Quantitative synchronization of Collatz trajectories
Abstract
We study simultaneous trajectories of the ordinary Collatz map. A balanced coupling of the shortcut trajectories gives a uniform absorption estimate of order √((1 + log(1 + |d|))/h) for offset d and shortcut horizon h. Its realization preserves both the ordinary clock and the number of halvings. Consequently Gao's finite-horizon dyadic coalescence proportions tend to one, as first proved publicly by Said Duran, and the balanced deficiency is Θ(k−1/2) at horizon k, under Gao's original requirement that merger occur before either trajectory reaches 1. The upper bound is deduced from the pair estimate; the matching lower bound comes from a converse clock dictionary from balanced ordinary merger to symbolic absorption. The leading constant is determined in a companion paper. For every prescribed L(n) = o(√(log n)) and every fixed log4 3 < c < 1, combining an earlier merger deadline with Korec's theorem gives a common value below nc for all starts n, n+1, …, n+L(n), at one ordinary time t ≤ 2 log2 n, for natural-density-almost every n. Combining adjacent-pair absorption with Tao's almost-bounded minimum theorem yields a simultaneous conclusion: for every prescribed L(n) = o(√(log n)) and every f(n) → ∞, logarithmic-density-almost every n admits one ordinary time t and one positive value y < f(n) shared by all starts n, n+1, …, n+L(n). Both arguments first prove early merger and then force the chosen descent witness to occur after merger. No convergence-to-one assumption is used, and the exceptional integer sets are not removed. The natural-density minimum law, proved in a companion paper, upgrades the arbitrary-threshold window statement to natural density for the attained minimum and its first ordinary clock. Companion papers: Brownian motion in Collatz orbits; Exact rates in Collatz synchronization; Gao's dyadic density-one coalescence conjecture; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.
Related
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- A Complete Solution to the Conjecture on the Inverse Collatz–Sinogowitz Irregularity Problem
- A variant of Collatz's conjecture over binary polynomials
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- Classification of the Orbits of Three Quadratic Collatz-type Maps
BibTeX
@misc{leen2026quantitative,
title = {{Quantitative synchronization of Collatz trajectories}},
author = {David Leen},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23268754},
url = {https://doi.org/10.5281/zenodo.23268754},
note = {CC BY},
}
RIS
TY - UNPB
TI - Quantitative synchronization of Collatz trajectories
AU - David Leen
PY - 2026
DA - 2026/10/09
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23268754
UR - https://doi.org/10.5281/zenodo.23268754
AB - We study simultaneous trajectories of the ordinary Collatz map. A balanced coupling of the shortcut trajectories gives a uniform absorption estimate of order √((1 + log(1 + |d|))/h) for offset d and shortcut horizon h. Its realization preserves both the ordinary clock and the number of halvings. Consequently Gao's finite-horizon dyadic coalescence proportions tend to one, as first proved publicly by Said Duran, and the balanced deficiency is Θ(k−1/2) at horizon k, under Gao's original requirement that merger occur before either trajectory reaches 1. The upper bound is deduced from the pair estimate; the matching lower bound comes from a converse clock dictionary from balanced ordinary merger to symbolic absorption. The leading constant is determined in a companion paper. For every prescribed L(n) = o(√(log n)) and every fixed log4 3 < c < 1, combining an earlier merger deadline with Korec's theorem gives a common value below nc for all starts n, n+1, …, n+L(n), at one ordinary time t ≤ 2 log2 n, for natural-density-almost every n. Combining adjacent-pair absorption with Tao's almost-bounded minimum theorem yields a simultaneous conclusion: for every prescribed L(n) = o(√(log n)) and every f(n) → ∞, logarithmic-density-almost every n admits one ordinary time t and one positive value y < f(n) shared by all starts n, n+1, …, n+L(n). Both arguments first prove early merger and then force the chosen descent witness to occur after merger. No convergence-to-one assumption is used, and the exceptional integer sets are not removed. The natural-density minimum law, proved in a companion paper, upgrades the arbitrary-threshold window statement to natural density for the attained minimum and its first ordinary clock. Companion papers: Brownian motion in Collatz orbits; Exact rates in Collatz synchronization; Gao's dyadic density-one coalescence conjecture; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.
ER -