Radical Orbit Decomposition of Polynomial Equations: A Conceptual Framework and Conjectural Trichotomy

This paper proposes a conceptual framework for understanding radical solutions of polynomial equations through the lens of orbit decomposition under Galois actions. We introduce the notions of radical orbits and radical atlases to describe how branch choices of radicals generate subsets of the solution set. We prove several foundational lemmas about the structure of the Chebyshev-Dickson family, including a fixed field characterization (Lemma 2.1) and a rigidity result based on semiconjugacy and functional decomposition (Theorem 3.1). We then propose a conjectural trichotomy (Conjectures 1-3) classifying polynomial equations into three types based on radical orbit structure: transitive (Type I), finite fragmentation (Type II), and obstructed (Type III). This work is intended as a research program rather than a complete classification. The conjectures and open problems presented here are offered as directions for future investigation.