Nonlinear Renormalization Group Enhanced Differential Algebraic Framework for the Three-Body Problem: From Microscopic Perturbations to Macroscopic Evolution

This paper establishes a constructive differential algebraic framework for the three-body problem by extending the nonlinear partial differential algebraic closure KNLPDE recently developed by the authors. We define the nonlinear ordinary differential algebraic closure K(3-body)NLODE as a differentially closed field extension constructed through a recursive adjunction process that incorporates: (i) fundamental solutions of linearized variational equations around two-body reference orbits, (ii) multi-index radical extensions Φ1/p, (iii) roots of unity ωp, and (iv) a constructive set of nonlinear special functions including Laplace coefficients, elliptic integrals, and hypergeometric functions arising from Fourier–Lagrange expansions. Wefurther merge this algebraic closure with the renormalization group (RG) theory, treating the homotopy parameter λ as an RG scale. We derive the exact β function governing the scale dependence of the combinatorial coefficients Γ(2,3)m,k (λ) and prove that the RG flow eliminates all secular terms to all orders in perturbation theory. The resulting renormalized solution exhibits only sub-polynomial error growth over exponentially long times, surpassing classical numerical integrators. We provide rigorous proofs for all conjectures and open problems from previous versions, converting them into theorems with complete multi-step derivations. Key new results include: (i) an exact formula for the dissipation coefficient γdiss at 2.5PN order, (ii) a rigorous DMRG convergence bound for lattice discretized three-body problems, (iii) a relationship between the maximal Lyapunov exponent and the β-function, (iv) a breather solution for mode coupling in Fourier space, (v) a phase transition thresh-old separating power-law from exponential decay of correlations, (vi) an anomalous stretched-exponential scaling at resonances, and (vii) a duality between time-scale and mode-space renormalization. All constructions are algorithmic; we present detailed pseudocode for adaptive RG-guided homotopy continuation with certified precision. Complexity analysis shows O(NλM 3max + NtM 2max) operations, where M is the number of basis modes, Nλ the number of homotopy steps, and Nt the number of time samples. The method is validated on several benchmark problems (circular restricted three-body problem, Pythagorean three-body problem, equal-mass periodic orbit) using interval arithmetic; residuals below 10−14 are achieved with M = 32 and GPU acceleration, outperforming standard DAC by a factor of 4–5 in speed at the same precision. We further generalize the RG framework to post-Newtonian approximations up to 2.5PN, to the N -body problem (N ≥ 4), and to non-perturbative regimes via lattice DMRG. Finally, we establish a categorical equivalence between the differential algebraic closure and the RG flow, proving that the closure is invariant under RG transformations.