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The Cylinder Horizon of the Accelerated 3x + 1 Map A Number-Theoretic Case Study in Descriptive Boundaries and Projectability Limits

descriptionPublicationkeyboard_double_arrow_right External research report Under curationPublisher:Zenodo
Authors: De Jesus, Elias;

doi: 10.5281/zenodo.20512207

The Cylinder Horizon of the Accelerated 3x + 1 Map A Number-Theoretic Case Study in Descriptive Boundaries and Projectability Limits

Abstract

We record an elementary, self-contained observation about the accelerated Collatz mapT (m) = (3m + 1)/2ν2(3m+1) on odd integers. Let Dk = ν2(3mk + 1) be the discharge (valuation)sequence of an orbit. We show that a positive integer can reproduce a fixed eventually-periodiclow-discharge valuation word—one whose mean discharge satisfies ¯d < log2 3—for at most¯d−1 log2 m0 + O(1) steps.

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