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

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

© Dr.(Prof.) Keshava Prasad Halemane,

Professor - retired (2017JAN31) from

Department of Mathematical And Computational Sciences,

National Institute of Technology Karnataka Surathkal,

Srinivasnagar, Mangaluru - 575025, India.

https://orcid.org/0000-0003-3483-3521

Independent Researcher

(no affiliations; no funding; no data used; no conflict of interests)

SASHESHA,  8-129/12  Sowjanya Road, Naigara Hills,

Bikarnakatte, Kulshekar Post,

Mangaluru-575005. Karnataka State, India

https://www.linkedin.com/in/keshavaprasadahalemane/

k.prasad.h@gmail.com

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) when x is a positive odd number; for achieving a Euclidean expansion shift to the coprime layer; avoiding the modulo-multiple-layer and also all the nilpotent layers; the trivial-cycle {(1⇐2⇐4)} being bypassed by an initialization-phase for the filtration scheme, starting with directly the modul-9 coprime-layer.

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.

Keywords: Binary-Exponential-Ladder; Modular-Periodicity; Arborescence; 

         CTUHSK-Theorem; CTUHSK-Generative-Parameters;

         Collatz-Thwaites-Ulam-Hasse-Syracuse-Kakutani (CTUHSK) Sequence;

         Collatz-Thwaites-Ulam-Hasse-Syracuse-Kakutani (CTUHSK) Conjecture;

         Order-isomorphism; Dedekind-Peano Axioms; Convergence;

         Halemane-Conjecture.

AMS MSC Mathematics Subject Classification:  11B83; 11Y55; 93A05; 93A99.

Conclusion

We established a two stage bijective mapping from the Collatz-domain to a meticulously designed structured system framework H with a careful reorganization & condensation :- in the first stage, a bijective mapping from the Collatz-domain to BELnet, that is the network of  Binary-Exponential-Ladders defined on the set of positive odd numbers; and then in the second stage, a bijective mapping from BELnet to the structured system framework H.  Then we established the order-isomorphism between the relevant component H^s (that is shown to be a well-ordered infinite linear sequence; or rather a hierarchy; or indeed an arborescence of the Binary-Exponential-Ladders having its root; with the invariant base element H₀ of the structured system framework H) and the set of positive integers; providing the necessary condition.  A sufficient condition is provided through a reductio-ad-absurdum argument (along with an exceptionally unique modular arithmetic characteristic property of the Collatz system) that establishes domain exhaustion; having already captured all the modular (MOD3) residue classes in H^s; and guarantees the non-existence of any extraneous objects like disjoint loops/cycles and/or divergent chains.  Since the Collatz (3n+1) Conjecture is restricted to the set of natural numbers, the existence of any non-standard ‘rogue’ and/or ‘ghost’ elements are of no concern here.

The entire dynamics of the Collatz (3n+1) system has been captured in 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 for a Euclidean expansion shift to the topmost coprime layer of that filtration scheme, avoiding the modulo-multiple-layer (zero layer) and also all the nilpotent layers with nilpotent elements (dead-end zero-divisors) like (6m-3) and (6m-0).  The trivial cycle {(1⇐2⇐4)} is bypassed by an initialization phase for the filtration scheme, starting with directly the modul-9 coprime layer.  This design for the system dynamics framework by itself provides an independent definitive proof of the Collatz conjecture, establishing that the entire map of the Collatz system dynamics is exactly represented by this novel algebraic structure defined by the set of all positive integers; neither missing any natural number nor including any extraneous elements; and therefore, providing both the necessary & sufficient condition simultaneously in one single sweep.

As a direction towards future research work, we have also presented a possible alternative approach in proving this convergence, using modular arithmetic for the conditions to be satisfied by the CTUHSK generative parameters or equivalently the exponents in a closed-form expression for generating the reverse Collatz sequence corresponding to any given positive odd number.

It is observed that the consistent & persistent application of the modular arithmetic characteristics associated with the local neighborhood of BELnet imposed by the CTUHSK function gives rise to the emergence of the global system property associated with the structured system framework that forms an arborescence (hierarchy) of BELnet.  This leads to an upper limit on the depth in the arborescence or the rank/level in the hierarchy that one needs to analyse in order to reach for the existence of a binary-exponential-ladder with the desired positive odd number as its defining base rung in BELnet.  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.