The Individuality Gap: Why Observational Uniqueness Does Not Establish Person-Specific Structure

Distinctiveness is routinely treated as evidence of individuality. A behavioural trace that nobody else shares, a profile that uniquely identifies its owner, or a sufficiently rich digital record is easily described as showing that the person is individually structured. This paper separates observational uniqueness — whether a representation is a singleton in a declared reference set — from predictive individuality — whether person-specific information improves prediction of unseen behaviour beyond an appropriate shared model — and from the still stronger claim of mechanistic individuality. Under logarithmic score the oracle person-specific gain is shown to be exactly the conditional mutual information between identity and the future outcome given the declared context, which supplies a canonical, model-class-free anchor for the second notion. The two notions are then proved variation independent: observational uniqueness is a functional of the marginal law of the representation, predictive individuality is a functional of the conditional law of the future given identity, and every pair of values in the full product is attainable. The identified set for the oracle gain, given the expected singleton proportion, is therefore the entire parameter space; the same holds under logarithmic score up to the information-theoretic ceiling of log n. Supporting results explain the asymmetry in practice: refining a representation cannot destroy existing singletons, expected uniqueness falls as the comparison class grows, and once collisions become rare the singleton proportion is pinned near one and carries almost no information about the amount of person-specific structure. A finite-sample threshold result protects the argument from its mirror image, showing that genuine person-specific heterogeneity can coexist with a personalized model that performs worse than a shared one. Fixed-seed simulations reproduce every analytic result, including a deterministic construction in which histories collide almost always while person-specific prediction is strictly valuable. The paper does not claim that human beings lack individuality, and it does not claim priority over the observation that identifiability and predictive value can come apart, which was made informally by Finn and Rosenberg (2021) and demonstrated empirically by Mantwill et al. (2022) within neuroimaging. What is offered is the general statement, independent of measurement modality. The deposit includes the manuscript, its LaTeX source, and a complete reproducibility package: fixed-seed simulation code, an independent verification script covering every formal claim, all generated data files and all figures. No external data are required.