What Does TwoNN Measure? Sample-Resolution Laws in High Dimensions

TwoNN estimates intrinsic dimension from ratios of first- and second-nearest-neighbor distances. In high dimensions, even nearest neighbors may lie outside the small neighborhoods underlying manifold-dimension interpretations. We study the pooled inverse-mean-log-ratio estimator as sample size and dimension grow together. For anisotropic Gaussian data under stated spectral assumptions, we prove concentration around an explicit function of sample size and covariance eigenvalues, accounting for dependence between ratios computed from the same sample. For uniform points on a noiseless sphere, polynomially many samples make TwoNN report a vanishing fraction of the true dimension. We also bound spherical mean-spacing approximation error and shallow Gaussian target-estimation error. Gaussian and spherical experiments test the phase laws. In a separate controlled Gaussian study, the trace-moment estimator closely recovers its oracle shallow target, while observed TwoNN retains finite-resolution and sampling error. In five CIFAR-10/ResNet-18 models, covariance-aligned rescaling fitted on disjoint images increases TwoNN by a median factor of 2.20 while preserving every prediction; random orientations with identical singular values give a factor of 0.995. Direct simulations from fitted global-Gaussian and class-Gaussian-mixture nulls compare neural TwoNN with those nulls: the mixture moves closer to real TwoNN at one tested layer and farther away at the other. TwoNN can therefore consistently track a sample-resolution-dependent target without recovering population manifold dimension. In the tested neural representations, its value should be interpreted together with coordinate controls and explicit distributional nulls.