Abstract
We introduce the Yat kernel
k_{b,\varepsilon}(\mathbf{w},\mathbf{x})=\frac{(\mathbf{w}^\top\mathbf{x}+b)^2}{\|\mathbf{x}-\mathbf{w}\|^2+\varepsilon},\qquad b\ge 0,\ \varepsilon>0, a rational hidden-unit primitive whose units are Mercer sections over a shared input/weight space. For
b\ge 0 the kernel is PSD; for
b>0 it dominates a scaled inverse-multiquadric (IMQ) in the Loewner order, yielding fixed-kernel universality, characteristicness, and strict positive definiteness on every compact domain. The polynomial numerator opens nonradial alignment channels absent from finite IMQ expansions, witnessed by the directional far-field trace
T_\infty g_\varepsilon(\cdot;\mathbf{w},b)(\mathbf{u})=(\mathbf{u}^\top\mathbf{w})^2. Algebraically, a second finite difference in the bias recovers any IMQ atom from three positive-bias Yat atoms exactly, sharp at three atoms in every dimension at exact pointwise equality. A trained shared-
(b,\varepsilon) Yat layer is therefore a finite learned-center expansion in a fixed universal characteristic RKHS, with closed-form norm
\boldsymbol{\alpha}^\top\mathbf{K}\boldsymbol{\alpha} and explicit diagonal
(\|\mathbf{x}\|^2+b)^2/\varepsilon driving a Rademacher generalization bound.
