I can provide some context on this question about applications of Caramello's theory to the Langlands program or other areas of mathematics.

First, let me clarify what Caramello's theory refers to. Olivia Caramello developed a systematic approach to using topos-theoretic methods to establish connections between different mathematical theories through "bridges" - these are ways of interpreting different theories as different "sites of definition" for the same topos. Her work builds on earlier ideas about Morita equivalence and categorical logic.

Regarding concrete applications to the Langlands program or other "mainstream" mathematical problems:

Current Status: As far as I'm aware, there have not yet been published papers that use Caramello's techniques to prove new results in the Langlands program or solve long-standing problems in number theory, algebraic geometry, or similar fields that were previously thought unrelated to topos theory. The connection Laurent Lafforgue mentions remains more at the level of a philosophical/structural observation rather than a worked-out technical development.

Why this might be the case:

1. The Langlands program is already formulated in highly sophisticated categorical language (derived categories, perverse sheaves, etc.)
2. Converting existing deep results into the topos-theoretic framework would require substantial technical work
3. The community working on Caramello's approach is relatively small compared to those working on Langlands

What exists instead:

• Caramello has shown how her techniques can recover and unify known results from different areas of mathematics
• There are applications within logic, model theory, and categorical algebra
• The theory provides new perspectives on known connections rather than entirely new results

The criticism you mention from BCnrd (Brian Conrad) reflects a common skepticism in the mathematical community about whether these topos-theoretic methods genuinely provide new insights into concrete mathematical problems versus providing a new language for existing results.

To directly answer your question: I'm not aware of papers meeting your specific criteria - proving results outside logic/set theory, not involving topoi in their statements, but using Caramello's techniques as an essential ingredient in their proofs.