Radical Orbit Decomposition of Polynomial Roots: A Framework for Understanding Limitations of Symmetric Lifting

This paper proposes a conceptual framework for interpreting radical solutions of polynomial equations as orbit structures induced by Galois actions on radical extensions. Building on previous work on Chebyshev and Dickson polynomials and symmetric rational lifting, the paper introduces the notions of radical orbits and radical atlases to formalize this perspective. The main contributions include: 1. A formal definition of radical towers, radical parametrizations, and branch automorphism groups2. The Radical Orbit Principle (Theorem 4.1) relating branch choices to Galois actions3. An orbit-Galois correspondence connecting radical orbits to coset spaces of subgroups4. A quantitative bound on orbit sizes arising from radical towers (Lemma 6.1)5. Worked examples including a quartic equation requiring multiple charts6. Conjectural principles: Orbit Decomposition Principle, Quintic Fragmentation, Dickson Transitivity Important disclaimer: This paper does not prove new solvability results. It provides a structural reinterpretation and a conceptual language for discussing radical solvability. The conjectures are proposed as interpretive principles, not as proven theorems. Keywords: radical orbits, radical atlas, orbit fragmentation, Galois theory, solvable polynomials, Dickson polynomials, Chebyshev polynomials This is Paper 9 in a series on numerically stable and radical solutions of polynomial equations.