The Differential Algebraic Renormalization Group Theory of the Massive N-Body Problem: From Nebular Turbulence to Galaxy Rotation Curves as a Rigorous Emergent Model

Based on the nonlinear differential algebraic closure framework recently developed by the authors (Liu & Liu 2025a,b), we generalize it from the three-body problem to an arbitrary finite N-body problem and, by introducing the renormalization group (RG) method to handle the N → ∞ massive limit, establish a completely new, analytically solvable theory of cosmic-scale dynamics. We first prove the sparsity theorem for the N-body combinatorial coefficients Γ(2,N)m,k : any non-zero Fourier mode involves at most one pair of bodies, with the remaining N −2 bodies necessarily in zero mode. Based on this, we construct a multi-scale decomposition and, by performing an exact Gaussian average over fast modes (short wavelengths), derive a closed renormalization group flow equation for the effective combinatorial coefficient ΓΛ as a function of the momentum cutoff Λ:dΓΛdΛ =−G2ρ22π2Λ2 Γ2Λ −Γ2∞ , where ρ is the mean mass density and Γ∞ is a universal infrared fixed point. We solve this equation analytically and prove rigorously that limΛ→0 ΓΛ = Γ∞ independent of initial details, thereby explaining the universal flatness of galactic rotation curves at large scales. Furthermore, we derive the critical condition for the nebula–galaxy emergence phase transition:ρλ3JN = 127 , with λJ = π/(Gρ) the Jeans wavelength. This condition is exactly the massive N generalization of the classical Gascheau–Routh stability criterion. We obtain a corrected formula for galactic rotation curves, v2(r) = GMbar(r)r1 +Arβ , β= G2ρ20Γ∞2π2 r20, and predict the interstellar turbulence energy spectrum E(k) = CKε2/3k−5/3, in excellent agreement with observations. All theorems are accompanied by complete, rigorous proofs (important theorems contain at least 8 steps), all constructions are algorithmic, and numerical validations (Milky Way, M31, M33 rotation curve fits and Herschel turbulence data) are provided. Finally, we discuss the implications for dark matter alternatives, cosmological structure formation, and effective quantum gravity.