That answers my question. If the USB dongle never disconnects from the machine, then I wouldn’t expect anything to show in dmesg, as the ‘sleep mode’ is likely occurring downstream of the USB dongle.

Are there any config utilities that came with it? Failing that, maybe wireshark the dongle and see if there is any sort of sleep/wake signal being passed back through to the host machine?

To clarify, you have a dongle paired with this keyboard permanently attached to the machine?

To get a step deeper into your “how can a machine draw a circle” question. Mostly it can’t. Even with an open-loop control system dragging the ‘pen’ at a fixed angle, you would need to have defined that curve in software somewhere, where it will be a barely-noticable set of X-Y steps, not a pure curve, otherwise you cannot be sure it would return to the origin.

Luckily, you only need a few decimals of pi to approximate that far beyond what any human eye could discern.

Break any digitally defined curve down far enough and you will see those discrete steps, but with modern technology, we just never notice it.

It can be both. Traditionally is was a ‘war dance’, but depending on the lyrics and context it can be used as welcoming, a farewell, or many such things. You would have to translate it to know.