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I'm considering building dev tooling on top of clojure.lang.Compiler/analyze. Is it part of Clojure's public API, or is it considered internal?
It's not considered part of the public API and will return internal classes in the Compiler, which may not be fully accessible in some cases (it has not been designed for this). Those classes also change regularly between releases.
Depending on your needs, you might want to consider https://github.com/clojure/tools.analyzer and https://github.com/clojure/tools.analyzer.jvm.