I speed built a missionary center on a world about 3 hexes into Thalan territory. I was still in their territory. Then I speed built a consulate. I was still in their territory. Then I speed built an Embassy in a triangle with the other 2 buildings. A few turns later, my planet has an influence radius of 26 and is beginning to culture convert a Thalan colony 11 hexes away.
The problem seems to be that level upgrades boost base influence by 1. So my missionary center is +1, +5 (Ancient wonder and 2 adjacencies), and +50%; my consulate is +.5 and +3, and for some reason my frozen world has +25% influence. Throw in approval and snuggler colonies, and I now have 19.6 growth per turn. I also have 2 numbers for influence, 283 and 235.
Level based adjacencies for influence should be % based, not a flat +1 per level.