This article develops a structural description of odd runs in the accelerated Collatz dynamics based on the intrinsic odd depth of line indices. Odd depth is an index‑immanent invariant: it is fixed by the arithmetic structure of the line index and is not created or modified by the Collatz orbit. Each odd run simply exhausts the intrinsic odd depth encoded in its starting index. A key structural result is that every odd step necessarily moves to a line index whose intrinsic odd depth is exactly one less than that of its predecessor. Thus the length of any odd run is fully determined by the odd depth of its initial index. Most class‑3 indices lie in fractal side branches and possess only small odd depth, producing short odd runs. Large odd depth occurs only on the staircase, whose indices become exponentially sparse. This dual mechanism — the rarity of indices with large intrinsic depth and the deterministic stepwise reduction of depth — confines long odd runs to a vanishingly small subset of the line‑index space. The article introduces the B‑function as the structural recursion underlying intrinsic odd depth and shows how odd‑depth–controlled dynamics supersede the traditional class‑controlled view of the accelerated Collatz map. This dual mechanism — the rarity of indices with large intrinsic depth and the deterministic stepwise reduction of depth — confines long odd runs to a vanishingly small subset of the line‑index space. The article introduces the B‑function as the structural recursion underlying intrinsic odd depth and shows how odd‑depth–controlled dynamics supersede the traditional class‑controlled view of the accelerated Collatz map.
Karl Huber (Mon,) studied this question.