= Solution
Retain fixed dimensionless structure constants and fixed $M$. Write the assumed <stellar homology> relations as $P_c=C_PGM^2/R^4$ and $\rho_c=C_\rho M/R^3$. The <equation of state> gives
$$
T_c=\frac{\mu_i}{\mathcal R}\left(\frac{P_c}{\rho_c}-K_{\rm nr}\rho_c^{2/3}\right)=\frac{\mu_i}{\mathcal R}\left(AG\frac{M}{R}-BK_{\rm nr}\frac{M^{2/3}}{R^2}\right),
$$
where $A=C_P/C_\rho$ and $B=C_\rho^{2/3}$. The first term is the temperature required for ordinary thermal <pressure> support; the second subtracts the support supplied by nonrelativistic <electron degeneracy pressure>.
As a function of $u=1/R$, this is a concave <quadratic polynomial>. Differentiation gives
$$
u_{\max}=\frac{AG}{2BK_{\rm nr}}M^{1/3},\qquad R_{\max}=\frac{2BK_{\rm nr}}{AG}M^{-1/3},
$$
and substitution proves
$$
\boxed{T_{\max}=\frac{A^2G^2}{4B\mathcal R K_{\rm nr}}\mu_iM^{4/3}\propto\mu_iM^{4/3}}.
$$
The scaling holds at fixed $K_{\rm nr}$ and homologous profile; changing the <mean molecular weight per electron> also changes $K_{\rm nr}$. At the maximum the ion thermal <pressure> and <electron degeneracy pressure> contribute equally. Subsequent contraction lets <electron degeneracy pressure> carry more of the load without raising the temperature; eventually the assumed positive-temperature branch approaches a cold, degenerate configuration.
\b[A sufficiently low-mass object never reaches the temperature needed for sustained <hydrogen burning>, so it becomes a cooling <brown dwarf> rather than a hydrogen-burning <star> on the <main sequence>.] The scaling explains a <hydrogen-burning minimum mass>; its numerical value additionally requires the composition, detailed structure and nuclear ignition conditions, which are not specified here.
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