As a side note, have you ever heard a coherent and/or useful explanation of the "orient" part? It seems like every time I hear about the OODA loop, that's the part that gets yadda-yadda-yadda'd over.
So, for example, if you can enumerate the possible transitions, do so.
Beyond OODA is probably closer to directly answering your question, though Violence of Mind appears to deal with concrete application of "orientation" to self-defense and violent confrontations as a "good guy". I think I recall hearing in a podcast Varg did that he actually talked with one of Boyd's colleagues when putting together Beyond OODA to learn more and make sure the content was spot on, which was a big motivation for my purchase of the book as I find OODA fascinating and the concept has been very influential in my life.
There is also this article that introduced OODA to me, and it goes fairly in-depth on the "orient" section: https://www.artofmanliness.com/character/behavior/ooda-loop/
"Orient" is "given all this information, what are the possible solutions"
Decide on one solution, and then act upon it.
Repeat ad infinitum.
(The OODA concept is more about modelling your opponent in a competitive game so that you can analyse the opportunities for disrupting them. A competitive game like “dogfighting” or “Cold War counter-intelligence” or “the market for office software products”. When you describe it as “repeat ad infinitum” it sounds like a neat piece of life advice for how to approach any problem (again, only competitive games!) but you’re missing almost the entire message and in essence saying something about as useful as “use your brain to solve the problem”. As long as you’re not asleep, it’s impossible not to be following OODA. Literally anything you do will satisfy it. That’s why it’s a good model for an opponent. And it’s also why it’s a great tool for convincing yourself you’re some kind of strategic genius because you can recognise these very normal things happening in your own brain.)
It seems many people want to take these sort of decision making frame works to new contexts or generalize them to a point they no longer make sense to market them.
My stab is specific to understanding the full picture is how you can move in it. How many movement choices do you have? Can you get back to a position? Does your moving cause others to move? Can you see places that are safe to experiment in?
For prior information, look for familiar analogs. Defensive positions. Offensive outposts. Well troden paths, etc.
That is, I suspect the asker wanted specific examples of what that means. I think it is fair to say that people learn in concretes, not in abstractions. Is why so few of us know what a semigroup is, after all.
But in the spirit of being a twit, your "answers" are hardly coherent, let alone concrete. They're just as hand wavy as the original asker was probably complaining about if they wanted a concrete answer.
> How many movement choices do you have? Can you get back to a position? Does your moving cause others to move? Can you see places that are safe to experiment in?
If you want to give a concrete answer instead of a hand wavy one like yours pick a real game, sport, or combat scenario and apply it. Here's a stab at it that's absolutely useless unless you generalize the concept back to my original answer:
BJJ (which a lot of people tie to OODA):
The observations are what my own body-awareness and my opponent's actions and position relative to me. Up to this point in the match I've gotten them into my guard, they put their weight and body just far enough back that I haven't had much luck getting more control. But I managed to bump them and trip them up, they slipped up, they just planted their left hand by my right shoulder (observation).
Orientation: Combining the observation and my training in BJJ, I know that this situation ripe for an arm bar or a triangle.
Decision: I will grab their arm and adjust my guard to pull off the next move, an arm bar.
Action: I grab their arm, but this isn't a turn based game and they move too.
Observation: Gripping my sleeve or collar or shifting their weight, they make an arm bar too hard.
Orientation: I'm still in a good position for a triangle.
Decision: I'll attempt the triangle, but maintaining proper control I can still shift back to the arm bar if they open themselves up to it again.
Action: Move my legs and their body to achieve the triangle.
---------
But while concrete, the only utility here is to point out its generality. Either the person gets it and understands the concept beyond BJJ (and this specific scenario) and maybe other combat sports or they don't. This was a longwinded way to get back to the core concept: Take the observations and combine them to feed into the decision process.
And if you really think that last sentence has anything to do with semigroup-level abstraction, I can't help you.
I also should have pushed back to your terms. I think it is surprisingly useful to constantly ask what that sentence would mean in different situations. Such that I plan on doing just that for the next few days. Specifically, what did you mean to synthesize new ideas?
Would love to have success getting my kids to try this. I love your narrative there, as it shows how rapid the progression can go. At least, that is my current read.
All of that gives you your present orientation, your position, in either a literal or figurative sense.
And then there's your "opponent" (if there's not one, OODA may not be the right mental model to use). In your observations of your opponent and repeated orientation you are building up a model of them, synthesis again. You start with any prior knowledge (have you encountered this opponent before?) or an assumption (maybe a worst case, or estimate based on sizing them up). Then you engage, and in the engagement you observe and determine their real capabilities, which feeds into orientation for rendering a more accurate model of the opponent.
Orientation is taking the existing knowledge, adding new knowledge or information, rendering a better model (hopefully). Then you decide based on that model, act on that decision, and observe.
Of course it's not actually linear, all of this is happening at the same time, or can be. You don't stop observing while you orient, decide, or act. And you don't stop acting while you observe, orient, or decide.