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Unfolding the Collatz Tree: An Indirect Structural Proof of the Collatz Conjecture
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DOI：10.1080/27684830.2025.2542052.png)
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Abstract
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We present an indirect structural proof of the Collatz Conjecture by constructing and analyzing an infinite directed tree based on the inverse dynamics of the Collatz map. First, we introduce a branch indexing scheme and prove by induction, supported by extensive computational checks, that every natural number appears in this tree. Second, we develop an algorithm to build a minimal connected subtree rooted at one that contains all natural numbers up to any given bound. Third, we show by contradiction and explicit construction that the only cycle in the tree is the trivial loop 1-2-4-1 and that every backward path from a node terminates at the root in a finite number of steps. Together, these results demonstrate that each natural number has a unique backward path to one that exactly mirrors its forward Collatz trajectory, thereby establishing the Collatz Conjecture for all natural numbers. This graph-theoretic framework recasts convergence as a matter of coverage, acyclicity, and reachability, pointing toward new avenues for further symbolic and algorithmic study.
Keywords:
Collatz conjecture
inverse tree
dynamical systems
integer sequences
Number theory
algorithm
Journal
R
IF:
1.1
Papers:
72
Citations:
0
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Cited Papers