The beauty implicitly assumes the math is correct and consistent, but expresses something much deeper.

Analogous to how music adheres to certain rules and structures (these rules and structures are allowed to vary across genres).

Another analogy is how a “beautiful play” in basketball should adhere to the rules of the game. For example, a beautiful dunk during a game implies that the player wasn’t travelling or carrying the ball.

My third analogy would be a “beautiful move” in chess. It’s implied you followed the chess rules to make that move otherwise it wouldn’t be beautiful.

Also answering a question. There's a youtube video about topology and solving .. I forgot but say elliptic curves equations. And when encoded in topological terms the problems simply vanishes (some would say the knot unties itself).

You can invent aesthetically pleasing theories and notations but if it doesn't help solving a problem.. it's a bit moot.

This is achieved merely by following the rules. But what are the most appropriate rules? It depends on the taste of the logician