  This paper introduces a comprehensive methodology for examining the stability
of dark matter (DM) halos, emphasizing the necessity for non-local
inter-particle interactions, whether they are fundamental or effective in
nature, to maintain halo stability. We highlight the inadequacy of vanilla cold
collision-less DM models in forecasting a stable halo without considering a
"non-local" interaction in the halo's effective free energy, which could
potentially arise from factors like baryonic feedback, self-interactions, or
the intrinsic quantum characteristics of dark particles. The stability
prerequisite necessitates significant effective interactions between any two
points within the halo, regardless of their distance from the center. The
methodology proposed herein offers a systematic framework to scrutinize the
stability of various DM models and refine their parameter spaces. We deduce
that DM halos within a model, where the deviation from the standard cold
collision-less framework is confined to regions near the halo center, are
unlikely to exhibit stability in their outer sectors. In our study, we
demonstrate that the issue of instability within DM halos cannot be addressed
adequately using perturbative quantum effects. This issue is less pronounced
for fermionic DM but suffers from a higher degree of severity when considering
bosonic DM. We find that halos made of bosons with notable quantum effects have
sharp edges, while those made of fermions show more diffuse boundaries
extending toward infinity. We also explore the broadest form of the effective
free-energy around a chosen mass profile.
