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 log₂3, built from the law ν of the terminal residue R(n) = Σ log₂(1 + 1/(3u)) over the odd values u of the orbit. Every such profile, on B or on an invariant subset of positive density, is an average of translates of the sawtooth (log 3/2)·2z mod log₂3. Hence the total variation along clocks is at most 1/2 and the ℓ¹ distance to slowly varying sequences at most 2 − √3, with equality exactly when the terminal law modulo log₂3 is a point mass. Given positive predecessor density, the starts whose orbit passes through 2j approach both bounds as j → ∞, without numerical input. On B, a slowly varying approximation exists only if ν is uniform modulo log₂3. A computer-assisted residue bound excludes this: the distance is at least 0.00718, and a positive proportion of central clocks is empty in every window [X, (1+δ)X) with δ < 0.254. That proportion is given exactly by the gaps of the support of ν. Builds on A Gaussian law for Collatz total stopping times (Zenodo 23046896). Companion paper: Brownian motion in Collatz orbits. The single numerical input is documented in the computational supplement.