I will explain how the Fargues-Fontaine curve appears from localizing invariants, including K-theory and topological cyclic homology, and indicate how the twistor projective line should appear analogously.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/10868