Trilinear Kernel Structure and Its Gravitational Realization

Abstract: We clarify the structural role of trilinear kernels in multidimensional integrable hierarchies and in stationary axisymmetric gravity. The Yu--Toda--Fukuyama (YTF) trilinear equation of Ref.~\cite{YuTodaSasaFukuyama:1998hierarchy} is shown to represent not a particular evolution equation but a universal kernel that generates the entire $(3+1)$--dimensional hierarchy by selecting commuting flows. The frequently quoted trilinear equation of Ref.~\cite{YTSF1998} is identified as one such flow of this kernel. We further show that stationary axisymmetric gravity corresponds to a projective realization of the YTF kernel rather than to any single flow. Imposing $\GL(2)$ covariance and homogeneity on the kernel leads uniquely to a gravitational trilinear kernel $\mathcal{Y}(τ_0,τ_1)$, whose vanishing reproduces the Ernst equation. The Tomimatsu--Sato family \cite{Tomimatsu1972} and related bilinear solutions are shown to arise as degenerate submanifolds of this projected trilinear structure, in agreement with the multilinear analysis of Ref.~\cite{Fukuyama:2025TS}. These results establish a unified structural framework linking multidimensional trilinear integrability, stationary gravity, and bilinear solution sectors, and clarify why trilinear kernels are both necessary and sufficient for describing soliton dynamics with projective geometry.