What question did this study set out to answer?

The aim is to demonstrate a local finite theorem for modulus 81 in the context of the 3x+1 map.

June 2, 20260 citationsOpen Access

The Local Modulus-81 Theorem

Key Points

The aim is to demonstrate a local finite theorem for modulus 81 in the context of the 3x+1 map.
Utilized a finite reverse-transport model associated with the 3x+1 map.
Applied exact integer and rational arithmetic for computational verification.
Included a reproducibility package comprising various audit scripts and output data.
Confirmed the local finite theorem but did not establish global convergence or higher-scale persistence.
Provided computational evidence through recomputed finite objects, including matrix and vector data.
Presented verified outputs and checks for pairing-law counterexamples but did not affirm the global 3x+1 conjecture.

Abstract

This record contains the manuscript and reproducibility package for The Local Modulus-81 Theorem. The paper proves a local finite theorem at modulus 81 for a finite reverse-transport model associated with the 3x+1 map. The result is local and finite in scope: it does not prove global convergence, higher-scale persistence, a forward odd-only bridge, or the global 3x+1 conjecture. The accompanying archive includes: PDF manuscript, LaTeX source, BibTeX file, audit scripts, audit outputs, SHA256 manifest, compile report, and verification report. The computational checks use exact integer and rational arithmetic to recompute the finite objects used in the paper, including the matrix/vector data, residual identities, witness values, dyadic channel checks, and the pairing-law counterexample.

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