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HyperShadow introduces a novel benchmark for detecting 3D projections of higher-dimensional spatial objects, challenging the conventional understanding of "4D" datasets. By employing advanced projection signatures and topology changes, the study achieves a remarkable 96.6% accuracy in distinguishing native 3D shapes from their higher-dimensional shadows, significantly outperforming existing methods. Additionally, a zero-parameter rigidity witness is introduced, achieving an AUROC of 0.982, providing a powerful tool for analyzing rigid motion in higher dimensions.
A staggering 96.6% accuracy in distinguishing 3D shapes from their higher-dimensional shadows reveals the limitations of traditional dimensionality estimators.
Machine-learning datasets labelled"4D"universally denote three spatial dimensions plus time. We introduce HyperShadow, the first public benchmark in which the fourth, fifth, and sixth dimensions are spatial: the task is to decide whether a 3D point cloud is a native three-dimensional shape or the projection, the"shadow", of a rigid object living in R^N (N = 4-6). We show this task is fundamentally distinct from intrinsic-dimension estimation: a shadow is still at-most-3-dimensional data, and standard estimators (TwoNN, Levina-Bickel MLE) reach only 71-73% accuracy. Detection instead requires projection signatures, density folds, filled volumes with characteristic radial profiles, and topology changes, which a 190k-parameter point network recovers at 96.6% accuracy across four corruption tiers, generalizing at 79-91% to object families never seen in training. On a temporal track of rigidly rotating objects we introduce a zero-parameter rigidity witness: the residual of the optimal rigid 3D alignment (Kabsch) between consecutive frames, which must vanish for any rigid 3D motion but cannot vanish for the shadow of a rigid rotation in R^N. This single interpretable statistic separates the classes at AUROC 0.982. All data are generated reproducibly from seeds; the dataset, models, and code are released publicly. HyperShadow makes no claim about physical reality; it is a controlled instrument for studying which observable statistics can certify incompatibility with a purely three-dimensional explanation.
