Central stellar temperature bound from the virial theorem ($T_c\ge ⟨T{⟩}_M>\mu m_{p}GM/(6k_{B}R)$)

For spherical hydrostatic equilibrium with negligible surface pressure, ideal gas support and constant mean molecular weight $\mu$, the stellar virial theorem gives $3k_B\int Tdm/(\mu m_p)=\int Gmdm/r$. Since $r<R$ throughout the interior, the right side exceeds $G{M}^2/(2R)$. Division by $M$ gives the stated mass-weighted temperature bound. The central bound additionally assumes that the central temperature is maximal. A central pressure lower bound alone is insufficient without controlling central mass density.