Exact Finite-Level Mixing Geometry in Accelerated Collatz Dynamics

Hiroki Kamanoi
Draft — May 2026

Paper B in a series on Collatz dynamics through finite Markov chain theory.

Overview

This paper establishes the exact finite-level mixing geometry for the accelerated Collatz map modulo $2^K$.

The central result is an exact arithmetic progression support geometry and an exact total variation formula for the Haar kernel. The paper isolates a fundamental separation between population-level geometric mixing and individual orbit arithmetic decorrelation.

Main Results

The paper establishes:

• exact arithmetic progression support geometry for the Haar kernel $P_K^{\mathrm{Haar}}$;

• exact total variation formula for $P_K^{\mathrm{Haar}}$;

• exact mixing time:

  $$T_\mathrm{mix}(K) = K - 1$$

• dual inverse-tree representation;

• distinction between exact Haar kernels and lift-averaged empirical kernels $P_{K,M}$.

Important Clarification

This repository does not claim a proof of the Collatz conjecture.

It also does not claim:

• orbitwise ergodicity or arithmetic decorrelation;
• asymptotic spectral rigidity.

The exact finite-level support geometry is proved rigorously for the Haar kernel. Numerical spectral observations concern lift-averaged empirical kernels $P_{K,M}$ and are not theorems.

Logical Structure of the Series

Paper	Core result	Status
A	Exact operator structure and renewal systems	Proved
B (this paper)	Exact TV mixing: $T_\mathrm{mix}(K) = K-1$	Proved
C	One-bit spectral jump $\delta_{K,1} \approx 0.29$; Open Gap Problem	Numerical / Open
D	Simultaneous scale coherence; conditional diverging mixing	Open / Conditional

Files

• collatz_mixing_paper.tex — LaTeX source
• collatz_mixing_paper.pdf — compiled draft
• REVIEW_NOTES.md — critical notes on the kernel distinction

Suggested arXiv Categories

Primary: math.DS
Secondary: math.NT, math.PR

License

MIT License

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Related work

This repository is part of a series on Collatz dynamics:

Repository	Description
collatz-renewal-structure	Paper A: exact renewal structure and operator theory
collatz-finite-mixing-geometry	Finite-level mixing geometry
collatz-spectral-amplification	Paper C: spectral amplification
collatz-scale-coherence	Paper D: scale-coherence rigidity

Author

Hiroki Kamanoi
hirokikamanoi@gmail.com
https://github.com/KAMANOI