David Leen

The mean relative maximum of Collatz trajectories

For the ordinary Collatz map C(n)=n/2 for even n and C(n)=3n+1 for odd n, let H(n)=sup over j>=0 of C^j(n), allowing positive infinity. This self-contained note proves that liminf as N tends to infinity of (1/(N log N)) times the sum over 1<=n<=N of H(n)/n is at least log(27/16)/...

Quantitative synchronization of Collatz trajectories

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 num...

Gao's dyadic density-one coalescence conjecture

We give a short proof of the density-one assertion posed by Gao for consecutive Collatz starts, independent of the first public proof, due to Said Duran (September 2026). Let C be the ordinary map and let H(n, m) count halving branches among the first m ordinary steps. Write Gk(n...

Exact rates in Collatz synchronization

Gao conjectured that his dyadic proportions of coalescing adjacent integers tend to one. We determine the exact deficiency: 1 − d̄k ~ β/√k, with β = √6·E[M]/√π and 1 ≤ E[M] < ∞, where M counts the excursions of the canonical adjacent-pair coupling. The balanced shortcut absorptio...

Brownian motion in Collatz orbits

For a positive integer n let m(n) be the least value of its Collatz orbit and τ*(n) the first time the orbit reaches it; both are finite for every n, and nothing is assumed about the Collatz conjecture. We prove that the descent of a typical orbit to its minimum is a Brownian mot...

The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits

The stochastic models of the 3x+1 problem of Lagarias and Weiss, Borovkov and Pfeifer, Kontorovich and Sinai, and Kontorovich and Lagarias treat the logarithm of an orbit as a random walk. The title is meant in the following sense. For natural-density-typical n, the model's predi...

Late synchronization and the geometry of Collatz families

We determine the geometry of Collatz trajectories conditioned on late synchronization. Their absolute odd-count separation converges, in the physical shortcut clock, to a Brownian meander continued to its first zero. One excursion carries asymptotically all the duration and heigh...

Tilted 3-adic Syracuse laws: absolute continuity, dimension and the limits of L²

Let g1, g2, … be independent with P(g = k) = θ(1 − θ)k−1 for k ≥ 1, and let ρ = Σj≥1 3j−12−(g1+⋯+gj) in the 3-adic integers. The law μθ of ρ is a 3-adic self-similar measure for the infinitely many maps x ↦ 2−g(1 + 3x), each contracting by exactly 1/3; at θ = 1/2 it is Tao's Syra...

Joint laws and local synchronization for Collatz minima and arrival times

Let m(n) be the least value on the ordinary Collatz orbit of n, and let τ(n) be its first-attainment time. We prove a spatial transfer theorem for the entire discrete pair (m(n), τ(n)). From uniform pair coalescence and a limiting minimum law, its empirical distribution becomes i...

Gaussian but not locally Gaussian: the teeth of the Collatz stopping-time histogram

Let B be the set of positive integers whose Collatz orbit reaches 1, and τ(n) the total stopping time. On B, τ satisfies a central limit theorem, but it is not locally Gaussian: in ℓ¹, the histogram of τ on B ∩ (1, X) is a Gaussian times a periodic profile Ω(k − log₂X) of period...