Your app does not update its grimoire information if it is running while you are doing things that are changing your grimoire in-game at the time. For it to update, you will need to stop the app from running (as I only have the iOS app, I will relay how that platform works: double click the home button, scroll to the running companion app, flick it up) and then start it up again after the relevant grimoire change to see the progress you have made.