On Generalized Hyperbolicity, Stability, and Shadowing for Linear Operators

What the math gives to ML

The paper gives a constructive pseudo-hyperbolic decomposition for invertible linear dynamics: a bounded projection separates directions that contract forward in time from directions whose inverse dynamics contract forward. This condition implies strong Lipschitz structural stability and shadowing, so bounded trajectory perturbations remain close to an exact trajectory. The transferable neural mechanism is a recurrent or state-space layer with explicitly controlled stable-forward and stable-backward channels. Its value is falsifiable: shadowing error should remain proportional to perturbation size below a spectral margin and should deteriorate sharply when either contraction bound reaches one.