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Synthetic populations, large sets of simulated people, are increasingly used in place of real people to pretest surveys, simulate users and communities, and evaluate AI systems. Each simulated person is defined by a persona, a written profile of fixed traits, demographics, and attitudes that a language model is instructed to act out. Every such system assumes that a person's identity is a stable attribute that behaves the same way in every situation the simulation presents. We show that this assumption is not a harmless simplification. Three decades of personality research find the opposite. How a given person behaves varies more across contexts than people vary from one another. Drawing on identity economics, which describes how different situations activate different identities and norms within the same person, we separate behavior into three parts, which are stable traits, the identity a context calls up, and the situation itself. We then prove, under two stated assumptions, that any simulation built on fixed personas, no matter how detailed the personas or how large the underlying model, cannot reproduce the part of a population's behavior that is specific to it and its circumstances. We propose a new evaluation criterion based on this result and a person-in-context design that can meet it in principle. We test one version on a naturalistic Reddit personality corpus and on a published clinical-trial recruitment survey of a community with documented distrust of medical institutions. The design does produce context-dependent behavior when tested directly, but in neither setting does it fully match the real population, and we show where it falls short. The consequence for practice is a risk that synthetic-population evaluations misrepresent the groups whose behavior is most shaped by context, a distortion that richer personas cannot correct.