Halemane’s Theorem on the Collatz (3n+1) System

Abstract

Halemane’s Theorem on the Collatz (3n+1) System states that the entire dynamics of the Collatz (3n+1) System and therefore the entire Collatz-Map defined by the transitive closure of the Collatz Function on the set of all natural numbers; can be exactly represented by a bounded finite small-sized ideal based graded filtration structure defined on a modulo-3 quotient semiring generated by the primitive-root 2; typically, with three layers; which guarantees the convergence to the trivial-cycle.  To facilitate further deeper study, this filtration structure can be subjected to a suitably designed dynamically evolving refinement mechanism.  This refinement is achieved with two unique features.  The first feature is the application of the global-affine-transformation, f(x)=(3x+1) when x is a positive odd number (as defined in the Collatz System); to achieve a Euclidean expansion shift to the topmost coprime-layer of that filtration-structure; while avoiding the modulo-multiple-layer (zero-layer) and all the intermediate nilpotent-layers with nilpotent-elements (dead-end zero-divisors) like (6m-3) and (6m-0).  The second feature is to bypass the trivial-cycle {(1⇐2⇐4)} of the Collatz-Map by an initialization-phase for this filtration structure, starting directly with the modul-9 coprime-layer.  The entire Collatz-Map (transitive-closure of the Collatz function) is exactly represented by each and every one of these dynamically evolving graded algebraic filtration structures; which is itself covers the complete set of all natural numbers; neither missing any natural-number nor including any extraneous-elements; providing both the necessary-&-sufficient condition simultaneously.

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.