I want to better understand why exactly is higher Arithmetic intensity better than a higher Communication-to-computation ratio. In my own words, communication is a lot slower than computation so we want less of it. If, on some hypothetical computer, communication was a lot quicker that computation would it be the opposite case? Would we then prefer a higher Communication-to-computation ratio and a lower Arithmetic intensity? If the communication and computation took the same amount of time would these ratios not matter?

@Eduardo Your understanding matches mine, I think all your hypotheticals are correct.

Even if our solution required many more arithmetic instructions at the sake of reducing memory instructions, thereby increasing instructions, arithmetic is faster and more energy-efficient than memory instructions, so arithmetic intensity is a good heuristic for good moves.