1. You can place the Building under construction even if it's matching colored die is a 3. Example: You play Artisans and have at least 3 influence with the legions. Artisans allows you to place Castle (A Noble Building) in-play under construction even if the Noble die is a 3. Under normal circumstances, if you played a Noble building when the Noble die was a 3, you would gain 3 levels of influence and then the text effect would be ignored because a 3 results in auto fail of the text and the building would be discarded instead of being placed under construction.
So since you're establishing that the word "Building" itself on the bottom of the card is part of the "text which does not resolve if you don't have influence in the trigger faction", requiring you to then discard the building instead of keeping it in play in your tableau -- then this means that you can ALWAYS keep buildings in play WHENEVER the dice-trigger faction die is zero?