Action at a Distance: A Universal Kernel from Polynomial Alignment and IMQ Distance

Inverse-multiquadric (IMQ) kernel sections decay at infinity. We study how quadratic alignment balances their inverse-square decay, adding persistent directional responses while preserving compact-domain approximation. The Yat kernel multiplies the IMQ kernel by a squared biased inner product. For fixed positive bias and distance offset, its RKHS continuously contains the IMQ RKHS and is universal on every compact domain. This global RKHS inclusion is strict: Yat sections with nonzero centers have a nonzero quadratic directional trace at infinity. Alignment also changes high-dimensional activation covariance. For a fixed number of orthogonal unit centers and fixed kernel parameters, isotropic Gaussian inputs give a rank-one rescaled limiting covariance for IMQ and a full-rank limit for Yat. Under uniform inputs on the radius-$\sqrt{d}$ sphere, both limits are full rank, identifying shared input-norm fluctuations as the Gaussian collapse mechanism in this regime. The connection is constructive: three positive-bias Yat atoms recover any IMQ section exactly, and three are necessary at nonzero centers when the distance offset is shared and biases may vary across atoms. On bounded cubes, rank-$r$ positive-semidefinite quadratics admit $r$-atom Yat approximations with growing center scales. Accurate IMQ approximation obeys explicit necessary coefficient-variation budgets for $d\ge 4$, without restrictions on width or center locations. Polynomial alignment thus preserves compact-domain approximation while changing global directional behavior and the resources used to represent functions.