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March 21, 2026Open Access

The Characteristic Ideal of the Collatz–Iwasawa Module and the Character Decomposition Bridge

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MEMatthew Eltgroth

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Overview

This work introduces the Collatz–Iwasawa module to analyze 2-adic integer behavior, suggesting a new perspective on the Collatz conjecture.

Key Points

  • The aim is to define the Collatz–Iwasawa module to study the algebraic relationship between the Syracuse return map and the tripling process.
  • Defined the Collatz–Iwasawa module as the cokernel of the operator difference between the Syracuse map and multiplicative action by 3.
  • Constructed a transition matrix on the orbits to compute finite-level Fitting ideals.
  • Analyzed the spectral properties through a character decomposition bridge established via Fourier analysis.
  • Demonstrated nonvanishing of the characteristic ideal for finite levels at the trivial character.
  • Showed that Fitting ideals factor under specific conditions involving 2-adic units.
  • Identified spectral gaps and positive-drift fraction convergence towards 1/2, supporting a finite-dimensional framework for the Collatz conjecture.

Cite This Study

Matthew Eltgroth (2026) studied this question.

synapsesocial.com/papers/69be38da6e48c4981c679938https://doi.org/10.5281/zenodo.19111857
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