System(3X±1)/4 Framework for the Collatz (3x+1) Problem: A Definitive Proof of the Collatz Conjecture

Abstract

We design a mathematical framework based on System(3X±1)/4; defined similar to the Collatz (3x+1) system, with slightly modified operations.  On similar lines, we define the CTUHSK(3x+1)/4 system, and show that to be exactly the same as the Collatz (3x+1) system.  We establish the convergence of System(3X±1)/4 to its trivial-cycle {(1⇐(2)⇐4)}; using the persistent local characteristic property of ‘odd-to-odd monotonicity’ in its defining relation.  Therefore, the transitive closure of the subset, say {1,5,3} under the inverse mapping of System(3X±1)/4; represented by the nodes of the system arborescence; is the entire set of positive integers.  We present a meticulously designed reversible (invertible) transformation (reduction); between System(3X±1)/4 and the CTUHSK(3x+1)/4 system; based on a well-defined pruning-&-joining operation on the system arborescence.  We show that every such pruning-&-joining operation preserves the connectedness property and even the binary(inward)arborescence structure of these systems.  We establish that the transitive-closure of the subset, say {1,5,3} under the inverse mapping of CTUHSK(3x+1)/4 system is exactly the same as the transitive-closure of that subset under the inverse mapping of System(3X±1)/4; which has been shown to be the entire set of all natural numbers.  This establishes the convergence of the CTUHSK(3x+1)/4 system and therefore the Collatz (3x+1) system, to the trivial-cycle {(1⇐2⇐4)}; thus providing a definitive proof of the Collatz Conjecture. 

This paper also presents Halemane-Conjecture on the Collatz sequence, that the upper bound on the number of odd (3x+1) operations required to reach its trivial-cycle {(1⇐2⇐4)} starting from any given positive integer and moving along the Collatz sequence, is the given number itself; the triad {(31⇐41⇐27)} being an exceptional limiting case.