= Minimum matching at a polytropic sonic point
{title2=$\min F=\min G$}
Put $y=\rho r^{4/(\gamma+1)}$ and $\alpha=4(\gamma-1)/(\gamma+1)$. The <polytropic stellar wind> equations become $F(r)=Cr^\alpha+GMr^{\alpha-1}=G(y)=\gamma Ky^{\gamma-1}/(\gamma-1)+\dot M^2/(32\pi^2y^2)$. At a regular <sonic point> both derivatives vanish. Matching their minima allows the two positive-$y$ branches to join with finite slope; unequal minima either leave a forbidden radial interval or separate the subsonic and supersonic branches. This reduces a singular differential-equation crossing to a geometric minimum comparison.
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