Selected effective classical constitutive regime

Use a smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold $(M,g,N,o)$ with signature $(-,+,+,+)$, the metric-selected connection and the independent orientation $o$, and a fixed bundle sector with smooth connection and smooth real curvature two-form $F$ satisfying $dF=0$, as in the nonlinear constitutive definition linked below. Define its invariants

$X:=-\frac{1}{4}{F}_{ab}{F}^{ab},Y:=-\frac{1}{4}{F}_{ab}({\star }_{g}F{)}^{ab},$

Choose fixed real parameters $(\alpha ,\beta )\mathrm{\ne }(0,0)$ and a dimensionless $\delta \in (0,1)$. Adopt, as a classical source-free constitutive model on field configurations whose invariant values lie in

${\mathcal{D}}_{\alpha ,\beta ,\delta }:=\{(X,Y)\in {\mathbb{R}}^2:\mathrm{\mid }\alpha X\mathrm{\mid }+\mathrm{\mid }\beta Y\mathrm{\mid }<\delta \},$

the Lagrangian density

$L_{\mathrm{e}\mathrm{f}\mathrm{f}}(X,Y):=X+\alpha X^2+\beta Y^2\mathrm{.}$

The corresponding constitutive tensor in the supplier's convention is

$H^{ab}=(1+2\alpha X)F^{ab}+2\beta Y({\star }_{g}F{)}^{ab}\mathrm{.}$

Here ${\star }_g$ is the Hodge map determined by the metric and independent orientation of the Lorentzian geometry. This postulate adopts the displayed constitutive assignment, but does not adopt or assert a solution of a nonlinear field equation.

The pair $(\alpha ,\beta )$ is fixed within a selected model but otherwise arbitrary; this item assigns no numerical values. If the components of $F$ have unit $\mathfrak{F}$, then $X,Y,L_{\mathrm{e}\mathrm{f}\mathrm{f}}$ have unit ${\mathfrak{F}}^2$ and $\alpha ,\beta$ have unit ${\mathfrak{F}}^{-2}$, making the domain condition dimensionless. This is the mathematical normalization of the supplier, not an SI action prefactor.

The domain ${\mathcal{D}}_{\alpha ,\beta ,\delta }$ is an adopted regime label, not a proved error bound. No quantum origin, QED matching, coefficient value, derivative-correction control, characteristic-cone condition, hyperbolicity, causality, or solution-existence claim is included. The formal nonlinear field equations, if separately derived under the supplier's compact-variation hypotheses, do not establish that this model describes nature.

Nonlinear electromagnetic constitutive model with a $C^2$ Lagrangian

This is a selected phenomenological classical model with no empirical premise. The arbitrary parameters and invariant-domain threshold are declared inputs. Any claim of low-energy quantum derivation, approximation error, semiclassical consistency, gravitational coupling, or observed validity requires separate hypotheses and evidence.