I don't buy that there is a paradox

I haven't analyzed it with bots, but one way this can happen is if 3-3 is a really important point to occupy given that there is an opponent stone on 4-4, but inefficient otherwise. (This hypothesis would be supported if the bots don't like to shoulder-hit an existing 3-3, although of course the situation with those two opposing stones is very different depending on who has the next move.)

I'll continue and try answering dfan's intermediate question: does LZ like shoulder hitting a 3-3, which should be particularly attractive when combined with a ni ren sei?

{snip}

This means the shoulder hit loses approximately 1,4% to enclosing one of Black's corners with a keima inside or outside. White recovers the loss she incured by playing two 3-3's.

What winrate do you get for White's top response to a) the outside keima, b) the inside keima, and c) the shoulder hit? Thanks.

For outside and inside keima, White's top choice and 2nd choice are the same: approach other corner, 3-3 other corner
The percentages are slightly different for similar playouts: 56,4 and 56,2 for outside keima, 56,1 and 55,9 for inside keima

For the shoulder hit White's top choice is the stretch towards the friendly stone and the second choice the stretch towards the enemy stone. The difference lies in the playouts though, as both get 57,3%

LeelaZero 15-block network (trained by bubblesId/bjiyxo)

So by this comparison Black prefers the outside keima by a small amount, and the winrate difference between the shoulder hit and the outside keima is 0.9% or 1.1%, depending on the number of playouts.

No. I'll explain better. From a starting position with 2 white 3-3 and 3 black 4-4, the following are

Black keima enclosure towards white (outside keima)
- White approaches other corner; White 56,4 Black 43,6
- White 3-3 invades other corner; Whte 56,2 Black 43,8
Black keima enclosure towards black (inside keima)
- White approaches other corner; White 56,1 Black 43,9
- White 3-3 invades other corner; White 55,9 Black 44,1
Black 4-4 shoulder hit
- White stretch towards white corner; White 57,3 Black 42,7
- White stretch towards black corner; White 57,3 Black 42,7

Black minimaxes and chooses the inside keima, upon which White approaches the other corner. White 56,1 Black 43,9

Thanks. So by this comparison Black prefers the inside keima to guarantee an estimated winrate of 43.9%, which is 1.2% better than Black's estimated winrate after the shoulder hit. Right?

The major paradox (apparent contradiction) in the AI revolution is that the bots like early 3-3 invasions but don't play 3-3 as an opening point.

I don't think there is a paradox: AI plays 3-3 to make a 4-4 stone inefficient, not because the 3-3 point has any special value by itself.

Also: AI actually plays 3-3, see AlphaGo's match with Ke Jie.

AI plays 3-3 to make a 4-4 stone inefficient