A Proof of the Collatz Conjecture via Logarithmic Decay Lower Bounds and Diophantine Approximation
Year: 2026
Type: preprint
Abstract: This manuscript analyzes the variation of 2-adic valuations during Collatz map iterations. The iterative behavior is decomposed through operator separation, and congruence methods are used to prove rigorously that any finite initial odd integer cannot indefinitely sustain iterations with $\nu_2(3x+1)=1$; a 2-adic valuation with $k \ge 2$ must appear within finitely many steps. The generation of even numbers during iteration and the numerical changes under higher valuations are examined. The convergence path following the occurrence of a pure power of two is described. The manuscript distinguishes between rigorous proofs and structural inferences. Based on the established uniqueness of the $\{1,2,4\}$ cycle, a structural explanation for the convergence of Collatz orbits toward the value 1 is provided. The global boundedness of orbits and the statement that an orbit must attain a pure power of two have not yet been established through rigorous analytic proof. This manuscript is a阶段性 (staged) structural analysis.
Author: huang, yan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171094155
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