Material response and stationary interfaces

Adopt as a modelling postulate of this framework the following coarse-graining convention for matter. Let the laboratory region $U\cong I\times \mathrm{\Omega }$, its frame basis $(e_1,e_2,e_3)$, its units, its fields and its sources be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values, The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis of the smoothing theorem cited in clause (b).

(a) [Averaging length, kernel and footprint] Fix $\mathrm{\ell }>0$ and nonnegative $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ with $\int \eta =1$, in the Euclidean test-function convention of Complex Lp classes and Euclidean test-function conventions. Put ${\eta }_{\mathrm{\ell }}(x)={\mathrm{\ell }}^{-3}\eta (x\mathrm{/}\mathrm{\ell })$, $K=\mathrm{supp}\eta$ and choose a declared finite bound $R_{\eta }>0$ with $K\subseteq {\bar{B}}_{R_{\eta }}(0)$. The kernel and length are fixed data; the imported mass-one mollifier definition The mollifier family generated by a unit-mass smooth bump does not require $K\subseteq {\bar{B}}_1(0)$. At $x$ the exact averaging footprint is $x-\mathrm{\ell }K$. Define the open interior averaging domain

${\mathrm{\Omega }}_{\mathrm{\ell },\eta }:=\{x\in \mathrm{\Omega }:x-\mathrm{\ell }K\subseteq \mathrm{\Omega }\}\mathrm{.}$

The sufficient condition $\mathrm{dist}(x,{\mathbb{R}}^3\setminus \mathrm{\Omega })>\mathrm{\ell }{R}_{\eta }$ implies membership. If the additional unit-support condition $K\subseteq {\bar{B}}_1(0)$ is declared, the original sufficient region $\mathrm{dist}(x,{\mathbb{R}}^3\setminus \mathrm{\Omega })>\mathrm{\ell }$ and the ball $B_{\mathrm{\ell }}(x)$ are valid special cases; unit support is not inferred from mass one.

(b) [Averaged fields and input regularity] For an open spatial domain $V$, locally integrable means measurable and absolutely integrable on every compact subset of $V$, componentwise for maps to ${\mathbb{R}}^m$, $m=1$ or $3$ (A locally integrable function on ${\mathbb{R}}^n$). First suppose $F:\mathrm{\Omega }\times I\to {\mathbb{R}}^m$ has a locally integrable spatial slice $F(\cdot ,t)$ for every $t$ under consideration. No integrability across the boundary of $\mathrm{\Omega }$ is required for the interior branch. Define, for $x\in {\mathrm{\Omega }}_{\mathrm{\ell },\eta }$,

$F_{\mathrm{\ell }}(x,t)={\int }_{\mathrm{\Omega }}{\eta }_{\mathrm{\ell }}(x-y)F(y,t)dy\mathrm{.}$

The compact footprint lies in $\mathrm{\Omega }$, so this integral exists absolutely, and for each such $t$ it is smooth in $x$ on ${\mathrm{\Omega }}_{\mathrm{\ell },\eta }$ with

${\mathrm{\partial }}_x^{\alpha }{F}_{\mathrm{\ell }}(x,t)={\int }_{\mathrm{\Omega }}({\mathrm{\partial }}^{\alpha }{\eta }_{\mathrm{\ell }})(x-y)F(y,t)dy$

for every multi-index; these are applications of the pinned mollifier smoothing supplier Convolution with a mollifier is smooth, and derivatives pass under the integral sign after localization: near a fixed $x_0\in {\mathrm{\Omega }}_{\mathrm{\ell },\eta }$, compactness of $x_0-\mathrm{\ell }K\subset \mathrm{\Omega }$ gives a small closed ball $Q$ of averaging centres whose footprints have compact union inside $\mathrm{\Omega }$. Choose $\zeta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ equal to one near that union (Test function cutoffs and euclidean localization); the zero extension of $\zeta F(\cdot ,t)$ is globally locally integrable and its mollification agrees with the displayed average on the interior of $Q$. This also proves that ${\mathrm{\Omega }}_{\mathrm{\ell },\eta }$ is open, without assuming that the zero extension of $F$ itself is locally integrable. For each admissible fixed $t$ the spatial convolution is measurable in $x$ and independent of a.e.-equal spatial representatives (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree); no joint $(x,t)$ measurability follows from the every-slice hypothesis alone. Joint $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega }\times I)$ input alone instead gives these slice statements for almost every $t$, by local Fubini on compact spatial/time exhaustions (Fubini's theorem for L^1 functions on a sigma-finite product, A countable union of measure-zero sets has measure zero, by countable choice, Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line); it gives no every-time slice claim. A claim for every $t$ must explicitly supply locally integrable slices for every $t$ (for example jointly continuous fields supply that condition). A spacetime a.e. class alone determines the resulting averages only for almost every $t$; changing an exceptional time slice may change its pointwise-time assignment. Spatial smoothing asserts no time differentiability or joint smoothness without separately stated time hypotheses.

[Compatible extension branch] If a declared extension $\widehat{F}(\cdot ,t)$ agrees with $F$ a.e. on $\mathrm{\Omega }$ and is locally integrable on an open larger spatial domain $V$, use the same formula there for all $x$ whose compact footprint lies in $V$. If $V={\mathbb{R}}^3$ and every declared slice is globally locally integrable, the average is defined and spatially smooth on all ${\mathbb{R}}^3\times I$. If only joint local integrability of the extension is supplied, the global spatial conclusion is for almost every $t$ unless the every-slice hypothesis is added. The zero extension is an allowed choice only when it meets this local-integrability condition: equivalently, at each declared $t$, ${\int }_{Q\cap \mathrm{\Omega }}\mathrm{\mid }F(y,t)\mathrm{\mid }dy<\mathrm{\infty }$ for every compact $Q\subset {\mathbb{R}}^3$. Mere local integrability inside $\mathrm{\Omega }$ does not imply that boundary condition. All these extension and per-time conditions are part of the declared configuration, with the original interior branch retaining every locally integrable spatial input.

(c) [Macroscopic reading and scale separation] The framework reads its macroscopic fields and source densities as such averages only in situations where a separation of microscopic and observation scales is stated for the problem at hand; the averaging length and the actual kernel footprint (or its radius bound $\mathrm{\ell }{R}_{\eta }$) are part of that statement, and no canonical value of $\mathrm{\ell }$ is asserted. This is a formulation choice for the macroscopic description, not a consequence of the vacuum equations.

(d) [Boundary handling] For requested averages at $x\in \mathrm{\Omega }$, boundary handling concerns the locations where $x-\mathrm{\ell }K$ is not contained in the original spatial domain. The region $\{x\in \mathrm{\Omega }:\mathrm{dist}(x,\mathrm{\partial }\mathrm{\Omega })\le \mathrm{\ell }{R}_{\eta }\}$ is a sufficient enclosing boundary layer, with $\mathrm{\ell }$ in place of $\mathrm{\ell }{R}_{\eta }$ only in the declared unit-support special case. Every requested average at an exterior location $x\mathrm{\notin }\mathrm{\Omega }$ must separately satisfy the extension/domain conditions of (b); the interior boundary-layer statement makes no claim to enclose exterior locations. In the interior layer a zero-extension average is used only under the compatible-extension hypothesis of (b). Otherwise one must declare another locally integrable extension, a boundary-layer model with its own well-definedness hypotheses, or restrict the averaging domain to the exact interior footprint domain. No boundary model is supplied by local integrability alone. An extension does not automatically extend a microscopic field equation across the original boundary; any averaging of such equations must either declare their validity on all used extended footprints or retain the resulting boundary distribution terms. These boundary qualifications do not impose a constitutive law.

(e) [No constitutive content] Coarse graining reorganises the description; it does not by itself produce a constitutive relation between averaged fields and averaged sources, and no linear, local or instantaneous response is asserted by this postulate. Such a response is the separate assumption post-em-material-closure, and the derivation of macroscopic field equations from the vacuum equations is the separate result pthm-em-macroscopic-maxwell.

This is an adopted modelling assumption, not a theorem of the framework and not an empirical law: it fixes the kernel, the averaging length and the boundary/scale conventions with which this framework reads macroscopic matter. The mathematical content used in clauses (a)-(b) is discharged by the named published suppliers under the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ convention; the averaging, scale-separation and boundary conventions themselves are not deduced from the vacuum equations. Clauses (c)-(e) record scope: a scale separation must be declared for each problem, the footprint (not merely a radius bound) enters the boundary layer, and the postulate supplies neither a constitutive law nor a field equation. The item pthm-em-macroscopic-maxwell uses this convention to formulate macroscopic fields and sources, and post-em-material-closure adds the material response; neither is proved or assumed here.

Let the laboratory region $U\cong I\times \mathrm{\Omega }$, its frame basis $(e_1,e_2,e_3)$, its units, its fields $E,B$ and its source densities $\rho ,J$ be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values, The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and let the averaging convention be that of Declared averaging kernel, macroscopic scale assumptions and averaged fields, in whose sense the objects below are macroscopic. Any convolution used to instantiate that reading is taken on its declared exact interior-footprint domain, or under a declared locally integrable compatible extension covering every used footprint. Its every-time use requires every-time locally integrable spatial slices; joint spacetime local integrability alone supplies only the almost-every-time branch. The joint $C^2$ regularity specified below is a separate model hypothesis, not a consequence of spatial smoothing.

(a) [Dipole moments, polarization and magnetization] Fix the following moment convention in the frame basis. For a supplied closed rectangular box $K\subset {\mathbb{R}}^3$, a reference position $r_0\in {\mathbb{R}}^3$, a continuous charge density $f:{\mathbb{R}}^3\to \mathbb{R}$ supported in the interior of $K$, and a stationary current density $g:{\mathbb{R}}^3\to {\mathbb{R}}^3$ of class $C^1$ supported in the interior of $K$ with $\mathrm{div}g=0$, define the electric and magnetic dipole moments about $r_0$ by

$p[f;r_0]:={\int }_K(y-r_0)f(y)dy,m[g;r_0]:=\frac{1}{2}{\int }_K(y-r_0)\times g(y)dy\mathrm{.}$

Here $f$ carries $\mathrm{C}{\mathrm{m}}^{-3}$, $g$ carries $\mathrm{A}{\mathrm{m}}^{-2}$, $p$ carries $\mathrm{C}\mathrm{m}$ and $m$ carries $\mathrm{A}{\mathrm{m}}^2$; the cross product is that of The cross product in ${\mathbb{R}}^3$. Integrals are componentwise Riemann integrals of The Riemann integral of a bounded function over a bounded Jordan measurable set, finite by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set. These specify the normalization of source moments, with $r_0$ retained as part of the data; they impose no microscopic representation on the material below.

The material model specifies two maps $P,M:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of class $C^2$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. To say that they are electric and magnetic dipole-moment densities means that, for every compact Jordan-measurable volume $V\subset \mathrm{\Omega }$ and $t\in I$, the model assigns the material moment contents

$\mathcal{P}(V,t):={\int }_{V}P(y,t)dy,\mathcal{M}(V,t):={\int }_{V}M(y,t)dy,$

in $\mathrm{C}\mathrm{m}$ and $\mathrm{A}{\mathrm{m}}^2$, respectively, using the same finite componentwise integrals. Thus $P$ carries $\mathrm{C}{\mathrm{m}}^{-2}$ and $M$ carries $\mathrm{A}{\mathrm{m}}^{-1}$. These contents record the model's assigned material moments; they are not defined by applying the source-moment formulas to the total or bound sources restricted to $V$. Matching them to microscopic charge/current moments requires an additional material identification, not supplied here. The maps are data of the chosen material model, and no relation between $(P,M)$ and $E,B$ is asserted here.

(b) [Bound sources] Write $\mathrm{div}$ and $\mathrm{curl}$ for the divergence and curl of Divergence and curl of a $C^1$ vector field taken at fixed $t$, and ${\mathrm{\partial }}_t$ for the partial derivative of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ at fixed position, and define the bound charge density and bound current density by

${\rho }_b:=-\mathrm{div}P,J_b:={\mathrm{\partial }}_{t}P+\mathrm{curl}M\mathrm{.}$

Since $P,M$ are jointly $C^2$, ${\rho }_b$ and $J_b$ are continuous, indeed $C^1$, by the componentwise sum and product rules of Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$; ${\rho }_b$ carries $\mathrm{C}{\mathrm{m}}^{-3}$ and $J_b$ carries $\mathrm{A}{\mathrm{m}}^{-2}$.

(c) [Free sources] Define ${\rho }_f:=\rho -{\rho }_b$ and $J_f:=J-J_b$, in the same units as $\rho$ and $J$; these are the free charge and free current densities of the model. The decomposition $\rho ={\rho }_f+{\rho }_b$, $J=J_f+J_b$ is part of the chosen material model: the same total pair $(\rho ,J)$ with a different pair $(P,M)$ is a different model.

(d) [Displacement and magnetic intensity] Define the electric displacement and the magnetic intensity by

$D:={\epsilon }_{0}E+P,H:={\mu }_0^{-1}B-M,$

where ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and ${\mu }_0,c>0$ are the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; $D$ carries $\mathrm{C}{\mathrm{m}}^{-2}$ and $H$ carries $\mathrm{A}{\mathrm{m}}^{-1}$.

(e) [Status] This item fixes objects and the bound/free convention only. No field equation (the item pthm-em-macroscopic-maxwell), no constitutive relation (the item post-em-material-closure), no conservation of the bound pair (the item pthm-em-macroscopic-maxwell, clause (a)) and no identification of $P,M$ with microscopic dipole sums is asserted here.

Assume the smooth branch of the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields: fields $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ jointly of class $C^1$ with the additional derivatives required by each equation existing and continuous, and continuous source densities $\rho ,J$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current. Let $P,M$ and the derived objects ${\rho }_b,J_b,{\rho }_f,J_f,D,H$ be those of Polarization, magnetization, bound and free sources, and the fields D and H, with $P,M$ of class $C^2$, and use the operators of Divergence and curl of a $C^1$ vector field and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. Then:

(a) [Bound-source conservation] ${\mathrm{\partial }}_t{\rho }_b+\mathrm{div}{J}_b=0$ on $\mathrm{\Omega }\times I$. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the regular-distribution and classical-compatibility interfaces below. Identify continuous sources with their regular distributions using Regular distribution from a locally integrable function and Locally integrable functions embed in distributions. Then

${\mathrm{\partial }}_t{\rho }_f+\mathrm{div}{J}_f={\mathrm{\partial }}_t\rho +\mathrm{div}J$

as distributions on $U$, with derivatives as in Distributional derivative: the free pair is distributionally conserved exactly when the total pair is. Under the adopted four vacuum equations both sides are zero distributions, by Distributional differentiation is continuous and commutes. These statements do not require classical derivatives of $\rho ,J$. Where the partial derivatives ${\mathrm{\partial }}_t\rho$ and ${\mathrm{\partial }}_{x_i}{J}_i$ exist, the same identity holds pointwise by differentiating the free/bound decomposition; in particular, if these derivatives are continuous on $U$, both pairs satisfy the classical pointwise continuity equation. The bound-source conservation equation remains classical and uses only $P,M\in C^2$.

(b) [Macroscopic equations] On $\mathrm{\Omega }\times I$,

$\mathrm{div}D={\rho }_f,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}H=J_f+{\mathrm{\partial }}_{t}D\mathrm{.}$

(c) [Interpretation and scope] Equations (b) are an exact rewriting of the adopted vacuum equations under the decomposition of Polarization, magnetization, bound and free sources, and the fields D and H: the two equations written in $E$ and $B$ are unchanged, and the $D$ and $H$ equations hold for every $C^2$ pair $(P,M)$, independently of any constitutive law. The classical bound identity of (a) uses the exchange of ${\mathrm{\partial }}_t$ with $\mathrm{div}$ for $C^2$ data; the free/total identity is distributional in the stated continuous-source class, with its classical interpretation only under the additional derivatives stated in (a). The distributional conservation conclusion embeds the adopted smooth solution only; no macroscopic field-equation claim is made for the weak branch of the Maxwell postulate, for singular or sheet sources or for discontinuous $(P,M)$; no constitutive relation and no empirical statement is made.

Adopt as a modelling postulate of this framework the following constitutive branches for the material described by the macroscopic fields of Polarization, magnetization, bound and free sources, and the fields D and H.

(a) [Material rest frame] The relations below are stated in a material rest frame: an inertial frame $(o;e_0,e_1,e_2,e_3)$ of the Minkowski model of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields (Inertial frames and inertial charts of the affine Minkowski model) in whose coordinates each modelled material element has fixed spatial position $x$, hence history $t\mapsto (ct,x)$ on the declared time interval. When this is the laboratory frame, the components below are laboratory components. Boosting the material changes both the fields and the rest observer; this postulate does not assert that the relations below hold with the same constants in every frame.

(b) [Elementary inertial-rest branch] In the rest frame, with constant symmetric positive-definite matrices $\epsilon ,\mu \in {\mathbb{R}}^{3\times 3}$ (the isotropic scalar cases $\epsilon =\epsilon \mathbf{1}$, $\mu =\mu \mathbf{1}$ with $\epsilon ,\mu >0$ included), the model takes

$D=\epsilon E,B=\mu H,J_c=\sigma E,$

where $\sigma \in {\mathbb{R}}^{3\times 3}$ is symmetric positive semidefinite and $J_c$ is the conduction part of the free current, $J_f=J_c+J_{\mathrm{e}\mathrm{x}\mathrm{t}}$ for a declared external free current $J_{\mathrm{e}\mathrm{x}\mathrm{t}}:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ in $\mathrm{A}{\mathrm{m}}^{-2}$. For a real matrix $T$, symmetric means $T^{\mathsf{T}}=T$, positive definite means $v^{\mathsf{T}}Tv>0$ for every nonzero $v\in {\mathbb{R}}^3$, and positive semidefinite means $v^{\mathsf{T}}Tv\ge 0$ for every $v$. Entries of $\epsilon ,\mu ,\sigma$ carry respectively $\mathrm{F}{\mathrm{m}}^{-1}$, $\mathrm{H}{\mathrm{m}}^{-1}$ and $\mathrm{S}{\mathrm{m}}^{-1}$, where $\mathrm{F}=\mathrm{C}\mathrm{/}\mathrm{V}$, $\mathrm{H}=\mathrm{T}{\mathrm{m}}^2\mathrm{/}\mathrm{A}$ and $\mathrm{S}=\mathrm{A}\mathrm{/}\mathrm{V}$; these make the three displayed relations dimensionally consistent. Local, linear and nondispersive (instantaneous) response is part of this restricted assumption. In the isotropic case an exact broadband interpretation of the constant-coefficient law additionally requires $\epsilon \mu \ge c^{-2}$, so that the characteristic speed $1\mathrm{/}\sqrt{\epsilon \mu }$ does not exceed the vacuum speed $c$; without that condition, and in the anisotropic branch, the constant-coefficient law is a declared finite-band effective phase-response model and carries no universal front-speed claim.

(c) [Causal memory branch] Alternatively, at each fixed material position $x$, declare a bounded continuous full history $E_x:\mathbb{R}\to {\mathbb{R}}^3$ in $\mathrm{V}{\mathrm{m}}^{-1}$ and its response $P_x:\mathbb{R}\to {\mathbb{R}}^3$ in $\mathrm{C}{\mathrm{m}}^{-2}$. Write $E(t)=E_x(t)$ and $P(t)=P_x(t)$ in the following formula; any field on a smaller time interval uses a declared full-history extension, not unspecified past values. The polarisation response is the causal convolution

$P(t)={\epsilon }_0{\int }_0^{\mathrm{\infty }}K(s)E(t-s)ds$

for a declared real matrix kernel $K:[0,\mathrm{\infty })\to {\mathbb{R}}^{3\times 3}$ with entries in $L^1(0,\mathrm{\infty })$ (The class $L^1(\mu )$ of integrable functions) and units ${\mathrm{s}}^{-1}$; the integral then converges absolutely because $\mathrm{\mid }{K}_{jk}(s)E_k(t-s)\mathrm{\mid }\le (\sup \mathrm{\mid }{E}_k\mathrm{\mid })\mathrm{\mid }{K}_{jk}(s)\mathrm{\mid }$. The convolution uses only the past history; its continuity and causality are proved in pthm-em-causal-dispersive-response, while passivity is an additional model condition. Spatial or temporal differentiability of the resulting fields requires separate hypotheses when used. A finitely-subtracted analytic response may instead be declared by a dimensionless scalar transfer function $\chi$ that is holomorphic on the open upper half-plane, continuous on its closure, of class $C^1$ on the real boundary away from a real subtraction point ${\omega }_0$, holomorphic on a neighbourhood of ${\omega }_0$ in $\mathbb{C}$, and satisfies $\mathrm{\mid }\chi (z)\mathrm{\mid }\le C(1+\mathrm{\mid }z{\mathrm{\mid }}^{n-1})$ on $\{\mathrm{Im}z\ge 0\}$ outside a compact set for some integer $n\ge 1$ and constant $C>0$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions). Here $z,{\omega }_0$ are numerical angular frequencies in ${\mathrm{s}}^{-1}$ and the growth bound uses those SI numerical coordinates. Require also $\chi (-\omega )=\overline{\chi (\omega )}$ for every real $\omega$, hence $\chi (0)\in \mathbb{R}$, so that the response is defined on real harmonic histories independently of their representation. For $\omega \mathrm{\ne }0$ the amplitude at that frequency is unique and reversing frequency conjugates it; the symmetry gives the same real output. At $\omega =0$ only $\mathrm{Re}e$ determines the input, and real $\chi (0)$ makes the output depend only on that input. For $e\in {\mathbb{C}}^3$, real $\omega$, and componentwise real part, declaring this transfer means adopting the harmonic relation $E(t)=\mathrm{Re}(e{e}^{-i\omega t})\mapsto P(t)=\mathrm{Re}({\epsilon }_0\chi (\omega )e{e}^{-i\omega t})$; upper-half-plane holomorphy alone does not construct a time-domain response for arbitrary histories.

(d) [Status] These are adopted model assumptions, one branch at a time; the framework does not derive them from the vacuum equations or from coarse graining, and no microscopic mechanism, material data or empirical validity is claimed. Ferroelectric and ferromagnetic response, hysteresis, nonlinear response and general nonlocal response lie outside the branches declared here.

The equations of The macroscopic Maxwell equations in terms of D, H and the free sources are formulated for arbitrary declared $C^2$ maps $P,M$; the branches above are the additional modelling assumptions by which a given medium is closed. Clause (a) fixes the frame in which the constant coefficients and kernels are declared, clause (b) is a restricted instantaneous branch with explicit positivity and the separate isotropic broadband condition, and clause (c) retains a causal memory response whose analysis is the separate result pthm-em-causal-dispersive-response. No branch is deduced from the vacuum equations, from the averaging convention of the item post-em-coarse-graining or from material data, and none is asserted to hold for any actual substance; the exclusions in (d) delimit the model.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis under which Distributional divergence and curl across a stationary interface and the surface conventions of Distributional Maxwell equations with point and stationary-sheet sources are stated, and let the laboratory region $U\cong I\times \mathrm{\Omega }$ and its units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values.

Let $S\subset \mathrm{\Omega }$ be a fixed compact embedded $C^2$ hypersurface with a chosen unit normal $n$ pointing from the side ${\mathrm{\Omega }}_{-}$ to the side ${\mathrm{\Omega }}_{+}$ (Surface distributions and interfacial jumps), let ${\sigma }_f:S\times I\to \mathbb{R}$ and $K_f:S\times I\to {\mathbb{R}}^3$ be continuous with $K_f(x,t)\cdot n(x)=0$ for all $(x,t)$ (a free surface charge density in $\mathrm{C}{\mathrm{m}}^{-2}$ and a tangential free surface current density in $\mathrm{A}{\mathrm{m}}^{-1}$, both stationary in $t$), and let ${\rho }_v:(\mathrm{\Omega }\setminus S)\times I\to \mathbb{R}$ and $J_v:(\mathrm{\Omega }\setminus S)\times I\to {\mathbb{R}}^3$ be continuous and locally integrable on $\mathrm{\Omega }\times I$. Let $E,B,D,H:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be locally integrable on $\mathrm{\Omega }\times I$ and of class $C^1$ on $(\mathrm{\Omega }\setminus S)\times I$, with the joint one-sided $C^1$ extension property: around every $(x_0,t_0)\in S\times I$, each side of each field is the restriction of a $C^1$ function on an open spacetime neighbourhood of the corresponding closed side of a smaller stationary graph cylinder, so that the fields and their first space and time derivatives extend continuously to that side, locally uniformly on compact cylinders. Write $[F]=F_{+}-F_{-}$ for the jump of the one-sided traces.

[Thin-interface hypothesis] Suppose that $E,B,D,H$ satisfy the four macroscopic equations of The macroscopic Maxwell equations in terms of D, H and the free sources in the distributional sense on $U$ with the source pair ${\rho }_f={\rho }_v+{\sigma }_f{\delta }_S$, $J_f=J_v+K_f{\delta }_S$, where the stationary-sheet pairings are those of Distributional Maxwell equations with point and stationary-sheet sources(c) and the equations are $\mathrm{div}D={\rho }_f$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$, $\mathrm{curl}H=J_f+{\mathrm{\partial }}_{t}D$ as identities in the sense of Distribution, the fields being identified with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and $\psi$ ranging over $C_c^{\mathrm{\infty }}(U)$ (Test function space d of an open set, Distributional derivative). This thin-interface solution hypothesis is a modelling assumption of the statement: it is the natural weakening of the smooth macroscopic equations to fields with a common stationary discontinuity, and it is not deduced from the smooth-region equations alone.

Then, pointwise on $S\times I$,

$n\cdot [D]={\sigma }_f,n\cdot [B]=0,n\times [E]=0,n\times [H]=K_f,$

with the Euclidean inner and cross products of the one-sided traces (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The cross product in ${\mathbb{R}}^3$); equivalently ${\sigma }_f=n\cdot [D]$ and $K_f=n\times [H]$ recover the free sheet densities from the field jumps.

[Weak-trace branch] If instead of the joint extension hypothesis the fields have the original one-sided spatial $C^1$ extensions at each time and, in addition, the piecewise first spatial and temporal derivatives are in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(U)$ and the one-sided field traces are measurable and locally integrable on $S\times I$ with respect to $dSdt$, then the same four identities hold as identities of sheet distributions and for $dSdt$-almost every $(x,t)$; since the traces are spatially continuous at each fixed time, they hold for every $x\in S$ for almost every $t$, and at every $(x,t)$ whenever the jump densities are additionally jointly continuous. Merely having one-sided spatial $C^1$ traces separately at every time does not justify the last upgrade.

[Scope] In the joint one-sided spacetime $C^1$ extension branch, stationarity and the joint extensions make ${\mathrm{\partial }}_{t}D$ and ${\mathrm{\partial }}_{t}B$ locally bounded near $S$ and hence locally integrable, contributing no delta terms. In the weak-trace branch the explicitly assumed locally integrable temporal derivatives provide this conclusion instead. No magnetic charge and no magnetic surface current are present in the sources; no dipolar sheet layer (a concentrated dipole distribution in $P$ or $M$, which would make the fields non-integrable) is included. No existence theorem for fields with singular sources is asserted, no constitutive law is used, and no empirical claim is made.

Assume the memory branch of the material-closure postulate Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in a material rest frame, and assume the hypotheses of the mathematical suppliers Causal convolution response, transfer function and dispersion relations and Dispersion relations for a finitely subtracted analytic response: Countable Choice for the smooth comparison route and, in the integrable branches, Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $E:\mathbb{R}\to {\mathbb{R}}^3$ be the electric-field history in the material rest frame, read componentwise in its spatial basis (The electric and magnetic fields as fixed-time spatial vector fields), and let $P$ be the polarisation response declared by the model.