Layerwise LQR

The exact Riccati formulation is used mainly as a reference rather than as a practical algorithm for modern networks. LLQR rewrites the exact dense geometry-aware update as a layerwise Linear Quadratic Regulator problem before imposing structure on an inverse preconditioner. The exact LLQR steepest-descent update is equivalent to a finite-horizon LQR problem under linearized layer dynamics. LLQR separates network dynamics from descent geometry in dense curvature matrices. LLQR is positioned as both a practical optimizer wrapper and a framework for studying scalable approximations to dense geometry-aware updates.