= Solution
Assume homologous, gas-pressure-supported <main sequence> stars with nearly constant composition and mean molecular weight $\mu$, radiative energy transport and approximately constant opacity $\kappa$. Hydrostatic and mass scalings give
$$
P_c\sim\frac{GM^2}{R^4},\qquad \rho_c\sim\frac M{R^3},\qquad T_c\sim\frac{\mu m_uGM}{k_{\rm B}R}.
$$
The <radiative diffusion> equation gives the luminosity scaling
$$
L\sim\frac{ac}{\kappa}\frac{RT_c^4}{\rho_c}\propto\frac{\mu^4}{\kappa}M^3.
$$
Therefore, under these explicit assumptions,
$$
\boxed{L\propto M^k,\qquad k=3.}
$$
The radius cancels. Radiation-pressure dominance in the most massive stars changes the relation towards the <Eddington luminosity> scaling $L\propto M$, so the cubic law is an idealized gas-pressure homology result rather than an exact universal massive-star law.
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