Radical Orbit Geometry of Polynomial Equations: A Classification Framework from Transitivity to Fragmentation

This paper builds upon the Radical Orbit Framework introduced in previous work, shifting focus from the existence of radical orbits to their classification. We propose a systematic categorization of polynomial equations based on the structure of radical orbits induced by radical parametrizations. Three fundamental types are proposed as working definitions: 1. Type I (Transitive): A single radical orbit covers the entire solution set. 2. Type II (Finite Fragmentation): The solution set decomposes into a finite number of disjoint radical orbits, requiring a radical atlas. 3. Type III (Obstructed, Conjectural): Within the studied framework, no finite radical atlas is known; this is conjectured based on structural constraints. We analyze structural constraints on the branch automorphism group A, relate orbit size to polynomial degree, and provide heuristic geometric interpretations via coverings and monodromy. A conditional obstruction result links orbit size to polynomial degree under a structural assumption. We also discuss the solvability properties of A and their conjectural connection to classical Galois theory. Case studies include Dickson polynomials (Type I), solvable irreducible quartics (Type II), and the general quintic (Type III, conjectural). The paper concludes with a research program: classifying polynomial equations by their radical orbit type, characterizing transitivity, and determining minimal atlas sizes. Important disclaimer: This work is a conceptual and organizational framework, not a source of new solvability theorems. All classifications are proposed as working definitions and conjectures to guide future research.