Collatz Descent Framework

A rigorous mathematical framework for analyzing the Collatz conjecture through valuation-based descent.

This repository presents a descent framework that reformulates the Collatz conjecture as a problem of infinite descent. The paper establishes several rigorous criteria under which a positive integer must descend to 1, and isolates the remaining unresolved step concerning infinite valuation patterns.

This repository does not claim to prove the Collatz conjecture. It presents a rigorous descent framework and isolates the remaining unresolved infinite-exclusion step.

Overview

The Collatz conjecture (also known as the 3x+1 problem) states that for any positive integer, repeated application of the map

• $T(n) = n/2$ if $n$ is even
• $T(n) = (3n+1)/2$ if $n$ is odd

eventually reaches 1. This paper introduces a valuation-surplus descent framework that analyzes why counterexamples cannot exhibit finite valuation patterns, and identifies the infinite-exclusion step as the remaining open problem.

What This Paper Proves

• The valuation-surplus descent criterion: if a Collatz trajectory has a finite set of valuation values, the number must descend to 1.
• Equivalence between the Collatz conjecture and the statement that every positive integer has a finite valuation set.
• Several sufficient conditions guaranteeing descent.

What This Paper Does Not Prove

• The full Collatz conjecture remains unproven.
• The infinite-exclusion step — showing that every positive integer has a finite valuation set — is not resolved and is explicitly stated as the remaining open problem.

Files

• collatz_descent_framework.tex — LaTeX source of the paper
• collatz_descent_framework.pdf — Compiled PDF
• references.bib — Bibliography in BibTeX format
• LICENSE — CC BY 4.0 license
• CITATION.cff — Citation metadata

How to Compile

pdflatex collatz_descent_framework.tex
bibtex collatz_descent_framework
pdflatex collatz_descent_framework.tex
pdflatex collatz_descent_framework.tex

References

• Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values.
• Lagarias, J. C. (1985). The 3x+1 Problem: An Overview.
• Additional references in references.bib.

License

This work is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).