Declared averaging kernel, macroscopic scale assumptions and averaged fields

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.