Connections I'm experimenting with
Not long, but wide.
It's been determined that 2 from the edge this is secure against one peep from the center; 3 or further and it's possible to force a ladder adjacent to the peep. For example:
Black 8 traps White 7 in a ladder. Note the play at is correct, as it protects and with tiger's mouths. If White has no ladder breaker, this becomes a perpetual Ko threat for White; therefor Black should eliminate the ladder at earliest convenience. Further, development down the line can bring a struggle by which White places a ladder breaker with a secondary function, which forces Black to choose; obviously, all rules of ladders apply here, and not killing a ladder presents a risk.
If White peeps at , Black can connect at A or capture-and-connect at B. A White peep at B creates an even worse situation, leaving Black a way to extend out a little and reduce White territory on either side.
White B to connect here looks like the better move after the peep at . This works for the center; but anything played from the third line or closer:
On the third line, this sequence kills White's invasion. If White plays A, Black threatens Atari and White B. White cannot create two eyes.
Here creates life: white at A or B produces 2 eyes, and likewise this is miai with 9 at B. Naturally, Black can't play in here to do anything but force the issue; Black can't even call this a Ko threat.
Thus, at the fourth line, White can live. Black should protect the edge side when threatened in this shape near the center. This is worth 30 points roughly.
There are two shapes I want to experiment with: