I think I may have managed min primary here, I haven't really checked period 5 or 6, so I could be wrong, and there may also be some wizardry to shave 1 or 2 area at period 4, but I'm very happy with this solve regardless, since I got to actually make some hexarm wizardry happen.

I quite like the way 3 * area + 7 * period works for this puzzle, I'd be curious to see if it's also good for other puzzles, it seems to lead towards very efficient machinery, similar to sum but a bit different.