Exact first-decrease densities for the Collatz map and their Sturmian fine structure

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 2^k 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 v_p of first-descent classes at each rank. The resulting densities a_n=v_n/2^⌈γn⌉, with γ=log⁡_2(3), are exact rational numbers, and the Collatz conjecture reduces to the identity ∑ₙ₌₁^∞ a_n=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