Beyond "Sufficiently Close": Geometric Guarantees and Failure Modes in Nonlinear Systems Root-Finding

Classical convergence theory for nonlinear systems guarantees rapid local convergence only when iterates lie in an uncomputable neighborhood of a regular root. In practice, solvers therefore rely on extrinsic safeguards—such as damping, trust regions, or acceptance tests—when such proximity cannot be certified a priori. We do not propose a new globalization mechanism; instead, we classify the intrinsic local geometry that determines when and why such safeguards become necessary. Using first- and second-order geometric information of the residual map, we identify a small set of fundamental guarantees governing rootward predictability, curvature dominance, and regularity, and define geometric regimes according to which of these guarantees hold or fail. This taxonomy distinguishes bounded overshoot from misdirection, clarifies when loss of predictability is unavoidable, and explains the structural role of safeguarding mechanisms. The framework is method-agnostic and does not rely on a priori certification of local convergence. It provides a unifying geometric lens for interpreting Newton-type methods and their safeguarded variants, offering a principled basis for failure analysis and safeguard design in nonlinear systems root-finding.