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 the Planck length lP, R (3/Λ)1/2 should be a fundamental constant. They lead to a dimensionless constant g ~ lPR−1 = (Gc−3Λ/3)1/2 ~ 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 characteristics. It can pass the observation tests and support the duality.