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

Abstract

This paper presents the Halemane’s Theory on the Collatz (3x+1) System; that is, the Collatz-Thwaites-Ulam-Hasse-Syracuse-Kakutani (CTUHSK) System; which asserts the convergence of the Collatz Sequence to the trivial-cycle {(1⇐2⇐4)}; thereby proving the Collatz Conjecture, a long-standing unresolved problem. 

The transitive closure of the subset {1,5,3} under the inverse Collatz function is shown to be the set of all natural numbers, asserting that the Collatz sequence starting from any given natural number converges to the trivial-cycle. 

An exact representation of the inverse Collatz function is achieved by a bounded finite small-sized ideal-based graded-algebraic-filtration-structure defined on a modulo-2 quotient-semiring generated by the pair of roots -1 and 3. 

The trivial-cycle of the Collatz-map is bypassed through an initialization-phase for this graded-algebraic-filtration-structure, starting directly with a 5-layered structure with the modulo-16 coprime-layer as its topmost layer.  This also facilitates a clear distinction among the six distinct possible combinations of the modulo-4 residue-classes and the modulo-6 residue-classes that are associated with any positive odd number; which is blurred in smaller structures. 

A layer-shifting global-affine-transformation, defined by the inverse Collatz function; results in a Euclidean shift to the topmost coprime-layer of this filtration structure; avoiding the modulo-multiple-layer (zero-layer) and all the intermediate nilpotent-layers with nilpotent-elements (dead-end zero-divisors). 

We establish that the transitive closure of the subset {1,5,3} under the inverse Collatz function is indeed the entire set of all natural numbers; covering all the relevant (modulo-3 & modulo-6) modular-residue-classes and also covering all the possible valid triplet-combinations of (1) input-values (2) operations and (3) output-values; governed by the modular-periodicity characteristics (resulting in the self-similar symmetry structure) of the Collatz system. 

Halemane-Conjecture states that the maximum number of odd (3x+1) operations required to reach the trivial-cycle {(1⇐2⇐4)}; starting from any given positive integer greater than one and moving along the Collatz sequence; is limited by that given number itself; with the triad {(31⇐41⇐27)} as an exceptional limiting case.