Language models prompted with cultural personas increasingly stand in for human respondents in cross-cultural research. Their responses separate personas cleanly, and that separation is read as evidence of a cultural point of view. We show that the separation is real, that the point of view is not, and that one criterion tells them apart. A trait is structure internal to one respondent that survives a change of measurement frame; a bias needs only group-specific item means. To test for the first, we represent a single response set as an Item--Dimension matrix and treat its correlation matrix as a point on the manifold of symmetric positive definite matrices. In humans this carries what a trait should: it reproduces across test--retest sessions sharing no items, order or context ($r=0.77$, $N=89$); on public NEO-PI-R data it identifies individuals at up to $76\%$ against a $0.4\%$ chance level ($N=263$); and it predicts GPA ($R^2=0.281$, $p=0.003$) where BigFive aggregates from the same responses predict nothing ($R^2=0.018$). In four frontier LLMs it returns nothing. Persona structure is readable only while every instance shares one item order: give each its own order and separation falls from $94.7\%$ to chance, while realigning instances to \emph{any} shared random order restores it to $82$--$84\%$. Responses generated independently item by item, with no latent structure, reproduce the entire pattern. The cultural signal is a group template, not a property of any instance, and alignment regimes differ only in which stereotype survives on the surface.

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