Collatz-Thwaites-Ulam-Hasse-Syracuse-Kakutani (CTUHSK) Theorem: Convergence of Collatz (3n+1) Sequence to the Trivial Cycle Proved

Abstract

This paper presents the Collatz-Thwaites-Ulam-Hasse-Syracuse-Kakutani (CTUHSK) theorem, which asserts the convergence of the Collatz (3n+1) Sequence to the trivial cycle {(1⇐2⇐4)}; thereby proving the Collatz Conjecture, a long-standing unsolved problem.  The proof is in two parts.  The necessary condition is provided by the order-isomorphism established between the relevant component H^s (with an invariant-base-element) of a structured system framework H and the set of positive integers.  The structured system framework H itself has been designed by a two-stage bijective mapping: (1) from the Collatz-domain to BELnet, that is the network of binary exponential ladders defined on the set of positive odd numbers; and (2) from BELnet to the structured system framework H.  The sufficient condition is provided by a reductio-ad-absurdum argument (along with an exceptionally unique modular arithmetic characteristic property of the Collatz system) that is used to demonstrate domain exhaustion; having already captured all the modular residue classes in H^s; logically excluding the existence of any extraneous elements or objects or sub-systems such as disjoint loops/cycles H^¥ and/or divergent chains H^& or even any/all non-standard objects, in H. 

An independent definitive proof is presented based on an exact representation of the Collatz system dynamics; using a dynamically evolving graded algebraic structure of ideal based filtration scheme with divisibility classes for modulo-3 quotient semiring generated by the primitive root 2; along with a filtration shifting global affine transformation f(x)=(3x+1) for a Euclidean expansion shift to the coprime layer; avoiding the modulo-multiple-layer and also all the nilpotent layers. 

Some directions for possible future research work on algorithmic, computational characteristics of the Collatz system have also been presented.  A situation has been identified wherein the emergence of global system properties through persistent local subsystem characteristics can be clearly demonstrated; Halemane Conjecture states that the maximum number of odd (3n+1) operations required to reach the trivial cycle {(1⇐2⇐4)} starting from any given positive integer and moving along the Collatz sequence, is limited by that given number itself; with the triad {(31⇐41⇐27)} as an exceptional limiting case.