ATI: An Ordered Operator Decomposition for Recursive Dynamics

We introduce the ATI framework, an ordered triadic operator decomposition for discrete time recursive dynamics in a Hilbert space, writing the update map as F = I◦T ◦A (alignment, threshold, continuation). The framework is not an algorithm but a structural lens that isolates functional roles and makes operator ordering explicit. We recall standard fixed-point theory guaranteeing weak convergence of relaxed iterations when F is nonexpansive and Fix(F) ̸ = ∅, and we show by explicit counterexample that reordering the same nonexpansive components can change both the fixed-point set and whether the limit lies in the constraint set Fix(T). We also illustrate how several classical projected and thresholded schemes fit naturally into the ATI form.