Number Theory

A τ-Vanishing Identity in the Collatz Problem: A Duality Between Base-6 Geometry and Base-2 Algebra

Authors: XinXin Li

We establish an exact algebraic identity connecting the cumulative near-conjugacyerror Eσ of a base-6 circle-rotation model of the Collatz map (HonarvarShakibaei Asli, 2026) with the terminal correction ε in the classical stopping-timeidentity. For a convergent Syracuse orbit, Eσ = 1 − log6(5n0+1)+ (log2 n0 + ε)·log6 2, in which the Syracuse stopping time τcancels exactly via the change-of-base formula log2 6 · log6 2 = 1.Consequently Eσ = O(1) ⇔ ε = O(1) ⇔ Σ 1/ni−1 = O(1), so theboundedness of the reciprocal sum becomes the central open question. The identity isverified to machine precision (≤ 2×10−15) for every tested n0 ≤ 77031.

Comments: 5 pages. CC BY 4.0.

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Submission history

[v1] 2026-08-13 12:06:32

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