>>Through all of this, don't get me wrong, the rigorous application of rationality

Much of this is just talking to oneself and testing it to see if our idea holds under test conditions.

I was once watching a video on how chess grandmasters think and work. Most of it is-

1. Do we know a pattern of moves, even if done, in series that is known to score some win/check. If so, lets do it.

2. Are any pieces under attack, If gone can effect point 1. eventually? If yes, lets protect them.

3. What can all possible moves of our pieces prevent opponent from having successfully execute their own point 1. And can we force opponent into point 2? Lets do it.

Basically every our move and its possible outcomes(Known through prior study of patterns of previous games seen), every move of our opponent.

A strong internal monologue and testing imaginary moves.

Math is just this except over paper.