Exact entropy identity in the selected bulk branch

On a nonempty open region of the smooth $(-,+,+,+)$ Lorentzian model of Specified bulk extended entropy and relaxation law, assume an actual solution of its source-free stress and particle balances and corrected relaxation law. Use the state maps and units of Rest fluid state, frame and geometric current definitions and Thermodynamics-GR fields, currents, reference normalization and units, with smooth future unit observer congruence $u$ (the frame, not a chart or tetrad), ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, $n,T,c>0$, positive $C^1$ coefficient fields $\zeta ,\tau$, and $\beta =\tau \mathrm{/}\zeta$. The state function and map are $C^2$ and $u$ is smooth as in Rest fluid state, frame and geometric current definitions; take $\mathrm{\Pi }$ smooth as required by Corrected bulk relaxation and exact extended entropy, with the positive coefficients $C^1$ as in the postulate. Thus the displayed stress and current and $S^a$ are $C^1$; require the balances and corrected law to hold pointwise. The solution remains in the application's declared open near-equilibrium regime. With ${\mathsf{T}}^{ab}=\epsilon u^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab},J^a=n{u}^a,S^a=(s-\frac{\beta {\mathrm{\Pi }}^2}{2T})u^a,$ the exact pointwise identity is ${\mathrm{\nabla }}_a{S}^a=\frac{{\mathrm{\Pi }}^2}{\zeta cT}\ge 0.$ This is conditional on an actual solution and does not assert general solution existence.

There is also an explicit homogeneous solution under the additional application condition that the declared open regime contains constant zero-gradient reference data with $\mathrm{\Pi }=0$. Let the selected material's state domain contain $(v_0,{\sigma }_0)$ with $n_0=1\mathrm{/}{v}_0>0$ and $T_0=e_{{\sigma }_p}(v_0,{\sigma }_0)>0$, and let the regime contain those constant state data. On $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,x^1,x^2,x^3)$, Minkowski metric $g=-(d{x}^0{)}^2+{\sum }_{i=1}^3(d{x}^i{)}^2$, and future direction ${\mathrm{\partial }}_{x^0}$, choose constant positive ${\zeta }_0,{\tau }_0$ and set $u={\mathrm{\partial }}_{x^0}$. For some $\delta >0$ seconds take the open region $O=(-c\delta ,\mathrm{\infty })\times {\mathbb{R}}^3$, put $t=x^0\mathrm{/}c$, and define $\mathrm{\Pi }(t)={\mathrm{\Pi }}_0{e}^{-t\mathrm{/}{\tau }_0},t>-\delta \mathrm{.}$ Choose ${\mathrm{\Pi }}_0$ sufficiently small that the full field and first-derivative data stay in the declared regime throughout $O$. Then the source-free balances and corrected law hold, and the entropy production is strictly positive at every finite $t\ge 0$ when ${\mathrm{\Pi }}_0\mathrm{\ne }0$, and zero when ${\mathrm{\Pi }}_0=0$.

The argument is a conditional model deduction, not an observation or a general nonlinear existence, causality, or stability theorem. No empirical premise is used.