Rough Ideal Statistical Convergence in Neutrosophic Normed Spaces

This monograph develops a rigorous framework for rough ideal statistical convergence of order alpha in neutrosophic normed spaces. It integrates four complementary approaches to generalized convergence: rough convergence, statistical convergence, ideal convergence, and neutrosophic norm structures. A confidence-calibrated formulation is introduced to ensure that the roughness radius retains its geometric meaning in canonical norm-induced neutrosophic models. Central to the theory is the ideal-essential radius functional, whose sublevel sets characterize rough ideal statistical limit sets. Its continuity, convexity, stability, and minimal-radius properties are investigated, together with conditions for the existence and uniqueness of statistical centers. The study further establishes perturbation estimates and transformation principles for sums, scalar multiples, nonlinear maps, operators, products, shifts, moving averages, and finite differences. Representation theorems and bounded-error decompositions are obtained under explicitly stated properties of the underlying ideal. Applications include inexact fixed-point iterations, uncertain discrete dynamical systems, robust sensor fusion, and resistance to sparse faults. Finite-sample trimmed-radius estimators, an exact one-dimensional algorithm, and a mixed-integer conic formulation extend the framework to computational settings. Examples, counterexamples, exercises, and research directions demonstrate both the scope of the theory and the importance of carefully calibrated definitions and hypotheses.