Soliton cores at the center of fuzzy dark matter (FDM) halos provide a promising way to distinguish FDM from other dark matter models. However, the relation between solitons and their host halos remains contentious. Here, we rigorously examine this soliton-halo relation (SHR) using a rich set of cosmological simulations across various FDM particle masses, halo masses, and redshifts. We explicitly demonstrate thermal equilibrium between solitons and surrounding halo granules, energy equipartition within halos, and an FDM concentration-mass-nonisothermality relation. For each FDM halo, we confirm that its density profile outside the central soliton matches a collisionless N-body simulation from the same initial condition, serving as stringent numerical convergence tests. Our refined SHR agrees well with virialized halos in simulations, with a $1\sigma$ deviation of less than 30%. These findings not only reaffirm the SHR proposed by Schive et al. [Understanding the core-halo relation of quantum wave dark matter from 3D simulations, Phys. Rev. Lett. 113, 261302 (2014)] but also offer a more comprehensive understanding that extends its applicability.

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• Dark matter
• Galactic halos
• Astrophysical & cosmological simulations