Unified Solving of Exponential Diophantine Equations via Hierarchical Differential Algebraic Closure: A Complete Constructive Framework for Structured Subclasses

This paper presents a complete constructive framework for solving specific subclasses of exponential Diophantine equations through hierarchical differential algebraic methods. Unlike previous non-constructive approaches, we provide explicit computational procedures with rigorous error bounds for equations satisfying certain structural conditions. Our key contributions include: (1) Constructive definitions avoiding transfinite recursion for well-conditioned systems; (2) Explicit solution formulas with computable basis functions and coefficient polynomials for equations with dominant monomial structure; (3) Complete numerical analysis with proven error bounds under appropriate regularity conditions; (4) Experimental validation demonstrating high-precision accuracy (residuals < 10−20) for structured problem classes; (5) Rigorous reconciliation with classical impossibility results. The framework achieves polynomial complexity for systems with low treewidth and symmetry, while honestly acknowledging the fundamental limitations imposed by undecidability results.