AI 中文总结
该研究证明了通用Beta入射角的平方余弦律，通过两种方法证明并推导其刚性、可扩展性及相关准则，给出正交坐标下的缩放F律。
AI 中文摘要
设U是单位球面$\boldsymbol{\text{S}}^{p-1}$上的哈尔均匀分布随机变量，$a_1,\boldsymbol{\text{…}},a_k$为任意非零向量，$w_1,\boldsymbol{\text{…}},w_k$为单纯形权重。定义$g(U)=\boldsymbol{\text{Σ}}_{j=1}^k w_j\frac{a_j}{a_j^\top U}$，$N(U)=\frac{g(U)}{\boldsymbol{\text{∥}}g(U)\boldsymbol{\text{∥}}}$。我们证明通用入射律$\boldsymbol{\text{\textbraceleft}}U^\top N(U)\boldsymbol{\text{\textbraceleft}}^2\boldsymbol{\text{\textasciitilde}}\boldsymbol{\text{Beta}}\boldsymbol{\text{\textless}}\frac{1}{2},\frac{p-1}{2}\boldsymbol{\text{\textgreater}}$，该律与方向的数量、排列、秩、超完备性以及权重均无关。因此，U的一个确定性、通常非哈尔函数具有与独立哈尔方向相同的平方余弦律。一种证明结合了Herglotz–Cauchy边界原理、具有一个公共相位的哈尔随机二维平面，以及精确的Beta–Cauchy正切投影等价性；第二种证明是正定Pillai–Meng恒等式的特例。平面结构引出逆命题：平面条件柯西律可恢复正性，而对于符号测度，精确的相位抵消赤字等于两倍的隐藏负质量。这在严格弱于单射的相位归一化条件下产生局域到全局的刚性，并无条件排除负原子。该律在几乎处处倒数可积的概率测度下可扩展，我们通过精确的Wiener–Dini带级数刻画该条件，证明有限香农熵是可数权重的尖锐通用准则，并给出聚类测度的熵–几何扩展。所有一维豪斯多夫测度为零的紧支撑集均为容许集，而非零可求长弧分量会在正哈尔测度集上导致发散。在正交坐标下，该定理还给出无权重的缩放F律，用于从固定单纯形向量到$\boldsymbol{\text{Dirichlet}}(\frac{1}{2},\boldsymbol{\text{…}},\frac{1}{2})$向量的皮尔逊散度。
英文摘要
Let $U$ be Haar-uniform on $\mathbb S^{p-1}$, let $a_1,\ldots,a_k$ be arbitrary nonzero vectors, and let $w_1,\ldots,w_k$ be simplex weights. Define \[ g(U)=\sum_{j=1}^k w_j\frac{a_j}{a_j^\top U}, \qquad N(U)=\frac{g(U)}{\|g(U)\|}. \] We prove the universal incidence law \[ \{U^\top N(U)\}^2\sim\operatorname{Beta}\!\left(\frac12,\frac{p-1}{2}\right), \] 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 $\operatorname{Dirichlet}(1/2,\ldots,1/2)$ vector.
