Galaxies as stochastic systems: why the next breakthrough in galaxy evolution requires one hundred million spectra

Each galaxy is observed only once along its life, making galaxy evolution fundamentally an inverse statistical problem: time-dependent physics must be inferred from ensembles of single-epoch snapshots. To move beyond descriptive scaling relations toward physical regulation mechanisms of star formation, quenching, chemical enrichment and black hole growth, galaxies must be treated as realizations of a stochastic process whose hyper-parameters (e.g., correlation timescales, burstiness, duty cycles) are inferred hierarchically. This demands both depth and scale: continuum S/N sufficient for absorption-line ages and chemistry, and samples far larger than those in SDSS, DESI, 4MOST or MOONS, which provide either depth or size but not both across $0<z<3$. Once the relevant axes of mass, redshift, environment, structure and evolutionary phase are populated, the requirement naturally rises from $10^7$ to $\sim10^8$ galaxies. This is the regime where stochastic hyper-parameters can be well constrained and where comparisons to simulations and cosmological forward models become limited by theory rather than observations. We outline the science enabled by such a programme and the corresponding requirements for a future ESO wide-field spectroscopic facility capable of delivering tens to hundreds of millions of rest-UV-optical spectra over $0\lesssim z\lesssim3$.