Classical finite-regularity charged-scalar system and relative action

In a smooth chart, inversion of a nondegenerate $C^3$ matrix is $C^3$; the Christoffel formula therefore gives a $C^2$ connection and the curvature formula a $C^1$ tensor. Smooth coordinate changes preserve these regularities. The ordinary chain and product rules give $F,P\in C^1$, the stated continuous Euler–Lagrange expressions, and $C^1$ current and stress. Symmetry of $g^{-1}$ makes $g^{ab}\overline{P_a}{P}_b$ real and $\mathrm{Re}(\overline{P_a}{P}_b)$ symmetric; the Maxwell quadratic is also symmetric. The coordinate transformation laws follow by the chain rule, so all displayed equations compare global tensors. Commuting the continuous second partial derivatives of $A$ gives $dF=0$.

Let $K=\mathrm{supp}\omega$. Apply [F2] and multiply $\omega$ by its finitely many cutoffs. Each continuous coefficient has compact support strictly inside its chart, so zero extension is continuous and has a bounding rectangle; [F4] defines its finite integral. On a chart overlap the top-form coefficient transforms by the determinant of the smooth coordinate change. Its orientation sign converts that determinant to the absolute determinant, so [F3] gives the same signed integral in either chart. For two finite localizations $({\chi }_i)$ and $({\tau }_j)$, insert ${\chi }_i\omega ={\sum }_j{\chi }_i{\tau }_j\omega$ and the corresponding identity with $i,j$ exchanged. Every product has compact support in the chart intersection. Finite linearity from [F5] and the overlap identity make both sums equal to the same double sum. Thus ${\mathcal{I}}_M$ is well defined, linear and local. Empty support gives the empty sum zero.

Each action density has continuous coefficients. Configurations agreeing off a compact set have density difference supported in that set, including the curvature term, because their derivatives agree on its open complement. Step 1.2 therefore defines every stated relative action. For three mutually compactly differing configurations the density differences add on the union of their compact supports, so their relative actions add. On smooth fields the construction is precisely the same finite chart integral and the formulas in [F1]; hence it agrees with the original smooth relative action.

On the union $K$ of the three direction supports, nondegeneracy and the chosen time cone persist under a uniformly small perturbation: use finitely many chart neighborhoods of the compact set and take the minimum of their positive parameter bounds. The attained set $\mathrm{\mid }\phi {\mathrm{\mid }}^2(K)$ is a compact subset of the open set $\mathcal{U}$, so uniform convergence of $\mathrm{\mid }\phi +\epsilon \eta {\mathrm{\mid }}^2$ keeps it inside $\mathcal{U}$ for small $\epsilon$; outside $K$ nothing changes. The resulting metric is $C^3$ and the matter fields are $C^2$, with $\delta g_{ab}=-g_{ac}{g}_{bd}{q}^{cd}$. The action densities and their parameter derivatives are continuous on compact parameter-chart rectangles. Consequently these directions are actual variations of the declared class, not only formal symbols.

A real $C^3$ function $\chi$ gives $A^{\mathrm{\prime }}=A+d\chi \in C^2$ and ${\phi }^{\mathrm{\prime }}=e^{i\lambda \chi }\phi \in C^2$. The product rule gives $P^{\mathrm{\prime }}=e^{i\lambda \chi }P$ and $F^{\mathrm{\prime }}=F$, so the action density is unchanged. Pullback by a smooth orientation-preserving diffeomorphism also preserves the classical class and the density formulas. For compactly supported transformations, step 1.2 and [F3] give invariance of the relative action. These are mathematical invariances, not a physical adoption or a solution-existence theorem.