Ultra-Radical: Geometric Continuity of Master Series Beyond the Convergence Radius

This paper presents a geometric criterion for selecting branches of the ultra-radical that ensures continuity of roots of algebraic equations during analytical continuation beyond the radius of convergence using the Master-J method. For each root of an algebraic equation beyond the convergence radius, we construct a unique power series that maintains continuity of the given branch. The proposed approach provides a deterministic algorithm for analytical continuation, including standards for resolving ambiguities in degenerate cases (sector boundaries). The method naturally generalizes to equations with arbitrary coefficients and number of terms, opening pathways for its application to a broader class of transcendental equations.