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, thus proving the Collatz Conjecture, which has been a long-standing unsolved problem.  The proof is based on the isomorphism established between a carefully designed structured system framework H (with an arborescence of binary-exponential-ladders BELnet defined on the set of positive odd numbers) 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 that is the set of natural numbers :- the first stage being, from the Collatz-domain to the network of binary exponential ladders BELnet, and the second stage being, from BELnet to the structured system framework H.  Also a reductio-ad-absurdum argument is used to demonstrate domain exhaustion; logically excluding the existence of any extraneous objects, such as disjoint loops or divergent chains.  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 or computational or dynamic characteristics of the Collatz system.