Latent-variable models on spheres and hyperbolic spaces usually draw a Gaussian in the tangent space at a base point and push it onto the manifold. On these spaces the distance from the base point is the coordinate that carries meaning: depth in a hierarchy, the angle of a rotation, the deviation of a protein frame from a reference. We show that the standard construction silently replaces whatever distance distribution the modeler intended with a fixed one, a scaled chi law whose shape no setting of the scale can change. We then solve the reverse problem. Given the intended distance distribution, we derive in closed form the tangent density that realizes it, prove it is the only isotropic choice with chart-independent likelihoods for a broad class of charts, and prove a lower bound with explicit constants on what ignoring the problem costs a variational autoencoder. Experiments backed by an exact per-run normalization audit confirm that the compensated prior is invariant to the chart and stable across scales, every wrapped baseline we train collapses to the boundary of its chart, curvature becomes recoverable where the wrapped prior fails and protein-orientation likelihood improves from 2.58 to 0.87 nats at identical accuracy.

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