Incremental stability in $p=1$ and $p=\infty$: classification and synthesis

All Lipschitz dynamics with the weak infinitesimal contraction (WIC) property can be expressed as a Lipschitz nonlinear system in proportional negative feedback -- this statement, a ``structure theorem,'' is true in the $p=1$ and $p=\infty$ norms. Equivalently, a Lipschitz vector field is WIC if and only if it can be written as a scalar decay plus a Lipschitz-bounded residual. We put this theorem to use using neural networks to approximate Lipschitz functions. This results in a map from unconstrained parameters to the set of WIC vector fields, enabling standard gradient-based training with no projections or penalty terms. Because the induced $1$- and $\infty$-norms of a matrix reduce to row or column sums, Lipschitz certification costs only $O(d^2)$ operations -- the same order as a forward pass and appreciably cheaper than eigenvalue or semidefinite methods for the $2$-norm. Numerical experiments on a planar flow-fitting task and a four-node opinion network demonstrate that the parameterization (re-)constructs contracting dynamics from trajectory data. In a discussion of the expressiveness of non-Euclidean contraction, we prove that the set of $2\times 2$ systems that contract in a weighted $1$- or $\infty$-norm is characterized by an eigenvalue cone, a strict subset of the Hurwitz region that quantifies the cost of moving away from the Euclidean norm.