Individuality Cost in Bits: Optimal Shared Descriptions for Partial Unique Naming

How much shared information is needed to make a prescribed fraction of keys uniquely identifiable? We study partial unique naming: a message selects a map from a set of n keys to available slots, and a key succeeds when its slot contains no other key. Under unlimited input-independent public randomness and unrestricted computation, we determine the exact asymptotic worst-input expected prefix-message rate for a required expected singleton fraction. For n slots and a sparse universe, the rate is $D_2(\alpha\|1/e)$ above singleton fraction $\alpha=1/e$, and zero below it, where $D_2$ is binary relative entropy in bits. Thus full uniqueness costs $\log_2 e\simeq1.4427$ bits per key, while 90 percent singletons cost about 0.8956 bits per key. Matching converses and selection bounds extend to dense integer-ratio universes and expanded slot ranges. For fixed interior targets, expected and hard singleton quotas both have total cost $nR+\tfrac12\log_2 n+O(1)$. We also determine the joint density-range zero-rate frontier and prove positive-rate results in explicit saddle domains. Deterministic covering and finite-public-seed bounds clarify the role of shared randomness. Exact fallback identities, serialized local-search experiments, and contextual observation-selection tests distinguish ideal information rates from implementation costs. Naming, ranking, retrieval and membership are treated as separate information contracts.