Collatz-Hasse-Syracuse-Ulam-Kakutani (CHSUK) Theorem: Convergence of the Collatz Sequence to the Trivial Cycle

Abstract

This paper presents the Collatz-Hasse-Syracuse-Ulam-Kakutani (CHSUK) theorem, which asserts the convergence of the Collatz Sequence to the trivial cycle, thereby proving the Collatz Conjecture, a long-standing unsolved problem.  The proof is based on the isomorphism established between the relevant component H^s 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.  A reductio ad absurdum argument is used to demonstrate domain exhaustion; logically excluding the existence of any extraneous and/or non-standard objects, such as disjoint loops H^¥ and/or divergent chains H^& in H.  The proof uses only the most fundamental Dedekind-Peano axioms and modulus arithmetic properties of the Collatz system.  Some directions for possible future research work on algorithmic, computational and/or dynamic 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; with {(31←41←27)} and {(1←5←3)} as exceptional and limiting cases.