Positive-square radial pulsation energy ($K=\frac{1}{3}{\int }_0^{R}p{r}^2[4r^2({\xi }^{\prime }{)}^2+(3\gamma -4)(r{\xi }^{\prime }+3\xi {)}^2]dr$)

Integrate the pressure-gradient term in the weighted stellar pulsation Rayleigh quotient by parts, with regular centre and zero surface pressure. Writing $\chi =r{\xi }^{\prime }+3\xi$ uses $3{\xi }^2+2r\xi {\xi }^{\prime }=({\chi }^2-r^2({\xi }^{\prime }{)}^2)/3$ and gives the displayed expression. Thus radial instability requires $\gamma <4/3$ somewhere in the positive-pressure interior. A constant trial function gives $K=3{\xi }^2{\int }_0^R(3\gamma -4)p{r}^{2}dr$, so strict radial stability requires this weighted integral to be positive, in particular $\gamma >4/3$ somewhere. When $\gamma =4/3$ throughout, constant $\xi$ is an exact zero-frequency homologous mode: the star is marginal, rather than strictly stable. Local threshold crossings alone do not determine stability for a mixed profile; the full Rayleigh-Ritz variational principle does.