On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective

What the math gives to ML

The paper identifies a sharp, computable threshold for persistence of a differentiable normally hyperbolic invariant graph under a C1 perturbation. For the dissipative twist map studied, normal contraction is governed by lambda in (0,1), and a perturbation with C1 norm below (1-sqrt(lambda))^2 preserves a C1 graph, while equality can create a unique graph with nondifferentiable points. The transferable mechanism is to build recurrent or state-space layers with an explicitly contracting auxiliary state and constrain the learned residual's value and Jacobian within a contraction-dependent budget. This yields a falsifiable smoothness and long-horizon stability transition rather than an untargeted regularization heuristic.