Binary Structure as a Carrier of Dynamic Properties of Numbers in the Collatz Process

This paper presents a systematic numerical study of the Collatz process from the perspective of binary structure. We introduce a family of numbers of the form n = a * 2^R - 1, where a is an odd integer representing the binary "head" and R is the length of the tail of ones. For each odd a up to 10001 and for R = 2 to 6, we compute the absolute peak of the Collatz trajectory. Based on the resulting sequences of peaks, we propose a classification into three types: exponential growth, ladder of attractors, and anomalous behaviour. We analyse the binary patterns associated with each type, identify structural invariants, and compare the results with a ternary analogue n = a * 3^R - 1. The findings suggest that the binary geometry of a number — specifically the arrangement of zeros and ones — carries essential information about its dynamic behaviour in the Collatz process.