Black tries to make an eye in sente with . Of course, black might ignore
and run out into the centre, but in that case white can take the corner.
From here there are three ways to proceed.
But after this , it's back to the main lines of Gokyo Shumyo section 1 problem 52/Solution, although with a different move order. So white is alive. But it's complicated enough that I may have missed something!