Struggling to understand life and death here

Hi guys,

Apologies if this is tagged wrong or in the wrong place.

I’ve been going over this puzzle for a little while now and I feel like I am missing something.

The puzzle set I am doing say that white is dead. How so?

I understand that if white plays inside at A12 or D13 then he is dead but what if white doesn’t?

If black fully surrounds then throws-in to capture then white captures and has a 4-long eye space which is live.

So how does black kill here?


Im so bad, thanks very much for the quick and useful answer & materials.