Remark 7.57.
For analytic rings, Scholze states that the two congruences are equal; see [Scholze 2026, Lecture IX]. Under this additional assertion, the Gestalt-valued functor does factor through the further localization used to define analytic stacks. Marc expects the equality to hold more generally, but the preceding argument proves only the displayed inclusion.
References
- Peter Scholze. Geometry and Higher Category Theory. 2026.