Description

Companion code to:

Negative Drift and State Instability in a Bitwise System Equivalent to the Collatz Conjecture

and

Deterministic Limits and Ergodic Properties of a Bitwise Syracuse Map

This repository contains code and scripts to reproduce results in the above research papers regarding the Stochastic Stability of the Collatz Conjecture. It implements an arbitrary-precision bitwise map simulation capable of analyzing trajectories exceeding 10^7 bits and statistical analyses of bit-span dynamics, and a Ulam Method estimation of the transfer operator's spectral gap.

Mathematical Significance: The Zero-Noise Limit

This framework addresses the Collatz Conjecture by reducing it to a specific problem of stochastic stability in the zero-noise limit.

The Reduction

The code implements a bitwise dynamical system $U(n)$ which acts as a perturbed irrational rotation on the 2-adic integers. We define the dynamics as: $$T(x) = T_{asymp}(x) + \epsilon_j(x)$$ Where $T_{asymp}$ is a pure rotation (ergodic but non-mixing) and $\epsilon_j$ is a state-dependent perturbation that decays as the bit-span $S \to \infty$.

The Open Question

The central contribution of this work is the formal reduction of the conjecture to a single open question of ergodicity: Does the invariant measure of the system remain absolutely continuous in the limit as $|\epsilon| \to 0$?

• ACIM Stability: We prove that if the invariant measure possesses any continuous probability density function (not necessarily Benford), the low-order bits of the orbit must be uniformly distributed.
• Negative Drift: A uniform distribution of low-order bits necessitates a global negative drift in the bit-span metric ($E[\Delta S] \approx -0.41$), rendering divergence impossible.
• Singular Collapse: Divergence is therefore only possible if the invariant measure collapses onto a singular support as the perturbation vanishes.

Empirical Validation

This repository provides the tools to test this stability. The collatz executable can simulate trajectories exceeding $10^7$ bits, demonstrating that the system's mixing properties and drift metrics remain scale-invariant even as the perturbation magnitude approaches $2^{-10,000,000}$. This suggests the system is stochastically stable and the absolutely continuous measure persists.

Requirements

This code requires:

• The GNU MP BigNum library (GMP), available here: https://gmplib.org/
• The Eigen3 library for linear algebra, available here: https://eigen.tuxfamily.org
• The Spectra library for large scale eigenvalue problems, available here: https://spectralib.org/

If you are using a Ubuntu based system you should be able to install all dependencies with apt:

sudo apt install libeigen3-dev libspectra-dev libgmp-dev

Building

The supplied Makefile should build everything for you:

make

Running

Main program

The primary executable that computes the bit-span metrics can accept the number of random starting values and the maximum starting value size, in bits:

./collatz [num values] [num bits]

The defaults are "num values" = 1000000 and "num bits" = 128. You can run singular trajectories with massive starting values such as:

./collatz 1 1000000

Recreating the plots

If you want to recreate the plots from the papers, or to generate the data for your own exploration, use

make generate-plots

or

make generate-data

respectively. Expect it to take about 20 minutes. Note that plot generation requires R with ggplot2 and dplyr packages. Installing these on Ubuntu based systems can be accomplished with:

sudo apt install r-base r-cran-ggplot2 r-cran-dplyr

Generated plots include:

• benford-dist.eps - Plots the distribution of the fractional position of iterates (Paper 2, Figure 2)
  • The data for this plot is fractional-position.csv
• autocorrelation.eps - Plot the autocorrelation of the fractional positions over sequence lags (Paper 2, Figure 3)
  • The data for this plot is autocorrelation.csv
• spectral-gap.eps - Log-log plot of the spectral gap vs input bit size (Supplement, Figure 1)
  • The data for this plot is ulam-gap-out.csv
• delta-L-freq.eps - Semi-log plot of LSB shift distribution (Supplement, Figure 2)
  • The data for this plot is delta-L-freq.csv