  In this paper, we apply the theory of Bridgeland stability conditions, which
was originated from string theory, to study the derived category of coherent
sheaves on Fargues--Fontaine curves. This leads us to consider the
quasi-coherent sheaves $\mathcal{O}(\theta^{\pm})$ via the convergents of an
irrational number $\theta$.
  We define the continuum envelop $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$ to be
the smallest abelian subcategory in $\mathrm{QCoh}(X_{FF})$ containing
$\mathrm{Coh}(X_{FF})$ and $\mathcal{O}(\theta^{\pm})$. We study the
homological algebra of $\mathrm{QCoh}_{\mathbb{R}}(X_{FF})$ via Farey diagrams.
We show that the homological property of $\mathcal{O}(\theta^{\pm})$ depends
heavily on the arithmetic property of $\theta$. The Fargues--Fontaine curves
present strong similarity with complex elliptic curves in this point of view.
