Proof of the Collatz Conjecture via Binary Towers
Year: 2026
Type: preprint
Abstract: This paper presents a proposed proof of the Collatz conjecture for all positive integers based on the structure of binary towers. The argument works directly with binary representations and ordinary carry operations. Its basic mechanism is the identity b + (b + 2b) = 4b, which allows an incoming block to pass through a two-level tower in exactly two carries. The proof develops this local mechanism into a global construction using three finite integer descent arguments. These establish the attainment of successive binary scales, the completion of finite buffers between towers, and the merging of the initial components. The final stage produces a power of two, followed by repeated halving to 1. The approach does not use probabilistic assumptions, statistical estimates, iteration trees, heuristic arguments, or exhaustive computational searches. The preprint is available in English and Russian.
Author: Glukhov, Alexander S.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225789045
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