Probing Saturon-like Limits in QCD Systems

High-occupancy QCD matter enters a saturated regime when its entropy or occupancy approaches the unitarity bound $\sim 1/\alpha$, the ``saturon" criterion. We test this criterion for protons and nuclei at small $x$ using analytic and numerical solutions of the BK equation. From these solutions we construct the gluon occupancy $N_g(x)$ and a thermodynamic entropy $S(x)$ via an Unruh-like temperature $T = Q_s/(2\pi)$ and an emergent gluon mass $M_g \sim Q_s$. For protons, both $N_g$ and $S$ rise toward small $x$ yet stay below $1/\alpha_s$ in our baseline setup. For nuclei, by contrast, the nuclear entropy $S_A$ attains the $1/\alpha_s$ benchmark in a small-$x$ window where the proton does not. This singles out nuclei as the natural environment to search for saturon-like behavior and motivates precision small-$x$ measurements and high-occupancy $pA$ and $AA$ collisions.