A Structural Study of Parity Vectors in the Collatz Conjecture

抄録

The Collatz conjecture is examined using parity vectors (PVs). Parity vectors are classified according to d, the number of occurrences of 1, rather than by their length. We show that all unconverged and Glide PVs with a fixed finite value of d can be uniquely arranged into sequences of a specific form. Each unconverged PV is then divided into sub-PVs of the same length as the corresponding Glide PV, and we find that no unconverged sub-PVs exist for this value of d. The argument applies separately to each finite value of d and does not address the existence of infinite PVs. Accordingly, the results do not establish the Collatz conjecture, but rather provide a new structural viewpoint on PVs that complements earlier analyses based on PV length.