Array superdirectivity is traditionally derived through singular optimization of densely spaced antenna arrays. In this paper, we show that the phenomenon admits a geometric interpretation as a concentration effect induced by spectral collision. As the spacing of an $M$-element linear array tends to zero, the exponential family generated by a linear array undergoes a spectral collision, and the associated finite-dimensional subspaces converge in reproducing kernel to a polynomial jet space. The maximum achievable array gain equals the diagonal evaluation of the reproducing kernel, and is therefore governed by the reciprocal Christoffel function. For the classical flat $L^2([-1,1])$ geometry, the Christoffel--Darboux kernel exhibits boundary concentration, yielding the quadratic $M^2$ superdirective law as a direct consequence of kernel asymptotics. This viewpoint separates intrinsic gain limits from numerical conditioning and identifies superdirectivity as a manifestation of a more general concentration mechanism. The framework further shows that the classical $M^2$ scaling is not universal: alternative spectral geometries produce different concentration laws through their associated Christoffel asymptotics. The results establish a direct connection between superdirectivity, reproducing kernels, orthogonal polynomials, and concentration phenomena arising from singular spectral limits.

翻译：阵列超方向性传统上通过对密集排列天线阵列的奇异优化导出。本文表明，该现象可被几何解释为谱碰撞诱导的集中效应。当$M$元线性阵列的间距趋近于零时，该阵列生成的指数族发生谱碰撞，相应的有限维子空间在再生核意义下收敛至多项式喷流空间。最大可实现阵列增益等于再生核的对角求值，因此由互反克里斯托费尔函数控制。对于经典平坦$L^2([-1,1])$几何构型，克里斯托费尔–达布核表现出边界集中，从而通过对核渐近性质的直接推导获得二次型$M^2$超方向性定律。该视角将固有增益极限与数值条件数分离，并揭示超方向性是更一般集中机制的表现形式。核框架进一步表明，经典$M^2$标度律并非普适：替代谱几何通过其关联的克里斯托费尔渐近行为产生不同的集中定律。研究结果建立了超方向性、再生核、正交多项式与奇异谱极限引发的集中现象之间的直接关联。