Toward an Observable Algebra for de Sitter Space: Gap Protection and Modular Dressings
We investigate the relation between fundamental complements and regional operator algebras in global de Sitter space. For a finite union of open arcs on the time-symmetric circle of global dS$_2$, we show that the fundamental complement is determined by the largest complementary gap: a gap contributes precisely when its angular length is at least $\pi$, and at most one gap can do so. In higher-dimensional global de Sitter space, the corresponding criterion is containment of an open hemisphere. We then study the operator-algebraic consequences in an abstract M\"obius-covariant chiral conformal net. Each regional algebra is crossed with its vacuum modular flow using a shared auxiliary clock. We derive an exact criterion for inclusions between the resulting continuous cores and show that vacuum correlations obstruct this criterion when the larger region contains an additional spacelike arc. A compatible family can nevertheless be obtained on the finite Boolean algebra generated by a fixed collection of separated arcs by using a split product state. Motivated by the holograms prescription of Bousso and Penington, we introduce a representation-dependent commutant model for regional algebras. When the protected gap overlaps a fixed reference arc, the assigned algebra is a proper von Neumann subalgebra of the reference Type-II$_1$ factor. For an explicit two-arc family, adding an arbitrarily short antipodal component removes the fundamental complement and changes the assigned algebra discontinuously to the full reference algebra.