Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements
Abstract
Let U be Haar-uniform on 𝕊p−1, let a1,…,ak be arbitrary nonzero vectors, and let w1,…,wk be simplex weights. Define g(U)=∑j=1k wjaj/(aj⊤ U), N(U)=g(U)/|g(U)|. We prove the universal incidence law {U⊤ N(U)}2∼Beta (1/2,(p−1)/2), independently of the number, arrangement, rank, or overcompleteness of the directions and of the weights. Thus a deterministic, generally non-Haar function of U has the same squared-cosine law as an independent Haar direction. One proof combines a Herglotz--Cauchy boundary principle, a Haar-random two-plane with one common phase, and an exact Beta--Cauchy tangent-projection equivalence. A second proof specializes the positive-semidefinite Pillai--Meng identity. The planar structure leads to converses: plane-conditional Cauchy laws recover positivity, while for signed measures an exact phase-cancellation deficit equals twice the hidden negative mass. This yields local-to-global rigidity under a phase-norming condition strictly weaker than injectivity and an unconditional exclusion of negative atoms. The law extends to probability measures under almost-sure reciprocal integrability. We characterize this condition by an exact Wiener--Dini belt series, prove finite Shannon entropy to be the sharp universal criterion for countable weights, and give an entropy--geometry extension for clustered measures. Every compact carrier of zero one-dimensional Hausdorff measure is admissible, whereas a nonzero rectifiable arc component forces divergence on a set of positive Haar measure. In orthogonal coordinates, the theorem also gives a weight-free scaled F law for Pearson divergence from a fixed simplex vector to a Dirichlet(1/2,…,1/2) vector.
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BibTeX
@misc{liu2026universal,
title = {{Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements}},
author = {Tianle Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00603},
eprint = {2609.00603},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00603},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements
AU - Tianle Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00603
UR - https://arxiv.org/abs/2609.00603
AB - Let U be Haar-uniform on 𝕊^p−1, let a_1,…,a_k be arbitrary nonzero vectors, and let w_1,…,w_k be simplex weights. Define g(U)=∑_j=1^k w_ja_j/(a_j^⊤ U), N(U)=g(U)/|g(U)|. We prove the universal incidence law {U^⊤ N(U)}^2∼Beta (1/2,(p−1)/2), independently of the number, arrangement, rank, or overcompleteness of the directions and of the weights. Thus a deterministic, generally non-Haar function of U has the same squared-cosine law as an independent Haar direction. One proof combines a Herglotz--Cauchy boundary principle, a Haar-random two-plane with one common phase, and an exact Beta--Cauchy tangent-projection equivalence. A second proof specializes the positive-semidefinite Pillai--Meng identity. The planar structure leads to converses: plane-conditional Cauchy laws recover positivity, while for signed measures an exact phase-cancellation deficit equals twice the hidden negative mass. This yields local-to-global rigidity under a phase-norming condition strictly weaker than injectivity and an unconditional exclusion of negative atoms. The law extends to probability measures under almost-sure reciprocal integrability. We characterize this condition by an exact Wiener--Dini belt series, prove finite Shannon entropy to be the sharp universal criterion for countable weights, and give an entropy--geometry extension for clustered measures. Every compact carrier of zero one-dimensional Hausdorff measure is admissible, whereas a nonzero rectifiable arc component forces divergence on a set of positive Haar measure. In orthogonal coordinates, the theorem also gives a weight-free scaled F law for Pearson divergence from a fixed simplex vector to a Dirichlet(1/2,…,1/2) vector.
ER -