Self-force, runaways and finite-size regularizations are not supplied by this page

Every cited conditional result retains its stated hypotheses. In particular, the point-source and causal-wave comparisons below assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) through their source suppliers; this qualification supplies no additional field or self-interaction model.

Record, with no speculative content, the limits of the radiation model built on this page.

(a) [Self-energy] For nonzero charge $q\mathrm{\ne }0$, the truncated electrostatic field energy of the stationary idealized point charge diverges as the inner cutoff is removed (Divergence of the point-charge field energy and the limits of mass inference): the point source's own field carries infinite energy in this framework's idealization, and any finite-energy treatment of a point-like charge requires modifying the source or the field near the singularity, or an explicit cutoff, none of which is supplied here. The $q=0$ branch has zero field energy and no divergence.

(b) [Undefined self-interaction; nonzero and zero source branches] The framework's source model Distributional Maxwell equations with point and stationary-sheet sources pairs distributions against compact smooth tests and expressly excludes arbitrary products of distributions. An explicit nonzero singular self-field is the stationary Coulomb field of The localized Newtonian potential and the ideal point-charge Coulomb field(d): for a constant trajectory $z(t)=x_0$ and $q\mathrm{\ne }0$, $\mathrm{\mid }{E}_q(x)\mathrm{\mid }=\mathrm{\mid }q\mathrm{\mid }\mathrm{/}(4\pi {\epsilon }_0\mathrm{\mid }x-x_0{\mathrm{\mid }}^2)$ for $x\mathrm{\ne }{x}_0$, which tends to infinity as $x\to x_0$. Its point-source charge distribution is supported on that stationary worldline. For $q=0$, the point-source pairings of Distributional Maxwell equations with point and stationary-sheet sources(b) give ${\rho }_{\mathrm{p}\mathrm{t}}=J_{\mathrm{p}\mathrm{t}}=0$ for every admitted trajectory; in the stationary Coulomb source-field branch, $E_q=0$ off $x_0$ extends smoothly by zero, with $B=0$. Thus that zero-source branch has no self-field singularity and no nontrivial self-interaction. This states the source-field branch, not uniqueness among fields with an additional free field.

The moving pure point-source branch retains a precise general obstruction: if $q\mathrm{\ne }0$, $z$ is an admitted $C^1$ proper trajectory, and locally integrable fields satisfy the distributional Maxwell equations with exactly its point-source pair (Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources, Distributional Maxwell equations with point and stationary-sheet sources(b),(e)), there is no $C^1$ extension of the electric field through any interior worldline event that still satisfies those sourced equations. To see this, suppose such an extension exists near $(x_0,t_0)=(z(t_0),t_0)$ and write $g=\mathrm{div}E$. Its distributional divergence is the regular distribution of this continuous $g$ by Distributional differentiation is continuous and commutes, under the stated Countable Choice convention. On a fixed compact box inside that neighbourhood, $\mathrm{\mid }g\mathrm{\mid }\le M<\mathrm{\infty }$ by A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value. Choose a spatial bump $0\le {\varphi }_r\le 1$, equal to one on $B_{r\mathrm{/}2}(x_0)$ and supported in $B_r(x_0)$; continuity of $z$ gives a time interval $J_r$ about $t_0$, inside that fixed box's time interval, with $z(J_r)\subset B_{r\mathrm{/}2}(x_0)$. Choose a nonzero nonnegative smooth temporal bump ${\eta }_r$ supported in $J_r$, with $A_r=\int {\eta }_{r}dt>0$ (Compactly supported scaled Euclidean bumps). The compact smooth test ${\varphi }_r(x){\eta }_r(t)$ has charge pairing $q{A}_r$, whereas its pairing with $g$ has modulus at most $M(2r{)}^3{A}_r$: the spatial support lies in the side-$2r$ cube, and iterated integration of the continuous compactly supported integrand gives this bound (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, Riemann–Lebesgue comparison for distribution test integrands, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$). The sourced Gauss identity would therefore give $\mathrm{\mid }q\mathrm{\mid }\mathrm{/}{\epsilon }_0\le M(2r{)}^3$ for every sufficiently small $r>0$, an impossibility. This is a failure of a $C^1$ sourced extension, not a claim that every moving field norm blows up; the explicit norm blowup above is the stationary Coulomb case.

For nonzero singular self-fields, evaluating the field on its own point-source support or forming its product with the distributional source is not supplied by these postulates. A vanishing component of the current does not remedy the singular charge-field product. More generally the prescribed moving-source framework supplies no arbitrary distribution-product operation, regularization or self-force equation; its emitted-power results below do not introduce one. No moving-source norm blowup is inferred from the stationary example, and no equation of motion is derived in either charge branch.

(c) [What the emitted-power items do and do not give] Instantaneous-rest Larmor power and uniform low-speed comparison for a prescribed point charge and Relativistic emitted power of a prescribed point charge at a general emission event are prescribed-source statements: they compute the emitted power of a stipulated worldline at a stipulated emission event, per unit laboratory emission time, using the retarded-time conversion factor in their weighted far-zone flux definition. At an instantaneous-rest emission event that factor is one, so the power also equals the fixed-sphere flux specified in Instantaneous-rest Larmor power and uniform low-speed comparison for a prescribed point charge(e). These results determine no worldline evolution, no back-reaction and no mechanical energy-loss rate of the particle; identifying the emitted power with such a loss rate requires an additional mechanical balance assumption.

(d) [Deferred, not claimed] Extended-charge or finite-size regularizations of the self-field, the well-posedness of any self-force dynamics, runaway solutions and an electromagnetic derivation of rest mass are outside this page and are neither stated nor proved here. No experiment, measurement, statistical estimate or precision is reported or implied, and nothing in this remark is a refutation of any physical theory.