Adopted retarded solution selection: the physical branch carries no incoming free field

Adopt as a physical assumption of this framework the following solution-selection rule for the classical vacuum model on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the constants and SI units of Framework-local SI units, the vacuum constants and the exact/measured status of their values (so $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$).

(a) [Formulation] In the potential formulation of Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators with the conventions of Scalar-vector potential pairs, gauge transformation, Lorenz divergence and the wave expression, for a $C^2$ potential pair in Lorenz gauge the sourced Maxwell system of The microscopic vacuum Maxwell equations: smooth and weak branches is equivalent to the decoupled sourced wave equations $W\phi =\rho \mathrm{/}{\epsilon }_0$, $WA={\mu }_{0}J$, where $W=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$. The wave expression $W$ is invariant under $t\mapsto -t$; where explicit retarded or advanced constructions for a prescribed history exist, they give distinguished particular solutions built from the corresponding forward or backward fundamental kernels. Every other solution differs from a chosen particular solution by a source-free addition; retarded and advanced particular solutions also differ by such an addition. This describes the affine free-solution freedom, not a claim that arbitrary histories have exactly two solutions or that either construction always exists.

(b) [Adopted rule] The framework adopts the retarded branch with no free field added: the physical potential pair of the model is the retarded one, that is, the one whose support and propagation lie in the forward light cones of the source and which carries no field arriving from the past independently of the source. For the zero-incoming model the selected sourced solution has no independently prescribed homogeneous addition. If nonzero Cauchy data are instead prescribed, their compatible homogeneous addition is a separate initial-data model; it is not equivalent to setting every free addition to zero. The advanced construction and independent incoming free fields are excluded by the present retarded selection.

(c) [Status and scope] This is an adopted extra temporal condition carrying the model's arrow-of-time content. It is not a theorem, not a consequence of the time-symmetric wave or Maxwell equations (which admit advanced and free solutions as mathematical objects), and not a deduction from experiment; it asserts no existence, uniqueness, regularity, support or decay for any source class, all of which must be supplied separately by theorems of this page. It selects a solution only of the models to which it is applied; it introduces no product of distributions, no source model, no self-force, no radiation reaction, no measured signal and no statistical claim. The retarded compact-source branch to which this selection applies is developed on this page as a conditional theorem, and the point-source branch is developed later on this page; neither existence statement is part of this postulate. For the separately declared point-source branch assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as in its exact source/field interfaces, and adopt the same no-free-addition retarded selection in the weak equations of Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources, for the locally integrable field and proper point-current types of Distributional Maxwell equations with point and stationary-sheet sources. This extends the selection's model scope explicitly; neither existence nor an arbitrary convolution of noncompact distributions is postulated.

The classical treatments cited above display the retarded and advanced Green functions of the wave operator and then identify the choice of the retarded branch as additional structure imposed on the time-reversal-symmetric equations; they support the formulation of clauses (a)-(b) and the honest non-theorem status recorded in clause (c). They are not evidence for the physical correctness of the choice, and this postulate reports no experiment, measurement or statistical inference. The exact interfaces of the wave form, the gauge conventions, the smooth Maxwell system and the separately adopted weak singular-source equations are those of the cited framework items; the weak point-source extension is an explicit physical selection on the already-adopted singular-source equations, not a deduction from the smooth-only potential theorem, and a consuming theorem must construct its actual history integrals and prove weak admissibility.