Pulsars also have a rotational period that provides a reference point for interpreting arithmetic notation.

Yeah, and it's one of the nearby stars in reality.

It's also sort of the subject - and title! - of this great short story: https://clarkesworldmagazine.com/macleod_12_17_reprint/

Thank you!

It has been a little while since I read the book, but I think you have the cause of the curve wrong. These things are moving at such a high percentage of the speed of light that there isn't much visible curve from the movement of the planet. The curve is instead caused by them leaving the star at the pole (which leads to a different nitpick of the visualization since the curve isn't shaped to represent that). It's theorized that the Astrophage do this to make is easier to find their way as exiting the plane of the solar system reduces the chances of there being any occultation blocking their view of their destination.