= Solution
Assume <stellar homology>: fixed dimensionless profiles, uniform fixed composition, negligible <radiation pressure>, <ideal gas> support, and radiative transport throughout the model. <Mass conservation> gives $\rho_c\propto M/R^3$. The <hydrostatic pressure support equation> gives $P_c\propto GM^2/R^4$; combining with $P\propto\rho T$ gives $T_c\propto M/R$.
Integrating specific <proton–proton chain> energy generation over mass gives
$$
L_{\rm nuc}\propto M\rho_cT_c^4\propto M^6R^{-7}.
$$
For <radiative diffusion in a star>, $T_c/R\propto\kappa_c\rho_cL/(R^2T_c^3)$, hence $L_{\rm rad}\propto RT_c^4/(\kappa_c\rho_c)$. With the <Kramers' opacity law> $\kappa_c\propto\rho_cT_c^{-7/2}$,
$$
L_{\rm rad}\propto\frac{RT_c^{15/2}}{\rho_c^2}\propto M^{11/2}R^{-1/2}.
$$
Equating generated and transported luminosity gives $M^{1/2}\propto R^{13/2}$, so the <radiative homology with proton-proton burning and Kramers opacity> relation is
$$
\boxed{M\propto R^{13}.}
$$
The proportionality constants are fixed only within the adopted homologous family, not for arbitrary evolving stars.
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