Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) as carried by the declared distributional and surface source objects of Distributional Maxwell equations with point and stationary-sheet sources. In the flat classical laboratory model of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the positive constants and SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values, adopt the following additional physical assumption for models using the declared point or stationary-sheet sources: the locally integrable field pair and the volume-plus-singular source pair of Distributional Maxwell equations with point and stationary-sheet sources(a)-(d) must satisfy its solution predicate (e). Explicitly,

$\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$

are required in ${\mathcal{D}}^{\mathrm{\prime }}(U)$, meaning that all four residual pairings defined there vanish for every compact smooth test. This is a physical adoption of the classical vacuum idealisation in that source class, not a theorem about sources and not a consequence of the original locally integrable source branch of The microscopic vacuum Maxwell equations: smooth and weak branches. When the singular part vanishes, the adopted identities coincide with that earlier weak branch.

Only source and field tuples satisfying these identities are models of this postulate; the postulate does not assert existence for every prescribed source pair, and compatibility conditions remain consequences to be checked. Sources are the exact point and stationary-sheet types defined earlier, including their properness or endpoint and tangential-current qualifications. No moving sheet, line current, magnetic source, product of distributions, self-force, regularisation, matter dynamics or material constitutive law is introduced. This postulate reports no experiment or measurement and does not infer empirical certainty from its mathematical formulation.

The adoption of the vacuum field equations for idealised point and stationary sheet sources is the standard classical idealisation of the cited treatments: the point-particle delta current and the surface charge and current dictionaries are presented there as modelling assumptions of the same type. The formulation, the field and source types, the units and the distributional scope are the framework's own, as recorded in the cited definitions.