A union-bound tolerance rule and the information price of a fitted orientation in spatial-coincidence tests on the sphere

Duplicate deposit — do not cite this record. The same manuscript was deposited twice within two minutes on 24 August 2026 by two parallel automated sessions of the same author. The record of reference is 10.5281/zenodo.22077512; please cite that identifier. The content of the two records is the same manuscript. This note is added rather than the record deleted, because a published Zenodo record cannot be withdrawn by its owner; a request for removal has been recorded on the author’s side. Version 2 (24 August 2026) — correction. This version corrects an arithmetic misattribution in the orientation table of version 1. Version 1 reported M = 1713 distinguishable orientations and a price of 10.7 bits at a tolerance of 250 km; that pair is arithmetically correct but belongs to a tolerance of 362 km. At 250 km the correct values are M = 5.20 × 103 and 12.3 bits, and the implied Bonferroni threshold is 9.6 × 10−6 rather than 2.9 × 10−5. The correction is stated explicitly in the text (Remark 1) rather than made silently, because version 1 is publicly deposited and citable. It is not in the direction favourable to the author’s own hypothesis: the corrected price of a fitted orientation is higher, not lower. The manuscript has also been reorganised for a journal of mathematical statistics and is under consideration there. Abstract. A test of non-random coincidence between n observed directions on the sphere and a fixed finite configuration F of N points requires two quantities that the applied literature normally supplies by convention rather than by derivation: the angular tolerance at which a coincidence is declared, and the accounting for a rotation of F fitted to the same data. Both are given here in closed form. First, a tolerance rule: by subadditivity of measure, the probability that a uniformly distributed direction falls within angle δ of some point of F is at most N(1 − cos δ)/2, with equality if and only if the caps are pairwise disjoint; requiring this to be at most α for a level fixed in advance gives δmax = arccos(1 − 2α/N). Only the configuration size and the level enter, so the tolerance is not tunable after the fact and a referee can reproduce it independently. Two consequences follow: the admissible tolerance falls as N−1/2, so refining a configuration without tightening its tolerance strictly weakens the test; and the conventional terrestrial tolerance of 250 km corresponds, for N = 62, to an implied level α = 0.0239, which is an inference from the tolerance rather than a choice. Second, an orientation price: using the Haar law of the rotation angle on SO(3), the number of orientations distinguishable at resolution ε is M(ε) = 6π/(gε3) for a configuration with symmetry group of order g, and a fitted orientation costs log2 M bits of evidence. For icosahedral symmetry at ε = 250/6371 this is M = 5.20 × 103 and 12.3 bits; a rotation-invariant criterion costs nothing. Neither result depends on the substantive hypothesis that motivated it, and both hold for an arbitrary finite configuration and an arbitrary finite symmetry group. The rules are applied to the author’s own earlier analysis, where they remove a positive-looking result.