digit_sum_parity_cycles.pdf

This paper investigates a novel digit-based iterative process on the integers, called the digit-sum parity transformation. For any integer ( n ), the transformation computes the sum of its digits ( s(n) ) and then applies ( f(n) = n + s(n) ) if ( n ) is odd, or ( f(n) = n - s(n) ) if ( n ) is even. Focusing on multiples of nine, the research combines computational experiments, mathematical analysis, and data visualization to reveal a remarkable phenomenon: every sequence generated by this process rapidly converges to a stable 2-cycle (a pair of numbers that repeat indefinitely).The study provides:A formal definition and theoretical analysis of the transformation.Computational evidence for all multiples of nine up to 100,000, showing that no sequence diverges or forms longer cycles.Visualizations in the form of directed graphs, illustrating how all trajectories are funneled into distinct 2-cycles.A proof outline explaining why only 2-cycles can exist for this transformation on multiples of nine.A comprehensive literature review situating this work within the context of digital root dynamics, automata theory, and number theory.This discovery uncovers a hidden self-organizing structure in the arithmetic of multiples of nine and opens new avenues for research in digit-based dynamical systems. The paper is suitable for researchers in mathematics, dynamical systems, and computational number theory, as well as educators interested in visual and algorithmic explorations of number patterns.