Google Play tries to "update" my F-Droid v0.5.7 app to v0.5.6

[glevv]

Gemini tells me it is somehow connected to versionCode check

[TheMysteriousGhost]

Gemini is right that it's versionCode. Android and the Play Store only compare the versionCode, never the version name, and the two builds use completely different numbering:

• F-Droid builds Rethink with its own scheme, 10 × VERSION_CODE + an ABI digit, so v0.5.7 there is 692 / 693 / 698 / 699 (693 on arm64). You can see that in F-Droid's metadata for com.celzero.bravedns.
• The project's own builds (Play and website) add an ABI multiplier of 10,000,000: 3 × 10,000,000 + VERSION_CODE for arm64 and universal builds (the androidComponents block in app/build.gradle.kts). So v0.5.6 from Play is a versionCode around 30,000,000.

From the Play Store's point of view, its 0.5.6 is therefore much "newer" than your F-Droid 0.5.7.

It can't actually replace it, though. F-Droid signs Rethink with its own key, so the Play build has a different signature and Android won't install it over yours. Just ignore it, or open Rethink's page in the Play Store, tap ⋮ and untick Enable auto update, and keep updating through F-Droid. Switching between the two would mean uninstalling first, so back up your settings before you ever do that.