Halemane’s Theory on the Collatz (3x+1) System

Abstract

This paper presents certain details about the development of Halemane’s Theory on the Collatz (3x+1) System.  The concept of binary-exponential-ladder BEL has been introduced, to enable effective merging/absorption of repeated application of the Collatz even-operator (x/2) whenever valid, within the corresponding BEL.  The set of all binary exponential ladders has a direct bijective mapping with the set of all positive integers.  Every positive odd integer corresponds to the unique defining-base-rung of a BEL; whereas every positive even integer corresponds to a non-base higher rung of the corresponding BEL.  The immediate-successor relation is defined as a unique one-to-one binary-relation on the set of binary exponential ladders, using the condensed compact Collatz (odd-to-odd) function.  Therefore, the network of binary exponential ladders BELnet provides an exact mathematical representation of the Collatz system.  Note that the inverse of the condensed compact Collatz (odd-to-odd) function defines the immediate-predecessor relation, which is shown to be a one-to-many mapping.

A structured system framework H has been designed merely as a well-organized condensation of BELnet by exploiting the partial order relation in BELnet; to achieve a linear strict order in the structured system framework H.

An order-isomorphism is established between the relevant component H^s (with an invariant-base-element H₀) of the structured system framework H and the set of positive integers.  This provides the necessary condition that starting with any integer in H^s the Collatz sequence converges to the trivial-cycle contained in BEL(1) which is the invariant-base-element or the sink-node H₀ in H^s.

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.

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 upper limit on the number of odd (3x+1) operations required to reach the trivial-cycle {(1⇐2⇐4)} starting from any given positive integer and moving along the Collatz sequence, is the given number itself; the triad {(31⇐41⇐27)} being an exceptional limiting case.