Central obstruction to an outer halo mass offset ($M_h(r)\simeq V_{\infty }^{2}r/G-M_{\star }$)

Subtracting a finite stellar mass from the distant spherical mass required for a flat galaxy rotation curve gives the displayed asymptotic offset. Extending it to $r=0$ would give negative enclosed mass, so it cannot define a physical inner halo. Keeping only its positive leading term instead gives a singular isothermal halo with density proportional to $r^{-2}$ and a nonzero limiting central circular speed. These are different extrapolations, and neither is forced by an outer rotation curve.