The electromagnetic bridge: what a charged-particle coupling must add to the framework, what the force branch then yields, and what is not imported

The particle side of the framework is complete on its own terms: labels with $C^2$ worldlines parametrized by frame time and fixed positive masses (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses); a force branch in which the force map is declared model input with standing continuity and state-local-Lipschitz hypotheses (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance); motions, unique maximal motions and the continuation criterion (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder, Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain); the mechanical quantities and their balance theorems, conditional on the declared interaction structure (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy, Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries); and the operational observable model through which predictions at observation times are read (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error). The framework contains no charge, no field and no coupling law.

A charged-particle coupling is a new physical assumption, not a consequence of the framework, and it must supply at least:

1. Charge attributes. To each label an attribute with a declared dimension and unit. The dimension group of the framework is generated by length, duration and mass alone (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework clause (a)); the dimension and unit of charge, and of the components of any field used, must therefore be declared by the coupling and cannot be inferred from this framework.
2. Field maps. For an electric/magnetic vector-field presentation, specify maps $\mathbf{E},\mathbf{B}:{\mathrm{\Omega }}_{\mathrm{E}\mathrm{M}}\times I_0\to {\mathbb{R}}^3$ in the declared inertial frame, with ${\mathrm{\Omega }}_{\mathrm{E}\mathrm{M}}\subseteq {\mathbb{R}}^3$ open and $x_i\in {\mathrm{\Omega }}_{\mathrm{E}\mathrm{M}}$ for every configuration and label on the coupling domain. At fixed time these are spatial vector fields; their component dimensions, units and regularity are additional declared data. The fields may be prescribed external maps without a model of their sources or any boundary or radiation conditions. If instead they are determined by field equations, the independent field model must supply its source data and whatever initial, boundary or radiation conditions its formulation requires. In either case, the total particle force must satisfy the standing regularity hypotheses below on the chosen domain; singular source loci must be excluded where those hypotheses fail, and fields requiring a distributional force need a separate solution notion. A presentation by potentials likewise requires their exact maps and the rule constructing the fields or particle force from them.
3. A coupling law. An added force map $F^{\mathrm{E}\mathrm{M}}:I_0\times Q\times ({\mathbb{R}}^3{)}^N\to ({\mathbb{R}}^3{)}^N$ with components $F_i^{\mathrm{E}\mathrm{M}}$ of force dimension $\mu \mathrm{\ell }{\tau }^{-2}$, read in newtons, for instance a declared function of the fields, the charge and the label's velocity in the chosen frame, as in the Lorentz force of the cited electromagnetic treatment. The coupling law is adopted: the framework neither derives it nor tests it.
4. Field-side bookkeeping. If statements about the energy or momentum of the combined system are wanted, the field contributions and their balance with the particles must be declared as separate objects; this framework contains no field energy or field momentum.

The particle ODE results apply once the coupling supplies the closed time-state force map $F^{\mathrm{E}\mathrm{M}}$ of item 3 on the same domain as $F$, and $F+F^{\mathrm{E}\mathrm{M}}$ satisfies the standing hypotheses of the force branch (continuity and state-local Lipschitz, locally uniformly in time). Then Local existence and uniqueness for the Newtonian initial-value problem, with the quantitative Picard-Lindelöf cylinder and Maximal Newtonian motions and the compact-continuation criterion: finite-time breakdown requires escape from every compact subset of the time-state domain give local existence, uniqueness, maximal motions and the continuation criterion for that particle model. If the fields are sourced by evolving labels, their values along a coupled solution are not automatically prescribed time-space maps or functions of the instantaneous particle state. A closed force map must be supplied by a justified reduction; otherwise the independent field equations and particle equations require a separate coupled solution theory, to which these particle ODE theorems alone do not apply.

The balance laws recorded in Status audit and scope of the balance page: conditional conservation, open systems, frames, potential normalization and empirical boundaries retain their individual hypotheses. In particular, a coupling force can be added wholly to the external contribution of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (i) only when $F_i^{\mathrm{E}\mathrm{M}}(t,x,v)=H_i(t,x_i,v_i)$ for declared single-label maps $H_i$ on open domains containing all these arguments, with the standing regularity. Prescribed fields independent of the evolving particle labels, with a local coupling law depending only on $(t,x_i,v_i)$ and fixed charge attributes, have this form if the resulting $H_i$ satisfies that regularity. Then replace $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}$ by $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}+H_i$ in an existing external-plus-pair presentation. More generally, reuse of that presentation requires a declared identity $F_i^{\mathrm{E}\mathrm{M}}(t,x,v)=H_i(t,x_i,v_i)+{\sum }_{j\mathrm{\ne }i}{K}_{ij}(t,x_i,x_j,v_i,v_j),$ with the component domains and regularity of that clause; add $K_{ij}$ to the corresponding pair map. Antisymmetry, centrality and any potential or residual-force presentation required by a particular balance must be checked for the resulting total force. Regularity alone implies none of these restrictions. The work-energy identity for the total force retains its force-model hypotheses without requiring an external-plus-pair presentation.

No field-momentum theorem is imported. With the electromagnetic force represented by single-label external maps as above, the particle mechanical momentum alone need not be conserved; the balance laws record the exchange through the external force, and the open-system boundary of Status audit and model boundaries of the force page: open systems, frame selection, breakdown and empirical scope applies. A conservation statement for matter plus field requires the field contributions of item 4 above as additional declared data, and is not a consequence of this framework.

No field equation, no value or dimension of any charge or coupling constant, no derivation or validation of a coupling law, no prediction of any electromagnetic phenomenon and no measured electromagnetic quantity is asserted. The observable models of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error acquire an electromagnetic readout only after a coupling has been declared; until then this framework is silent about electric and magnetic effects.