A key obstacle to developing a satisfying theory of galaxy evolution is the difficulty in extending analytic descriptions of early structure formation into full nonlinearity, the regime in which galaxy growth occurs. Extant techniques, though powerful, are based on approximate numerical methods whose Monte Carlo-like nature hinders intuition building. Here, we develop a new solution to this problem and its empirical validation. We first derive closed-form analytic expectations for the evolution of fixed percentiles in the real-space cosmic density distribution, {it averaged over representative volumes observers can track cross-sectionally}. Using the Lagrangian forms of the fluid equations, we show that percentiles in $delta$---the density relative to the median---should grow as $delta(t)proptodelta_{0}^{alpha},t^{beta}$, where $alphaequiv2$ and $betaequiv2$ for Newtonian gravity at epochs after the overdensities transitioned to nonlinear growth. We then use 9.5 sq. deg. of Carnegie-Spitzer-IMACS Redshift Survey data to map {it galaxy} environmental densities over $0.2<z<1.5$ ($sim$7 Gyr) and infer $alpha=1.98pm0.04$ and $beta=2.01pm0.11$---consistent with our analytic prediction. These findings---enabled by swapping the Eulerian domain of most work on density growth for a Lagrangian approach to real-space volumetric averages---provide some of the strongest evidence that a lognormal distribution of early density fluctuations indeed decoupled from cosmic expansion to grow through gravitational accretion. They also comprise the first exact, analytic description of the nonlinear growth of structure extensible to (arbitrarily) low redshift. We hope these results open the door to new modeling of, and insight-building into, the galaxy growth and its diversity in cosmological contexts.
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