Consider the function C(n) = {█(3n+1, n≡1 mod 2.@n/2, n≡0 mod 2.)┤ The 3x + 1 problem or Collatz conjecture asks if all trajectories iterating recursively contain one. This holds empirically but eludes theoretical proof. Remove even entries to obtain the Syracuse function, T(u), mapping odd numbers u , T(u) = (3u +1)/2r, where r maximal for T(u) odd. Here we show by expressing u as 4x ±1, a fractal regularity given by the amplitude of a periodic function passing through the x axis in the curves defined y=N cos⁡(πx/2^N ) ≡ N mod((x+2^(N-1))/2^N ,1), with ℕ the natural numbers. The expression describes the length of ascending segments R and with translation, r the number of divisions in T(u). Algebraic analysis reveals the Syracuse function’s image neatly partitions onto arithmetic progressions 6k ± 1, as a function of r. This confirms that all natural trajectories in the 3x + 1 problem contain one..