taking gote in error

Black gote  

(We assume that the region to the right is Black's territory.)

takes gote. White has made a local gain. Black's threat is at a, OC, which transposes to the next diagram.

Tewari  

If Black plays first in the original position, captures the stones. But now if , Black will not respond at . This shows why is gote in the above diagram.

Black sente  

By the sacrifice of , Black takes sente.

Question: In this position White plays elsewhere, has Black taken sente? Or to put it another way, is this a very good example of the subject in question. I don't say that I am right or wrong, only that I am unconvinced. Black's best local follow up seems to be at a, which is not obviously a must answer move.

Black gote  

Miai analysis follows. 2nd diagram is the questionable part, as black may have better alternatives to . This analysis presumes and are sente.

W gote: W=7  

Whether black plays or not can be disputed.

Let 1-2 and count the score. If W takes gote, W=6  

[This diagram has been edited. Previously, black mistakenly got the - exchange in sente, which is impossible.)

If B takes gote, either B=1 or W=2, ie. W=0.5.  

A(W=6, W=0.5) = W=3.25, which is less than W=3.5 for white. Black does not take a loss by answering white's first-line jump. QED.