Regional Curvature Taxonomy for Real-Domain Root-Finding

Across the broad literature on fixed-point iteration for root finding, an underlying geometric pattern can be observed that subtly directs iteration behavior, even in unstable regimes. Regardless of algorithm or function family, the local curvature profile in such regions assumes a uniform canonical form. In this paper, we explain why this form arises. Under standard one-dimensional, $C^2$, simple-root assumptions, we partition a neighborhood of the root by natural boundaries---the root, critical points ($f'(x)=0$), and inflection points ($f''(x)=0$)---so that within each region of fixed curvature sign, iterations exhibit either contractive or non-contractive steps. We derive this regional normal form, analyze iteration behavior in each region, and show how transitions across boundaries govern convergence and divergence. The resulting taxonomy is agnostic to any particular method and provides a compact, unified framework for understanding and comparing iteration behavior. We also provide illustrations showing how the framework supports function-space transformations that regularize adverse curvature, curvature-guided second- and third-order root approximations, and predictive localization within the taxonomy’s regional structure via cubic projection---without sampling.