Explicit Criteria for Finite-Time Identification in Sparse Primal--Dual Dynamics

We study when selected coordinates become and remain zero in a finite-dimensional sparse-regression primal–dual flow and its exact full resolvent iteration. Invariant balls centered at $(0,z)$, with $z$ strictly dual feasible, give computable deadlines without rank assumptions or optimizer knowledge. The certificates may be available at initialization. We determine the smallest invariant radius for the continuous flow within this family. For a prescribed positive feasibility margin, center selection is convex and admits an exact finite conic formulation. We bound the coordinate margins for all centers in this family and give a normalized correlated example attaining the bound. A composite Bregman Lyapunov estimate proves rank-free convergence without primal uniqueness and finite identification under strict dual margins. Under restricted injectivity, the discrete gap-based deadlines multiplied by the step size tend to the continuous bound for fixed data, observed state, and certificate parameters. Numerical experiments assess certificate timing. A noisy weak-form oscillator regression illustrates the difference between fitted and physical support.