Modern generative models can produce visually plausible images while violating exact identity constraints. Evaluating such failures requires distinguishing violations of the requested identity from permissible variation in appearance. We introduce Low-Entropy Visual Identity (LEVI), which characterizes visual identities determined by compact facts and explicit rules. The low-entropy requirement concerns identity uncertainty, while appearance may remain diverse. We formulate identity fidelity in terms of the probability that a conditional image distribution assigns to the identities permitted by the prompt. Visual Identity Spaces (VIS) formalize this distinction by specifying admissible identity values and which different realizations express the same identity. Building on this formulation, we develop Verifiable Visual Grammar (VVG), a declarative language whose typed primitives and compositional constraints describe identity-bearing structures. Using this grammar, we build a benchmark for image generation. It assesses each output against the task's identity requirements, drawing evidence from the final pixels. The curated benchmark contains 1,375 tasks spanning 120 base identity spaces across six domains. Among ten evaluated model configurations, the highest whole-task pass rate is 58.69%. These results reveal a persistent gap between visually plausible images and faithful realization of their specified identities. Project page: https://maxiao0234.github.io/LEVI/.

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