Constrained entropy criticality yields scoped TOV and Tolman–Klein relations

Assume the model preparation Supplied thermodynamic chart and gravitational entropy variation model, including a positive even $C^2$ profile pair with compact state-chart range, fixed $R>0$, $b\ge b_{\ast }>0$, fixed constraints $E,N_b$, and boundary normalization $\varphi (R)=\frac{1}{2}\log b(R)$. Use the exact $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ convention of Constrained entropy criticality, Tolman–Klein equivalence and TOV consequence. An admissible profile curve is $C^1$ in the sup-norm space $C^0([0,R]{)}^2$ on an open interval about zero, takes values in the even $C^2$ positive-b profile domain with range in a compact subset of the open state chart, and has a smooth derivative at zero compactly supported in $(0,R)$ and extended evenly at the center. Define constrained stationarity to mean that $S$ has zero derivative at the base profile along every such curve preserving both $E$ and $N_b$. Then constrained stationarity is equivalent to the existence of constants $\alpha ,{\mathrm{\Theta }}_K,{\mu }_K$ such that on $(0,R)$ $\frac{\mu }{\mathrm{\Theta }}=\alpha ,\mathrm{\Theta }{e}^{\varphi }={\mathrm{\Theta }}_K,\mu e^{\varphi }={\mu }_K,$ and the pressure obeys $P^{\mathrm{\prime }}=-(\epsilon +P){\varphi }^{\mathrm{\prime }}=-\frac{G(\epsilon +P)(M+4\pi r^{3}P\mathrm{/}{c}^2)}{c^2{r}^{2}b}\mathrm{.}$ Here ${\mathrm{\Theta }}_K=\mathrm{\Theta }(R)\sqrt{b(R)}$ and ${\mu }_K=\mu (R)\sqrt{b(R)}$. This is a conditional first-variation result in the adopted model, not a claim that nature selects such a profile or that it is an entropy maximum.