Exodus

6 minutes ago, Ammako said:

To be fair, if it's PID-related, then you still have a 1 in 16 chance to randomly get a valid PID for Ultra Shiny.

I forgot to mention one thing, and this is my fault, but the formula uses the 5-digit TID/SID, not the 4/6-digit ones from Gen 6-7. PKHeX shows you the 5-digit ones if you howver over the SID or TID fields.

The result of the formula using your Lucario's PID, TID and SID does indeed result in 0, so this is promising. If you'd like, make a copy of it, and increase the last digit of the PID by 1. This should change the result of the formula to 3, and then check if it still has the square shiny pattern, or only the regular shiny animation.

You nailed it! Okay, I had no idea about the 5/5 TID/SID. I ran it through aaaaand: 19832 xor 19832 = 0 

So, yup. I just generated a bunch more shiny charmanders with random PIDs, and one of them is ultra. Ran the numbers through and it also = 0

Well done!

3 minutes ago, Ammako said:

Yep, seems it's probably not linked to the PID, after all. Thanks for figuring it out!

Do check my edit above, though. I re-edited my three charmanders to try to make them all Ultra, then none were after that edit. So, whatever happened with my Lucario and (the first time I edited) Charmander, I dumbassed-lucked into it. There's something else, and Charismatic Mark may have nothing to do with it (my Lucario just happened to have it, and he's ultra). As difficult as it is for me to believe the chances, the Charmander must have had "it" to the first time. I rerolled the PID and encryption constants when I re-edited him, so your maths may very well be the cause. But my Lucario didn't meet your requirements laid out when I did the Xor calculations. Hm.

26 minutes ago, Ammako said:

Out of curiosity, do you still get the square pattern effect if the charismatic mark is on a non-shiny Pokémon, or does it only appear on shiny Pokémon?

Sorry, editing this due to one simple thing: it did not work on my third try. As difficult as it is to believe I stumbled with blind luck into it, altering my charmander again and reloading it did not produce ultra square sparkles this time.

I still have my ultra shiny Lucario, and I won't edit him again (so if I need to test some maths against his values or whatnot) so am happy to help use that to figure out what it may be. Very, very odd...

That seems like a lot of math. I made my Lucario an Ultra Shiny (square sparkles) by making it shiny and giving it the "Charismatic Mark" ribbon.

Edit: (I'll note that I did the calculations above, as outlined, to see if I just happened to luck into something, but it ended up being 765600 xor 19832 = 779224, so wasn't even close.)