$, [F1] gives $d{\star }_{\eta }F={\mu }_{0}c\mathcal{J}$. Therefore $d{F}_{\theta }=\sin (\theta ){\mu }_{0}c\mathcal{J}$, which is nonzero whenever $\sin (\theta )\mathcal{J}$ is nonzero. It then fails the adopted equation $d{F}_{\theta }=0$; restoring a duality symmetry would require changing the source sector, which this framework does not adopt. The theorem makes no claim that any material closure or boundary condition is preserved.