No More DeLuLu: Physics-Inspired Kernel Networks for Geometrically-Grounded Neural Computation

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 $\boldsymbolα^\top\mathbf{K}\boldsymbolα$ and explicit diagonal $(\|\mathbf{x}\|^2+b)^2/\varepsilon$ driving a Rademacher generalization bound.

Modern neural architectures are largely developed as distinct empirical constructions, despite recurring principles across convolutional, attention, state-space, and equivariant models. We introduce Product Interactions, an algebraic formalism that unifies these architectures under a common construction principle and turns their design into a systematic procedure. The core shift is representing data as algebraic elements and treating multiplication, rather than linear transformation alone, as a primitive architectural operation. Architectures are generated by composing multiplication operators, with input dependence organized by self-interaction order. Within this formalism, we derive convolutional and equivariant networks as symmetry-constrained first-order product interactions, recover Mamba from state-space models as an increase in self-interaction order from one to three, and establish attention as a third-order interaction from composing the same operator. Equivariance is incorporated by constraining the algebraic product via symmetry: translation equivariance yields convolution, while SO(3)-compatibility produces the Clebsch--Gordan product and tensor-field interactions. This perspective recasts architecture design as an algebraic construction problem: the choice of algebra defines admissible interactions, symmetry constrains their structure, and composition generates higher-order architectures. The framework is thus generative. Following these principles, we construct a simplified-algebra dynamical layer that improves both accuracy and throughput over Mamba on CIFAR-10. Experiments show that interaction order affects accuracy non-monotonically and symmetry constraints boost learnability. Product Interactions provides a mathematical methodology for analyzing existing architectures and systematically designing new ones.

While neural collapse (NC) predicts that a $K$-class-balanced classifier should organize terminal representations as a $(K-1)$-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on $S^1$; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of $\mathbb{Z}/P\mathbb{Z}$ and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an $O(1)$ advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a $Θ(K)$ advantage under Schatten or weight-decay surrogates, yielding a critical threshold $λ_{\mathrm{crit}} = Θ(1/K)$. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.