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 motion, independent of where the orbit ends. For n uniform in [2, X], the time to fall to or below n1−s (or to reach the minimum, if sooner), centered at μ s log n and scaled by σ√(log n), converges as a process in s ∈ [0, 1] to standard Brownian motion, jointly with and independently of m(n); here μ = 2/log(4/3) and σ² = 2(log 3)²/(log(4/3))³. The same holds on every arithmetic progression, in logarithmic density, and jointly with the residue that produces the teeth of the stopping-time histogram. Consequently half of all positive integers take at least μ log n steps to reach their minimum, and the fractions of the descent spent above its chord and behind its average schedule follow the uniform and arcsine laws. For fixed 0 < s < 1 the clock has Gaussian moderate deviations with relative error up to K√(log n log log n) for every K, and for s > 1/1001 also given a fate of positive density. Every set invariant under the map above some point has a natural density. The argument uses no numerical computation. This paper extends A Gaussian law for Collatz total stopping times (Zenodo 23046896). Companion paper: Gaussian but not locally Gaussian.