On the Motion of Numbers

This note reverses the usual direction of Euclidean decomposition. Instead of fixing a base and studying all products, it fixes a natural number and follows its appearances across growing multiplication tables.The first appearance of the number, the filtration of its admissible ordered factor pairs, the occupancy schedule of its Euclidean cell, and its trajectory in quotient–remainder coordinates are recorded. In this number-first reading, the Euclidean coordinate moves across bases while its internal family of factor witnesses grows by admission. An exact one-step law records the motion of the coordinate under n -> n+1.