A Structural Study of Parity Vectors in the Collatz Conjecture

抄録

We investigate the Collatz conjecture using parity vectors (PVs). We classify PVs according to $d$, the number of occurrences of 1, rather than by their length. For each finite value of $d$, we show that all unconverged and Glide PVs uniquely form sequences of a specific type. We then divide each unconverged PV into sub-PVs having the same length as the corresponding Glide PV and show that no unconverged sub-PVs exist for that value of $d$. This result holds for every finite value of $d$. The argument applies only to finite values of $d$ and therefore does not address the existence of infinite PVs. Consequently, the results do not establish the Collatz conjecture, but they provide a new structural approach for analyzing PVs beyond classifications based solely on PV length.