Snyder's model—de Sitter special relativity duality and de Sitter gravity

Between Snyder's quantized space-time model in de Sitter space of momenta and the dS special relativity on dS-spacetime of radius $R$ with Beltrami coordinates, there is a one-to-one dual correspondence supported by a minimum uncertainty-like argument. Together with Planck length $ell_P$, $Rsimeq (3/Lambda)^{1/2}$ should be a fundamental constant. They lead to a dimensionless constant $g{simell_PR^{-1}}=(Ghbar c^{-3}Lambda/3)^{1/2}sim 10^{-61}$. These indicate that physics at these two scales should be dual to each other and there is in-between gravity of local dS-invariance characterized by $g$. A simple model of dS-gravity with a gauge-like action on umbilical manifolds may show these characters. It can pass the observation tests and support the duality.