The MOND Depth Index and Dynamical Maturity Clock: Toward a Universal Classification of Galaxies and Star Clusters

Mass discrepancies in galaxies are empirically known to appear only below a characteristic acceleration scale a0. Here we show that this behaviour is not limited to galaxies: it extends continuously across the full hierarchy of self-gravitating stellar systems, from gas-rich dwarfs and spirals to massive early-type galaxies, and further down to compact stellar clusters. We introduce the Milgromian dynamics (MOND) depth index DM, together with dynamical maturity index T = tcross/tH, dynamical collisionality index T1 = tcross/trelax, with tcross being the crossing time, tH the Hubble time and trelax the median two-body relaxation time, and the MOND acceleration index A = abar/a0. We uncover a well-defined two-dimensional dividing surface in dynamical space. The "dark matter phenomenon" is found only in systems that are both in the deep-MOND regime (abar < a0) and collisionless (trelax > tH), while high-acceleration, collisional systems (abar > a0, trelax << tH), including globular clusters and UCDs, show no evidence for a mass discrepancy. This clean dynamical separation defines a new, physically motivated classification scheme for stellar systems, unifying galaxies and clusters under one framework. The observed division emerges naturally within the MOND framework and provides a useful diagnostic for examining how different gravitational paradigms account for the origin of the mass discrepancy.