The Charter and Constitution of the Empire of Taeliana outlines the rights, duties, and responsibilities of the Imperium and the Nobility, and the rights of the Populace. So yes, the Empire has a constitution.

The Populace (everyone) has a right to safety, security, and education. The Empire has a set of schools, and everyone older than 15 has what we today would consider at least a fourth-grade education in math and reading. The schools also serve to look for children with sharper minds or talents, to make sure those aren't wasted.

(yes, that is sort of taken from Imperial China. I thought it was a good idea.)

Safety and Security for areas are devolved from the Imperium to the Grand Dukes, the Dukes, the Counts, and finally the Barons. They report up. It's the responsibility of the various nobles to basically make sure they patrol their lands, have supplies in storage in case of disaster to make sure their people are taken care of, stuff like that.

Also mentioned in the Constitution is the right for a noble to be removed from their office if evidence is supplied that they are incompetent, corrupt, or unfit. The precise wording is "The first duty of a ruling noble is to leave their land better than it was when they took their seat."

Usually, their heir will file it with the Administratio Imperialis (the part of the Empire that handles the paperwork for the rest), but other people can if needed. The process is called Declaratio culpae ad meliorem, and at the end of Grey Swan, the remaining resident Corbeau children work together to put one together.

the Declaratio has never been filed against a sitting Emperor or Empress, because they don't tend to suck that badly. (I mean there was Maravor the Unwilling, but he was suffering from severe mental illness, and he was smart enough to marry well before it drove him to the point he could barely leave his bed.)

There are two reasons for that overall: 1) the Imperial Marriage Law tends to bring common sense into the Imperial Line (there are no morgantic marriages in Taelenia; marrying a member of the Populace doesn't make you resign), and 2) yeah, that's an Imperial Secret.

(also, yes, yarnzipan , I did borrow some stuff about the Empire from the Melcenes.)

heck yeah

I think it's the royal line of the Star-Kingdom of Manticore that has, as hard legislation, that the Heir must marry a commoner. Genetically sensible move.

...Or am I conflating with one of Doc Smith's post-mortem series? Drat. It's definitely not Barrayar.