= Solution
For spherical <hydrostatic equilibrium>, the <stellar structure equations> include
$$
\frac{dP}{dr}=-\frac{Gm(r)\rho(r)}{r^2},\qquad \frac{dm}{dr}=4\pi r^2\rho(r).
$$
Eliminate $\rho$ to obtain $dP/dm=-Gm/(4\pi r^4)$. Since $r(m)<R_\odot$ through the interior, integration gives
$$
P_c-P_s=\int_0^{M_\odot}\frac{Gm}{4\pi r(m)^4}\,dm>\frac{G}{4\pi R_\odot^4}\int_0^{M_\odot}m\,dm.
$$
With negligible surface <pressure>, this proves
$$
\boxed{P_{c,\odot}>\frac{GM_\odot^2}{8\pi R_\odot^4}\simeq4.48\times10^{13}\,\mathrm{Pa}.}
$$
This is a conservative lower bound; concentrating <mass> near the centre increases the required <pressure>.
For the <temperature> inference assume predominantly <ideal gas> support, a nearly fully ionized hydrogen–helium mixture with <mean molecular weight> $\mu\simeq0.61$, and the hottest region at the centre. A rigorous scale estimate follows from the <stellar virial theorem>, rather than substituting the mean <mass density> for the unknown central <mass density>. With negligible surface <pressure>,
$$
3\int P\,dV=-W,\qquad -W=\int_0^{M_\odot}\frac{Gm}{r(m)}\,dm>\frac{GM_\odot^2}{2R_\odot}.
$$
The <ideal gas law> gives $\int P\,dV=k_B\int T\,dm/(\mu m_p)$. The mass-weighted mean <temperature> consequently obeys
$$
\boxed{T_c\ge\langle T\rangle_M>\frac{\mu m_pGM_\odot}{6k_BR_\odot}\simeq2.35\times10^6\,\mathrm K.}
$$
This establishes several million <kelvin> under the stated physical assumptions. The central <pressure> inequality alone does not give a central <temperature> bound without an <equation of state> and information about <mass density> or an integrated argument.
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