Cancelling has nothing to do with it at Citi (for personal cards). You can reapply without cancelling (though eventually you'll need to cancel to avoid running into "you have too many cards from Citi already").

However, like I said, that's if you're doing nothing much else. That'll generate 5 to 6 pulls per 6 months, and the Citi limit is 6 pulls per 6 months (on whichever bureau or bureaus they pull), so that means to do this at this rate you must pretty much not be doing any other card apps (or credit apps of any kind) concurrently.

That's why I said it's a "theoretical" maximum rate.

However quickly you can apply, get the card, do the required $3k spend, have the miles post, cancel both the card account and the master account (business cards have two accounts, and they both have to be completely cancelled before you can apply again), and verify that the latter has happened. So while you can do this several times a year, I'm not sure how exaclty to calculate a theoretical maximum, since it depends on how quickly they send you the card (which may partly depend on which part of the country you live in?), whether you can always do the $3k spend the first day you get the card, how quickly your statement closes each time (which you can't necessarily control), how quickly you can get verification of the closure of both accounts, etc.

FYI, the Citi HH 50k churn maximum rate is easy to calculate because it doesn't require a cancellation or even completion of your spend before you apply for the next one. So you can be ovelapping a bit (completing your spend on the first card as you're waiting for them to send you the next card). The Citi AA Business churn maximum rate is hard to calculate because it does require everything to be done in the right order: apply, get the card, do the spend, wait for the statement that posts the bonus miles, cancel both accounts, verify that cancellation, and only then repeat.

The Citi application timing rules of no more than 1 card in any 8 day period and no more than 2 cards in any 65 day period are the only clocks that apply to either of these cards; however, the Citi HH 50K card doesn't require cancellation before reapplying, and so these clocks are all that matter, but the Citi AA Business does require cancellation before applying, and so you're not likely to be ready for a new application as soon as these clocks would say you would be.

(The Citi AA personal $95/year-or-less cards have an additional clock, of 12 to 18 to 26 months since you got your last Citi AA personal $95/year-or-less card. I have to say $95/year-or-less because this additional clock does not apply to the $400ish/year-not-waived Citi AA Executive card offers.)