J+1 group

The J+1 group.  

The J+1 group on its own is unconditionally alive.

Hane.  

If makes a hane, lives.

Placement.  

The placement at is answered by .

Another placement.  

This placement does not work either.

Endgame play

Endgame play.  

The usual endgame play by Black is the placement at . After , makes a descent, and is the only way to gurantee unconditional life. Now makes a throw-in, allowing to capture a stone.

Sente.  

If decides to capture , then is sente.

Seki.  

If connects, then and makes seki. White is definitely worse off in this diagram.

J+1 group with descent

The J+1 group with descent.  

When Black descends at , it changes things. To say the conclusion first, is sente, so White needs a move at a. (Anonymous: The other descent is not sente.)

Tesuji.  

The throw-in at followed by the placement at is tesuji. The White corner becomes a bent four in the corner dead shape.

White is dead.  

Subsequently, and are miai, and and are also miai.