Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective

Nicolò Bonacorsi; Matteo Bordoni

stat.MLMar 4, 2026

Bayesian Modeling of Collatz Stopping Times: A Probabilistic Machine Learning Perspective

Authors: Nicolò Bonacorsi, Matteo Bordoni

Organizations: Department of Applied Mathematics, Columbia University, New York, USA. · Department of Mathematics, Columbia University, New York, USA.

Abstract

We study the Collatz total stopping time $τ(n)$ over $n\le 10^7$ from a probabilistic machine learning viewpoint. Empirically, $τ(n)$ is a skewed and heavily overdispersed count with pronounced arithmetic heterogeneity. We develop two complementary models. First, a Bayesian hierarchical Negative Binomial regression (NB2-GLM) predicts $τ(n)$ from simple covariates ( $\log n$ and residue class $n \bmod 8$ ), quantifying uncertainty via posterior and posterior predictive distributions. Second, we propose a mechanistic generative approximation based on the odd-block decomposition: for odd $m$ , write $3m+1=2^{K(m)}m'$ with $m'$ odd and $K(m)=v_2(3m+1)\ge 1$ ; randomizing these block lengths yields a stochastic approximation calibrated via a Dirichlet-multinomial update. On held-out data, the NB2-GLM achieves substantially higher predictive likelihood than the odd-block generators. Conditioning the block-length distribution on $m\bmod 8$ markedly improves the generator's distributional fit, indicating that low-order modular structure is a key driver of heterogeneity in $τ(n)$ .

Figures & tables

Figure 1: Empirical distribution of $\tau(n)$ for $1\leq n\leq N$ (integer-aligned bins, width 2) with a KDE overlay computed on a large subsample to reduce noise. This motivates an overdispersed count likelihood.

Statistic	Value
$N$	10,000,000
$\tau_{\min}$	0
$\tau_{\max}$	685
$\widehat{\mathbb{E}}[\tau]$	155.272
$\widehat{\mathrm{Var}}(\tau)$	3814.045
$\widehat{\mathrm{Var}}(\tau)/\widehat{\mathbb{E}}[\tau]$	24.564

Table 1: Summary statistics for

\tau(n)

over

1\leq n\leq N

Figure 2: Scatter of $\tau(n)$ vs. $n$ (log- $x$ ). The mean increases slowly and is approximately linear as a function of $\log n$ , while the spread grows with $n$ ; banding suggests modular structure, motivating $\log n$ and $n\bmod 8$ as covariates.

Figure 3: Posterior predictive check for the hierarchical NB2-GLM (Model M3). The PPC matches the bulk well and mildly overestimates extreme right-tail mass.

Figure 4: Empirical block-length distribution $\hat{p}_{k}$ for $K=v_{2}(3m+1)$ (odd $m\leq N$ ) vs. geometric reference $2^{-k}$ on log- $y$ . This evaluates the “geometric $K$ ” heuristic.

Figure 5: Dirichlet posterior for $(p_{k})$ (log scale) with uncertainty bars, compared to the geometric reference $2^{-k}$ .

Figure 6: Posterior mean of $p_{k}$ conditional on the odd residue class $m\bmod 8$ (heatmap). This reveals systematic arithmetic dependence beyond an i.i.d. geometric law.

Figure 7: PPC overlay: empirical $\tau$ vs. G1 (geometric $2^{-k}$ ), G2 (global calibrated $p_{k}$ ), and G3 ( $p_{k}$ conditional on $m\bmod 8$ ). Conditioning reduces the bias in the mean and improves distributional agreement in the bulk.

Model	Log score (higher better)	$W_{1}$ distance (lower better)
NB2-GLM (M3)	$-272\,911.95$	$3.199$
Odd-block G2 (global $p_{k}$ )	$-1\,165\,983.43$	$17.594$
Odd-block G3 ( $p_{k}\mid m\bmod 8$ )	$-1\,079\,086.65$	$5.434$

Table 2: Quantitative comparison on held-out data using a proper log predictive score and a distributional distance. All scores are computed on

N_{\mathrm{test}}=50{,}000

held-out pairs, with the generative log scores estimated via

S_{\mathrm{MC}}=40

Monte Carlo replicates per test point.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Example trajectory in log scale for $n=27$ : deterministic Collatz vs. stochastic odd-block approximation.

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Exploring Collatz Dynamics with Human-LLM Collaboration

Edward Y. Chang

Computer Science, Stanford University · QuadriumAI

Dynamical Systems

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We present a comprehensive structural analysis of the Collatz conjecture through ~1014 computational experiments yielding 630 formal results. By systematically deploying 29 distinct mathematical paradigms--including transfer operator spectral theory, S-unit equations, p-adic interpolation, martingale methods, modular sieving, formal language theory, cascade algebra, discrete logarithm obstruction, and Diophantine approximation--we establish a Paradigm Exhaustion Theorem: every known framework for promoting distributional convergence ("almost all orbits descend") to pointwise convergence ("all orbits descend") encounters an irreducible structural obstruction when applied to the Syracuse map. On the unconditional side, we prove: (i) the Syracuse transfer operator has a uniform spectral gap for all M, implying equidistribution modulo any power of 2; (ii) any nontrivial cycle of length L satisfies D > 2^F for all L >= 3, giving ord_D(2) > F and F+1 distinct residues mod D; (iii) divergent starting points have natural density 0 and Hausdorff dimension ~0.68; (iv) the formal language of divergent-compatible v-sequences is not context-free; (v) cylinder-averaged density-1 convergence is proved unconditionally via spectral contraction on the invariant core I_2; (vi) a discrete logarithm triple filter achieves 100% cycle blockage for all L tested. We identify the Distributional-to-Pointwise Gap as the irreducible core and prove it equivalent to the divergence component. The modular sieve is permanently nonempty via the Mersenne Bypass. The present work is not a proof of the Collatz conjecture; it characterizes why the conjecture resists proof. The 29-paradigm exhaustion constitutes the most comprehensive structural survey of Collatz attack surfaces to date. Produced through human-LLM collaboration; see Section 12.

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