A random sequence sampled from a known continuous distribution is observed with the objective to choose an item with the overall rank one. A rejected item cannot be recalled and is immediately erased from the memory. Under this memory constraint, the choice problem is not amenable to recursive methods of optimal stopping and becomes a global optimisation task. We focus on a heavy-traffic form of the problem with infinitely many choice opportunities, which we state in terms of a planar Poisson process (PPP). Symmetries of the PPP are used to derive basic structural properties of the optimal stopping rule, including the balance at the boundary equation, and two key integral identities. Throughout, we make throrough comparison to the classic full-information counterpart of the problem, revisiting both discrete- and continuous-time models. The optimal value, stopping rule and other characteristics of the problem are determined analytically and approximated numerically with high precision.
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