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The Quantum Central Limit Theorem (QCLT) governs the emergence of classical behaviour in large ensembles of identical quantum subsystems through self-averaging of collective observables. For nonlinear observables, however, the strict inequality $\langle f(\hat{A})\rangle \neq f(\langle\hat{A}\rangle)$ generates residual corrections---controlled by the variance of the collective distribution---that survive coarse-graining at subleading order and persist into the classical regime. We analyse the consequences of this structure for macroscopic quantum systems evolving in an external potential. Applying the QCLT to the collective centre-of-mass dynamics of a large ensemble of identical subsystems in a Newtonian gravitational field, we show via the Ehrenfest theorem that the effective force law acquires an additional term proportional to $\langle r^{-2}\rangle - \langle r\rangle^{-2}$, which is strictly non-vanishing by Jensen's inequality whenever the radial distribution has finite variance. For a physically motivated class of $R$-dependent mixed states---constructed from Gamma-distributed radial wavefunctions and justified by an open-system decoherence model---this quantum-statistical correction generates an emergent enclosed-mass scaling (quasi-dark matter) \[\frac{\Delta M(R)}{M_b} \sim \frac{R}{\kappa}, \qquad \kappa \ll R \ll R_{\mathrm{corr}},\] corresponding to an isothermal effective density profile $\rho_{\mathrm{eff}} \propto R^{-2}$. No modification of the gravitational interaction nor new matter fields are introduced; the effect is a structural consequence of nonlinear expectation-value algebra under coarse-graining. As an astrophysical application, this emergent mass scaling reproduces flat galactic rotation curves and the baryonic Tully--Fisher relation $V_{\mathrm{flat}} \propto M_b^{1/4}$, with the onset of flattening predicted at $R \sim \kappa \propto \sqrt{M_b/\Sigma_0}$---a galaxy-by-galaxy, falsifiable prediction absent from both $\Lambda$CDM and MOND. The framework further predicts a surface-density-dependent Radial Acceleration Relation, \[g_{\mathrm{obs}} \propto \sqrt{G\Sigma_0}\,\sqrt{g_{\mathrm{bar}}},\] testable against the SPARC dataset. Beyond $R_{\mathrm{corr}}$, identified with the \ion{H}{I} truncation radius, the correction saturates and Keplerian decline is restored. Relativistic extensions and cosmological implications are left for future work. These results demonstrate that quantum statistical coarse-graining alone can generate nontrivial emergent scaling behaviour at macroscopic scales, and highlight a structural link between limit theorems in many-body quantum mechanics, open-system decoherence, and emergent classical dynamics.