A Resolution of the Collatz Conjecture
Year: 2025
Type: article
Abstract: The system is built from a reverse rule that assigns every odd number a single valid predecessor.This predecessor is determined strictly by the number’s residue class, so there is never any ambiguity or branching.Because of this, every odd number has a unique finite chain of predecessors, and all such chains eventually reach the number 1.There are no cycles or alternative routes because each number belongs to exactly one predecessor chain. These predecessor chains organize the odd numbers into disjoint arithmetic progressions, sometimes called “rails.”Each rail originates from 1 and expands outward by a fixed linear rule.No two rails overlap, and together they account for every odd number without leaving any gaps. The forward rule of the system is exactly the inverse of the reverse rule.Every even number reduces immediately to its odd core, and every odd number moves to the next element on its rail.Because the reverse rule reaches 1 in finitely many steps for every odd number, the forward rule does the same.Thus, within this closed system, every positive integer eventually reaches 1. This system is internally complete:All numbers are included, every number has a unique place within the structure, the structure is fully determined by residue classes, and the forward and reverse motions exactly mirror one another. The framework developed in this paper does not modify the Collatz map; it exposes the arithmetic structure that is already implicit within the classical iteration. Thus the system is not an analogue or model, it is the internal dynamical structure of the Collatz function itself, expressed in explicit arithmetic form. No probabilistic assumptions are used; the behavior follows entirely from the arithmetic structure of the rails and their disjoint, exhaustive nature. Changelog: Added Lemma 5.38, Corollary 5.39, Lemma 5.40, Theorem 5.41. These additions explicitly establish the forward–reverse rail correspondence and demonstrate that the rail-ancestry structure forms a well-founded Noetherian dependency relation, ensuring that all rails converge to the unique minimal point. V16 Added: final monotonic descent argument, clarified 3.5, revised 4.16. V17: fixed typo.
Author: Spencer, Michael
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7087283293
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