Restricted local equilibrium parcel with adopted material state function

For a selected fixed-composition, single-material branch, adopt a local-equilibrium description on an open state domain $D\subset (0,\mathrm{\infty })\times \mathbb{R}$ with a supplied function $e\in C^2(D,\mathbb{R})$ and $T:={\mathrm{\partial }}_{{\sigma }_p}e>0$. Here $v$ is the rest volume per selected material marker, ${\sigma }_p$ is entropy per marker, and $e$ is energy per marker, including the rest-energy contribution; their types and SI units are those in Rest fluid state, frame and geometric current definitions and Thermodynamics-GR fields, currents, reference normalization and units. Here “comoving” means that local state values are referred to the observer-congruence rest spaces of Rest fluid state, frame and geometric current definitions; no orthogonal rest-slice foliation or finite-region volume integral is presupposed. For an admissible local continuum parcel, let $N_p>0$ be its real marker amount, $V>0$ its supplied proper-volume parameter, and $S\in \mathbb{R}$ its supplied total entropy, with $(V\mathrm{/}{N}_p,S\mathrm{/}{N}_p)\in D$. The quantities $V\mathrm{/}{N}_p$ and $S\mathrm{/}{N}_p$ have the per-marker state meanings and units of Rest fluid state, frame and geometric current definitions. This model assigns energy $E=N_{p}e(V\mathrm{/}{N}_p,S\mathrm{/}{N}_p)$. Treating the real marker amount as a particle count is part of the fixed-composition continuum model, not a claim about discrete microscopic transitions or a consequence of the mathematical marker definition.

This is an adopted constitutive/local-equilibrium premise for the stated material and regime. It does not derive $e$ microscopically, assert a measured equation of state, impose a spacetime conservation equation, imply a universal gas law, or make the local parcel energy additive with the energy of an arbitrary self-gravitating region. Rest energy belongs in $e$; if a rest mass $m$ is specified, its contribution is $m{c}^2$, not $m$ in SI units. No entropy-maximization, quantum Gibbs representation, or global gravitating thermodynamic limit is assumed.