Thank you so much! Looks like I'll be soft resetting for a while >_>If what I know is right, you have 1/2 chance of having the nature synchronized, and then a further bonus 1/25 that it simply comes out right, so you have 27/50 odds of nature. Assuming any three IVs are already perfect means that at least two of them will be relevant, the last one could be attack. In that case you have three more stats to roll 31 on. With 1/32 odds for each, all three comes up at 1/32768, of course the odds of this event being the actuality is 1/4, since two of the IVs are set to be not-attack. In the more common case that it chose not to roll attack, you only have 1/32 odds that you have to hit twice, a much more reasonable 1/1024.
So you automatically have 27/50 chances of nature with 3/4096 chance of it being pentaflawless and a further 1/131072 of it being fully flawless. Your final odds are (27/50)*((3/4096)+(1/131072)) which is (27/50)*(97/131072) or (2619/6553600).
The simple answer, about 0.04% or 1 in 2500.
Also, this just occurred to me: is there a shiny lock on the cover legends this time? I haven't seen anyone mention a shiny Xerneas or Yveltal yet. For that matter, what about Zygarde?