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 H^s which is the relevant component of a carefully designed structured system framework H and the set of positive integers.  The structured system framework H itself has been designed & constructed by a two-stage bijective mapping (with a meticulous reorganization and condensation) from the Collatz-domain :- the first stage being, from the Collatz-domain to BELnet, that is the network of binary exponential ladders defined on the set of positive odd numbers; and the second stage being, from BELnet to the structured system framework H.  The isomorphism between H^s and the set of natural numbers in turn leads to a reductio ad absurdum argument that is used to demonstrate domain exhaustion; logically excluding the existence of any extraneous objects, such as disjoint loops H^¥ or divergent chains H^& in H.  In proving the convergence, we have used only the most fundamental Dedekind-Peano axioms and modulus arithmetic properties of the Collatz system and have not used any algorithmic, computational or dynamic characteristics of the Collatz system.  Also, 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.