This paper addresses an open problem left by Terras (1976) in the probabilistic study of the reduced Collatz map T (n) =n/2 if n even, (3n+1) /2 if n odd. Terras showed that the stopping-time density is determined by counting admissible parity vectors — binary strings whose corresponding congruence class modulo 2ᵏ first decreases at step k — but left their exact enumeration open. We complete this programme by re-indexing admissible vectors by their number of odd steps (the rank p). Each admissible vector of length ⌈log_2 (3) p⌉ with exactly p odd steps corresponds to a unique residue class, and we derive an exact inclusion–exclusion recurrence for the count vₚ of first-descent classes at each rank. The resulting densities aₙ=vₙ/2^⌈γn⌉, with γ=log₂ (3), are exact rational numbers, and the Collatz conjecture reduces to the identity ∑₍=₁^∞ aₙ=1/2. The fine structure of the series is analysed through the Sturmian phase ε (n) =⌈γn⌉−γn, revealing eleven sub-band plateaux, contractive supercycle return products, and a Devil's-staircase profile. MSC 2020: 11B37, 11B83, 60C05
Théo Lesieux (Tue,) studied this question.