Timescale hierarchy in soliton cores. Comparison between the dynamical time t_dyn (blue solid line, ﹩{t}_{{\rm{dyn}}}\propto {M}_{s}^{-2}﹩) and the optically thin cooling time t_cool (red dashed line; this corresponds strictly to the optically thin cooling time defined in Equation (5), not the effective diffusion-limited time t_rad) as a function of soliton mass M_s. The top axis explicitly provides the corresponding virial temperature T_vir to guide physical intuition. At low masses (T_vir < 10⁴ K), the exponential suppression of atomic cooling in primordial gas leads to a stable regime where t_cool > t_dyn. As the soliton mass increases, the gas is heated above the atomic cooling threshold, causing a sharp drop in t_cool. A critical mass emerges at M_crit ≈ 2.8 × 10⁷M_⊙, identified exactly at the intersection where the condition t_cool = t_dyn is satisfied; this threshold is obtained analytically by solving t_cool = t_dyn using the cooling function adopted in our model (see Appendix B). Above this mass threshold, the system enters the cooling-dominated regime (t_cool < t_dyn). For the massive regime relevant to LRDs (M_s ≳ 10⁸M_⊙), the cooling time is orders of magnitude shorter than the dynamical time, inevitably triggering a catastrophic collapse.