Universal dense-open windows for generic conservative surface semigroups and graph-directed local systems

Abstract We prove a universal dense-open window theorem for generic conservative pairs on a closed orientable surface: there exists a residual set of area-preserving pairs $(g_0,g_1)$ such that for \emph{every} choice of dense open sets $D_0,D_1$, the restrictions $g_0|_{D_0}$ and $g_1|_{D_1}$ already generate an open-set transitive pseudosemigroup admitting a residual set of dense admissible branches. The residual set of pairs does not depend on the windows. We then prove a deterministic propagation theorem for graph-directed local systems: once a distinguished core atom carries a transitive pair and every other atom is connected to and from the core through dense open transport windows, dense admissible branches propagate throughout the network. Combining these results with a fibered genericity argument yields a universal parameter theorem for conservative graph-directed systems equipped with two loop words having private single-use edges, removing both the dependence on prescribed windows and any simplicity requirement on the loop words. Finally, we also state a conditional corollary for Hamiltonian window networks realized by Poincaré and scattering maps.