Specified classical charged complex-scalar model

On the smooth Lorentzian geometry and SI convention of Einstein–Maxwell primitive quantities and SI conventions, choose a fixed smooth bundle sector and the classical charged-scalar functional $S_{\phi }$ of Classical complex-scalar connection derivative and action. A configuration of this model is a smooth complex scalar section $\phi$ and the smooth connection and metric fields in that definition. In the action branch, take this actual functional as the selected matter action $S_m:=S_{\phi }$, with the same fixed-sector compact variations. The scalar is classical; the dimensionless field normalization, the positive coefficient $K_{\phi }$, the smooth real potential $V$, and the permitted coupling $\lambda$ are exactly those of Classical complex-scalar connection derivative and action.

In its circle-bundle branch, $\lambda {\kappa }_0=k\in \mathbb{Z}$ for the selected representation; in the separate trivial real-potential branch, $\lambda$ may be any real number. No quantum rest-mass identification, material-response claim, or existence theorem for the coupled equations is part of this definition.