The metric and the lattice: Identification limits of modality inference from questionnaire scores

We derive an exact, sharp lower bound for Hartigan's dip on lattice-supported questionnaire scores and characterise two further identification limits on latent-distribution shape. If q(2) is the mass of the second-largest score atom, then D ≥ q(2)/2. Across nine public collections (2.48 million records; 48 score series), all 48 reject under the diptest calibration at a common n = 12,000; 28 lie on the atom floor to four decimals, 34 are within one per cent, and the floor alone crosses the rejection threshold for all 48, with median sample-size crossing n* = 539. After within-cell perturbation removes the lattice, 47 series reject in none of 100 replications and one in all. Latent modality is also not invariant under the strictly increasing reparametrisations left free by a nonparametric item-response model; a five-standard-deviation mixture can be transformed to pass the dip test without changing ranks. Finally, fixing a metric does not identify the latent law: under a continuous finite item-response design, a relative-interior observable admits both atomic and atomless mixing distributions. Fisher-information arc length supplies an item-anchored coordinate, but finite-design identified uncertainty remains.