The web of swampland conjectures and the TCC bound

We consider the swampland distance and de Sitter conjectures, of respective order one parameters $\lambda$ and $c$. Inspired by the recent Trans-Planckian Censorship conjecture (TCC), we propose a generalization of the distance conjecture, which bounds $\lambda$ to be a half of the TCC bound for $c$, i.e. $\lambda \geq \frac{1}{2}\sqrt{\frac{2}{3}}$ in 4d. In addition, we propose a correspondence between the two conjectures, relating the tower mass $m$ on the one side to the scalar potential $V$ on the other side schematically as $m\sim |V|^{\frac{1}{2}}$, in the large distance limit. These proposals suggest a generalization of the scalar weak gravity conjecture, and are supported by a variety of examples. The lower bound on $\lambda$ is verified explicitly in many cases in the literature. The TCC bound on $c$ is checked as well on ten different no-go theorems, which are worked-out in detail, and $V$ is analysed in the asymptotic limit. In particular, new results on 4d scalar potentials from type II compactifications are obtained.