is there a way to do software verification where an implementation would periodically self sign a note saying it’s verified its build? and then also broadcast relay actions? Even, just to see the relay actions… not to move all trust of moderation to relays.

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Yep, it's called reproducible builds. Amethyst has this active. It means that anybody should be able to check if the APK I published matches the code on the repo by recompiling the app with the same code/parameters.