Hierarchical Differential Algebraic Closure Framework for the General N-Body Problem: A Comprehensive Symbolic-numeric Approach with Explicit Complexity Bounds

This paper presents a comprehensive extension of the hierarchical differential algebraic closure framework from the three-body problem to the general N-body gravitational problem. We develop a systematic polynomialization procedure that transforms the Newtonian gravitational equations into an exact multivariate polynomial system through the introduction of regularized auxiliary variables and constraint equations. The extended methodology rigorously preserves the geometric structure and conservation laws while handling collision singularities through Levi-Civita regularization. We provide complete symmetry group characterization for N-body systems, explicit dimension reduction formulas with corrected conformal symplectic treatment of scaling symmetry, and recursive Jacobi coordinate constructions. The hierarchical closure framework is generalized with detailed degree analysis and complexity bounds, explicitly addressing the Cayley-Menger relation degrees. Theoretical analysis demonstrates that the method achieves machine-precision accuracy while exactly preserving fundamental conservation laws at the algebraic level. Computational complexity classification establishes parameterized tractability for symmetric configurations, with explicit bounds for small to moderate N systems. Extensive mathematical validation confirms the correctness of all derivations and proofs, including numerical verification protocols and experimental performance analysis. The framework provides a unified symbolic-numeric approach to celestial mechanics with applications to planetary systems, star clusters, and gravitational dynamics.