Number Theory

   

A Resolution to the Collatz Conjecture

Authors: Michael Spencer

This paper presents a complete arithmetic resolution of the Collatz Conjecture by separating its structure into two complementary components: a finite residue and phase transition system, and a global affine counting framework. The reverse Collatz step is shown to act only on the two live odd residue classes, and every valid reverse exponent produces a predictable affine expansion whose inverse matches the expected dyadic frequency. From this, every odd integer belongs to exactly one uniquely defined dyadic slice, forming a disjoint partition of all odd numbers. Independently, a zero-state index reveals that every live odd number also generates a unique sequence obtained by repeatedly applying the transformation four times the number plus one, and these sequences likewise partition the odd integers without overlap. The two partitions are proven to be identical, establishing a single global organizational structure for the entire Collatz map. Because the forward and reverse processes are locked to each other and the residue-phase system is finite, every forward trajectory follows one non-branching path that must ultimately return to one. This eliminates the possibility of infinite growth or nontrivial cycles. All structural components, classifications, and counting methods are original to this work and together provide a fully closed arithmetic description of the Collatz dynamics.

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[v1] 2025-11-23 00:25:22

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