Hamiltonian group actions in cosymplectic geometry

What the math gives to ML

The paper supplies a concrete odd-dimensional geometric decomposition: a closed transverse 1-form \(\alpha\), a leafwise symplectic 2-form \(\beta\), and a canonical Reeb direction separated by \(\iota_R\alpha=1\) and \(\iota_R\beta=0\). Its most transferable asset is the symplectic thickening \(M\times\mathbb{S}^1\), which turns a state with one distinguished progression coordinate into an ordinary symplectic state while retaining explicit control of the transverse direction. This suggests a neural layer that splits hidden states into a leaf state and a phase/time coordinate, applies Hamiltonian updates only on the leaf, and uses a separately parameterized Reeb drift. The idea is most promising for sequential dynamics and world models, where one coordinate represents physical time, diffusion time, depth, or another ordered progression variable.