How is observing this pattern in someone else's brain any different, as a way of knowing, from observing your own brain doing the same thing? When "pure thought" tells you that 1 + 1 = 2, "independently of any experience or observation", you are, in effect, observing your own brain as evidence.

No no no. The difference between a priori and a posteriori is where the justification lies. You may be counting your fingers when you count 1 + 1. It may be that you won't be able to figure out the answer if someone cut off your fingers. In fact, it may be you won't be able to understand what 1 means if you didn't have your fingers. But the justification for 1 + 1 being 2 is not in your fingers.

So it may be that you are able to observe how your brain operates when you're counting 1 + 1. But even if your brain operated in a different way, 1 + 1 is still 2. If B is taller than A, and C is taller than B, C is taller than A. It may be that you're not able to understand this without three pencils. But C being taller than A is a priori knowledge.