Analytic Solutions of Renormalization Group Equations via Constructive Differential Algebraic Closure

This paper establishes a comprehensive theoretical framework for obtaining explicit analytic solutions to renormalization group (RG) equations through the systematic construction of specialized differential algebraic closures. Building upon recent advances in transcendental differential algebra and drawing inspiration from constructive approaches to nonlinear systems, we develop a hierarchical closure space K(M) RG that formally incorporates the mathematical structures inherent in RG flows. We prove that solutions to RG equations across quantum field theory, statistical physics, and quantum gravity can be expressed as convergent formal series within appropriately constructed differential algebraic closures. The solution construction involves a recursive algorithm that intertwines Taylor expansion of beta functions with multi-index combinatorial corrections. Key results include: (1) A constructive definition of RG differential fields and their hierarchical closures with complete existence proofs, (2) A theorem establishing the existence and convergence of formal solutions within these closures with explicit error bounds, (3) Derivation of multiindex combinatorial coefficients for RG systems with explicit formulas and rigorous properties, (4) Efficient algorithms for high-precision RG computation with complexity analysis, and (5) A rigorous validation framework with certified error bounds and detailed numerical examples. Numerical validation demonstrates spectral convergence with residuals typically below 10−25 across diverse physical systems. This work reconciles with known non-integrability results while demonstrating explicit solvability in extended differential closures, providing new mathematical foundations for renormalization theory.