The Ultra-Radical: Analytic Continuation, Branching, and Stability of the Principal Branch

We study the ultra radical — a multivalued solution to the equation y^a = 1 + a x y^b. Within the convergence radius |x| < R, each branch is given by the Master-J power series. For |x| ≥ R, analytic continuation requires switching to one of two conjugate series. We introduce a deterministic geometric criterion that selects the correct conjugate series for each branch index n, eliminating heuristic search and guaranteeing branch continuity when crossing the circle |x| = R. Key result: Only the principal branch (n=0) remains continuous when parameters a, b, and x vary smoothly. This includes critical limits a → 0 (transition to exponential equation) and b → 0 (transition to binomial root). Branches with n ≠ 0 exhibit oscillatory divergence at a → 0 and lose their identity in these limits. This structural continuity makes the principal branch suitable for applications where parameters may depend on system state, such as nonlinear media with field-dependent exponents or adaptive dynamical systems. The full abstract with mathematical notation is available in the PDF document.