> The actual, intended purpose, in a functioning society, is to {serve fair punishments and rehabilitate, enhance quality of health, etc}.

I'm not sure which history books you've been reading, but modern police were invented in the 17th and 18th centuries as brutal forms of population control. In America, the "actual, intended purpose" was to be a genocidal force that clears the land of Amerindians, to patrol the slaves, to terrorize the workers, and so on. In the South, you can trace the history of police departments directly to Confederate troops, who, immediately after losing the war, rounded up people and threw them in prison on fabricated charges. To a lot of people, it was as if the war never happened. The fact that there was no reign of terror in the south tells you everything you need to know. And the slavery continues to this day. So I have no idea what you mean when you talk about this fantastical "intended purpose".

There are several lines of evolution for modern policing.

The US instance is one.

Robbert Peele's "Peelian Principles" (UK) are another, which is how the UK ended up with a largely unarmed and civillian institution "policing by consent". (It also has its darker elements, but that's the thumbnail sketch.)

https://en.wikipedia.org/wiki/Peelian_principles

A French "gendarme" is literally a "man at arms", and is derived from military forces, representing a third tradition.

https://en.wikipedia.org/wiki/Gendarmerie

There are no "lines of evolution" like carcinization for modern policing. They were caused by the same social and political necessities and they serve the same function.

Citation requested.

By "actual, intended" purpose, I do not mean original. I mean commonly agreed upon purpose, that almost all would agree is the true goal to strive for in a {criminal justice system, healthcare, etc}. I specified "current" to differentiate it between the current purpose, and a hopeful future purpose, not a past one. I don't mean to imply that these systems have been "corrupted" from a virtuous initial state (because like you say, they have some dark origins). Though they certainly are corrupt.

We can trace the history of police all the way back to vikings, neanderthals or even to single celled organisms, but all that would be of little use to deal with the today's prison system.

I'm not naturalizing organized violence, you are. I'm talking about the history of one institution and its forms of reproduction. How are you going to "deal" with today's prison system without understanding how it reproduces itself (through the law, through the indoctrination of people such as yourself, etc)?

History of institution has no bearing on actions of individuals in this institution today. What has bearing is policies that exist today. Not history of those policies, but their actual current state.

For “intended purpose” of differing sectors of society, one may start, if one chooses, with Plato’s Republic.