This paper shows the need for the emergence of a universal minimum speed in the space–time by thoroughly investigating Dirac’s large number hypothesis (LNH). We realise that there should be a minimum speed V with the same status of the invariance of the speed of light c. However, V has gravitational origin. Hence, such a minimum speed forms a new kinematic basis in the space–time, leading to a new deformed special relativity (DSR) for the quantum world so-called symmetrical special relativity (SSR). Furthermore, we show that such a new structure of space–time (SSR) reveals a connection between V and a preferred reference frame S$_V$ of the background field, leading to the cosmological constant Λ, which can be associated with a cosmological antigravity. We also investigate the effect of the number of baryons N (Eddington number) of the observable Universe on the hydrogen atom. Finally, we show that SSR-metric plays the role of a de-Sitter (dS) metric with a positive cosmological constant, which could assume a tiny value.