= Solution
A <regular central expansion of the TOV equations> requires $m(0)=0$. Once $\rho_c$ is chosen, the <barotropic stellar equation of state> fixes $p_c=p(\rho_c)$. Smoothness at the centre forces the radial first derivatives of $p$ and $\rho$ to vanish. Expanding the <Tolman–Oppenheimer–Volkoff equation> gives
$$
\begin{aligned}
m(r)&=\frac{4\pi}{3}\rho_c r^3+O(r^5),\\
p(r)&=p_c-2\pi(\rho_c+p_c)\left(\frac{\rho_c}{3}+p_c\right)r^2+O(r^4),\\
\Phi(r)&=\Phi_c+2\pi\left(\frac{\rho_c}{3}+p_c\right)r^2+O(r^4).
\end{aligned}
$$
Since $dp/d\rho>0$, the <equation of state> is locally invertible, and these coefficients determine the corresponding <energy density> expansion as well. Integrating the regular solution outward gives the surface at the first zero of the <pressure>, when such a finite surface exists. The <ordinary differential equation> uniqueness theorem then fixes the interior and its exterior <mass> from $\rho_c$.
The additive constant $\Phi_c$ is a time-coordinate normalization, not a second physical parameter. Matching to the <Schwarzschild metric> with time normalized at infinity fixes it. Therefore \b[the regular stellar solutions, modulo time rescaling, have one physical parameter: $\rho_c$]. This statement uses a suitably smooth fixed <equation of state> and refers to central densities for which the solution actually forms a regular star; not every central <energy density> need give a finite-radius configuration.
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