A Gaussian law for Collatz total stopping times
Abstract
Let B be the set of positive integers whose shortcut Collatz orbit reaches 1, and let τ(n) be the first hitting time. We prove that B has a natural density h > 0 and that, for every real t, dens{ n ∈ B, n > 1 : (τ(n) − c log n) / √(v log n) ≤ t } = h Φ(t), where c = 2/d, v = 2(log 3)²/d³, d = log(4/3), and Φ is the standard normal distribution function. The proof combines a sum-conditioned arithmetic mixing theorem with a positive lower bound for the density of B. A single first-passage comparison, carrying a phase that records the elapsed time, gives both conclusions: at frequency zero it makes the basin proportions converge, and at frequencies of order (log n)^(−1/2) it produces the Gaussian factor. An elementary packing estimate transfers the Syracuse clock to the total stopping time. Consequently the convergent starts with τ(n) ≥ c log n have density h/2. This answers the exact-threshold infinitude question highlighted by Kontorovich and Lagarias and confirms, on the convergent basin, their prediction that half of all starts lie at or above this threshold. The same density holds after every o(√log n) change of threshold. We also obtain a joint law expressing asymptotic independence of the normalized clock, the starting location, and the initial power of two.
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BibTeX
@misc{leen2026gaussian,
title = {{A Gaussian law for Collatz total stopping times}},
author = {David Leen},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = sep,
doi = {10.5281/zenodo.23046895},
url = {https://doi.org/10.5281/zenodo.23046895},
note = {CC BY},
}
RIS
TY - UNPB
TI - A Gaussian law for Collatz total stopping times
AU - David Leen
PY - 2026
DA - 2026/09/29
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23046895
UR - https://doi.org/10.5281/zenodo.23046895
AB - Let B be the set of positive integers whose shortcut Collatz orbit reaches 1, and let τ(n) be the first hitting time. We prove that B has a natural density h > 0 and that, for every real t, dens{ n ∈ B, n > 1 : (τ(n) − c log n) / √(v log n) ≤ t } = h Φ(t), where c = 2/d, v = 2(log 3)²/d³, d = log(4/3), and Φ is the standard normal distribution function. The proof combines a sum-conditioned arithmetic mixing theorem with a positive lower bound for the density of B. A single first-passage comparison, carrying a phase that records the elapsed time, gives both conclusions: at frequency zero it makes the basin proportions converge, and at frequencies of order (log n)^(−1/2) it produces the Gaussian factor. An elementary packing estimate transfers the Syracuse clock to the total stopping time. Consequently the convergent starts with τ(n) ≥ c log n have density h/2. This answers the exact-threshold infinitude question highlighted by Kontorovich and Lagarias and confirms, on the convergent basin, their prediction that half of all starts lie at or above this threshold. The same density holds after every o(√log n) change of threshold. We also obtain a joint law expressing asymptotic independence of the normalized clock, the starting location, and the initial power of two.
ER -