Maybe I’m wrong, but I thought that Matrix was a protocol, while mautrix was a client framework for the Matrix protocol written in Golang. Mautrix bridges are bridges for Matrix based on the mautrix framework. Beeper uses the Matrix protocol with some bridges (and maybe other components) based on mautrix; it’s basically a proprietary commercial fork of Synapse and Element with better-integrated bridges. I have several mautrix bridges on my Matrix server, and every mautrix bridge I see (including gmessages) says it’s for Matrix.