Top panel: pressure vs. density for a DM-only model. Both the pressure and density are continuous, as per our phase transition requirements. At the core radius (black circle), the gradient jumps because of the discontinuity in the equation of state at the transition between the superfluid and normal phase of DM. The core radius is located very close to the rightmost part of the plot as the density is approximately constant within the core. Bottom panel: sound speed of DM (). The jump at the core radius (vertical line) results from the discontinuous equation of state at the boundary. This might be a consequence of our simplified model of the phase transition. Addressing this issue is best left for future work. Both panels have Λm³ = 0.3 × 10^-3 eV⁴/c⁸.