This work investigates the fundamental limits of \emph{perfect} covert communication assisted by an Intelligent Reflecting Surface (IRS). We first characterize the necessary and sufficient conditions for perfect covertness, defined as zero received energy at the warden (Willie), and provide a complete analytical solution for the two-element case. For general array sizes, we prove that the probability of achieving perfect covertness converges to $1$ almost surely as the number of reflecting elements increases. To enable practical implementation, we formulate the phase design problem as an interference-minimization problem. A key contribution is proving that the objective satisfies the \emph{strict-saddle} property. Consequently, we establish that gradient descent with random initialization converges almost surely to a global minimizer (zero interference), thereby identifying a feasible, perfectly covert configuration whenever one exists. Finally, to address physical limitations, such as imperfect CSI, we introduce the notion of \emph{operational perfect covertness}. In this setting, we assume a bounded CSI error model and an analogous constraint on Willie. We derive robust conditions guaranteeing that the warden's detection capability remains effectively nullified even when exact signal cancellation is infeasible.