Galaxy evolution is commonly described through the time evolution of observational statistics such as luminosity functions and stellar mass functions. However, these quantities are projections of an underlying multivariate galaxy state space rather than fundamental dynamical variables. We develop a unified framework in which galaxy evolution is formulated as the time evolution of a probability measure on the galaxy manifold. Representing galaxy states by latent variables $θ\in\mathcal{M}$ and the population by a density $ρ(θ,t)$, the evolution is governed by a general equation containing continuous transport and nonlocal jump processes. By reinterpreting manifold learning as the pushforward of measures, we distinguish observational, representation, and physical measures, and emphasize that manifold coordinates themselves need not carry direct physical meaning. In this picture, luminosity functions and stellar mass functions arise as projected observables of a single underlying dynamics, and generally do not form closed equations in observational space. The framework contains existing models as limiting cases: reduction to a single mass variable yields continuity-equation models, while additive post-merger states recover the Smoluchowski coagulation equation. We further show that luminosity-function evolution is naturally described within the Schechter family, whose apparent stability is interpreted as an effective consequence of projection. Since observables are projections of measures, inference of galaxy evolution becomes a statistical inverse problem of recovering manifold dynamics from data. This framework shifts the focus from fitting observed statistics directly to inferring the underlying state-space dynamics, thereby bridging manifold learning and physical theory.

翻译：暂无翻译