Birkhoff center and recurrent behavior of differentially positive systems on a homogeneous space

What the math gives to ML

The paper develops differential positivity on homogeneous spaces: tangent linearizations preserve a state-dependent homogeneous cone field, inducing an order relation on trajectories. Its main structural result is a Birkhoff-center dichotomy: each connected recurrent component, and the support of every invariant measure, is either strongly ordered or unordered. A transferable neural-network mechanism is to impose and monitor cone preservation in recurrent or state-space architectures, then use the resulting order structure to identify recurrent regimes and prevent hidden-state trajectories from developing uncontrolled transverse directions.