Iterated correlation matrices; Pearson correlation; nonlinear matrix dynamics; contraction dynamics; uncertainty quantification; predictive upper bounds; fuzzy upper bounds; dimension-independent convergence; adaptive stopping; empirical convergence laws.

This publication investigates the discrete dynamical system generated by repeatedly applying the Pearson correlation operator to square matrices. Extensive computational experiments involving 1,000 random initial matrices for each of 22 dimensions, ranging from 3 to 2,000, reveal four stable empirical properties: a strong dimension-independent first-step contraction, nearly monotone decay of successive matrix differences, uniformly bounded convergence times, and a universal V-shaped relationship between the step size and the contraction ratio.Building on this observed contraction geometry, the publication introduces a three-layer uncertainty-aware predictive model. The first layer provides a data-driven upper bound accounting for sampling variability. The second applies a tail-conservative correction for compounded model uncertainty. The third introduces a fuzzy upper bound with an adjustable acceptance level, allowing decision tolerance to be controlled explicitly. The resulting bounds preserve the empirical V-shaped geometry and are governed by interpretable parameters representing probability level, tail uncertainty, and decision tolerance.The principal contribution is therefore the identification of a dimension-independent contraction pattern in iterated correlation dynamics and its transformation into a reproducible predictive framework. The framework is intended to support adaptive stopping criteria, consistency checks, and uncertainty-sensitive control in iterative normalization and learning algorithms.

This publication investigates the discrete dynamical system generated by repeatedly applying the Pearson correlation operator to square matrices. Extensive computational experiments involving 1,000 random initial matrices for each of 22 dimensions, ranging from 3 to 2,000, reveal four stable empirical properties: a strong dimension-independent first-step contraction, nearly monotone decay of successive matrix differences, uniformly bounded convergence times, and a universal V-shaped relationship between the step size and the contraction ratio.Building on this observed contraction geometry, the publication introduces a three-layer uncertainty-aware predictive model. The first layer provides a data-driven upper bound accounting for sampling variability. The second applies a tail-conservative correction for compounded model uncertainty. The third introduces a fuzzy upper bound with an adjustable acceptance level, allowing decision tolerance to be controlled explicitly. The resulting bounds preserve the empirical V-shaped geometry and are governed by interpretable parameters representing probability level, tail uncertainty, and decision tolerance.The principal contribution is therefore the identification of a dimension-independent contraction pattern in iterated correlation dynamics and its transformation into a reproducible predictive framework. The framework is intended to support adaptive stopping criteria, consistency checks, and uncertainty-sensitive control in iterative normalization and learning algorithms.