When Is a Continuous-Time Optimization Model Dynamically Faithful to Its Discrete Algorithm?

Continuous-time models of optimization algorithms are usually justified by trajectory closeness as the step size vanishes. At a finite step size, however, a discrete update can possess qualitative features that no smooth ordinary differential equation (ODE) on the same state space can reproduce at the iteration time scale. We formalize this distinction through \emph{flow fidelity}: whether an update map can itself be a time map of a smooth ODE. A general orientation theorem gives an immediate obstruction: every ODE time map has strictly positive Jacobian determinant. For gradient descent on a scalar quadratic, this separates the classical stable interval into a flow-compatible regime $0