Current Research in Traffic Transportation Engineering(CRTTE)
ISSN: 3069-5538 | DOI: 10.33140/CRTTE
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Collatz Conjecture
Abstract
Yang Yanhong
State Space Definition: The Four Initial States Let the binary representation of the parity sequence for any integer n undergo the transformation T (Collatz map). After an initial fold or symmetry-breaking operation, the starting point \text{Start}(n) has exactly four distinct binary states, corresponding to the residue classes modulo 4: $$ \text{Start}(n) \in { \mathbf{0}, \mathbf{1}, \mathbf{10}, \ mathbf{11} }_2 $$ These represent the four possible "parent" states from which any number can descend under repeated application of the Collatz function.