H1>F2>G2>E2>E3>F1 is death as far as I can see 
EDIT: Oh no wait, lmao it’s just seki
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Yes, looks like seki. If there’s clear life and seki then naturally a book would recommend G1, right?
Most problems in this book have a single answer for life so I thought I might be missing something.
Furthermore, the seki is gote and leaves a huge yose at J1, so it’s not even clear which is better pointwise. (Writing this, I realise white can answer J1 and let you live, so playing H1 is giving the choice to white)
Anyway, thanks a lot for the double-check!
This order gives sente to white (so gote for black)
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Absolutely correct, thanks again.
Still I think the seki is in general inferior for white, because the yose on the right would be 7 points sente on an empty board, so white would rather play J1 in sente and let black live. But now we are too far from tsumego, I guess.