A Unified Operator Framework for Roots: Galois-Equivariant Flows, Transcendental Implicit Functions, and Puiseux-Stable Ramification

We present a general operator identity for roots of implicit equations that subsumes polynomial formulas (including the quartic branches in polar form), transcendental equations, and ramified algebraic families with fractional powers. For an equation $F(x;\mathbf{p})=0$ and a derivation-like operator $D$ acting on parameters $\mathbf{p}$.