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 predictions for the descent of the orbit of n to its minimum m(n) hold at the scale √(log n): the time to the minimum, centered at the model's mean μ log n and rescaled by σ√(log n), converges in distribution to the model's standard Gaussian, and the rescaled descent converges to the model's Brownian motion, independently of m(n) and of residues modulo any fixed q. On the convergent set this time is the total stopping time. Two predictions of the model are not proved: its extremal constant γRW ≈ 41.68, where every coefficient below 7.6269 is proved, and its prediction that almost every orbit reaches 1, which, given positive predecessor density (proved in a Lean development of Mazur, not refereed), is equivalent to the Collatz conjecture. We assemble, from results proved in companion preprints and in the work of Said Duran and of Mazur, a five-part theorem on the orbits of natural-density-typical integers that assumes nothing about the Collatz conjecture. (I) The descent of one orbit to its minimum is a Brownian motion, independent of the minimum, on every arithmetic progression, with Gaussian moderate deviations for the capped clock at every fixed fraction s < 1 of the descent. (II) At unit scale the stopping-time histogram on the convergent set is a Gaussian times a periodic profile of period log2 3, an average of translates of one sawtooth; its total variation along clocks is at most 1/2 and its ℓ1 distance to slowly varying sequences at most 2 − √3, with equality exactly when the terminal residue law modulo log2 3 is a point mass. (III) Consecutive integers coalesce, Gao's deficiency is Θ(k−1/2) with leading constant √6·E[M]/√π, late mergers have a Brownian-meander shape, and blocks of o(√(log n)) consecutive starts share their minimum and its exact first-attainment time. (IV) Every large octave [X, 2X) contains at least Xe convergent integers with stopping time at least c log n, for every c < 7.6269 with an exact window certificate and for every c < 7.503325 without any Collatz computation. (V) The tilted 3-adic Syracuse laws are absolutely continuous with finite relative entropy at tilt 1/4 (kernel-checked) and for every tilt in [1/2, 0.550024], which contains Tao's fair tilt; at every tilt θ their entropy dimension is min(1, h(θ)/(θ log 3)), and at no tilt θ ≥ 1/2 do they have a square-integrable density. Our main new result is a fate–phase theorem, proved in full. Over all integers, with no hypothesis on the Collatz conjecture and no positivity input, the histogram of the time to the minimum is, in ℓ1, a Gaussian times a periodic profile of period log2 3: the mixture, over the fates r = m(n), of profiles shifted by log2 r. On every set {m = r} of positive density the time to the minimum has the corresponding shifted local law. At the scale √(log n) the fates are indistinguishable; among the statistics considered, the clock sees the fate only through this lattice phase. We list which model predictions are now theorems, in which density, and which remain open; the open ones are the extremal constant, the slow-side moderate-deviation bound for the time to the minimum, and density-one convergence. Nothing here proves the Collatz conjecture.