Interior observations for differential equations with piecewise constant argument: compatibility, identifiability, and conditioning

For differential equations with piecewise constant argument on nonuniform intervals of constancy, an interior value is treated as an observation of the unknown preceding partition-point state. For linear systems, we characterize attainable data, derive exact first-order asymptotics for the smallest singular value of nearby weighted observation matrices, and obtain explicit lower bounds at positive observation separations that certify full column rank under simultaneous coefficient and observation-time uncertainty. For nonlinear equations, variational and projected Volterra estimates yield fold criteria, joint identification, a global Lipschitz graph representation of compatible data, quantitative residual--distance bounds, and noise estimates that distinguish components parallel and orthogonal to the linear observation space. A logistic population model illustrates inverse-branch coalescence and exact two-time identification. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/90/abstr.html