Fastest Possible Construction Builds

Manifest

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I've been searching up what the optimal use of leveling your headquarters is, and was wondering if there was some sort of calculator or already known research done into what is the fastest possible construction order to:

max out a standard offensive or defensive army
and/or to build your first noble train

Mostly what I am asking is how should you alternate leveling up your barracks and headquarters if resource cost was NO object and all your queues would always be constantly filled? Obviously you can't get headquarters to 30 as the ending building times would slow the building of barracks/stables, but you can't keep headquarters at 1 as not getting some levels would also slow the building of your barracks.

And you obviously can't just max headquarters, then max barracks, the most optimal time would be some alternation between the two right? I was just wondering what that would be or if there's some sort of time calculator for units AND buildings, or some research someone wrote up with hard data/calculations. I know there are quite a few variables, but just assuming general units, standard speed, no PP rushing etc. Sorry if this question was supposed to be easy to find, I did try searching for it.

2. If your troops are bringing in enough resources to fill up all queues, is it generally worth it (in any sense, not just for speed) to level up your resource buildings before you're maxed out on troops?