Gérard TAILLE wrote:
I simply assumed the quality of your theory is quite similar to Bill's one and I find this last theory quite valuable.
The quality is on the same level. For non-ko, non-pure-CGT endgame theory, I have written and proved ca. 6 times as many theorems, but a significant part of them would have been impossible without his prior informal, conceptual ideas. We both have studied ko theory. He has provided several times more pure-CGT endgame theory than me. So on the pure theorem count, I win hands down. Overall, our achievements play in the same league.
Quote:
Now, seeing this strange theorem I have serious doubt concerning your theory. Assuming it exists locally both a gote option and a sente option I cannot believe that you can ignore the sente option only on condition the temperature is high.
For me either you use another hidden assupmtion on the position or you made a mistake in your proof. I hope it is the first case!
"We study a local endgame with one player's gote and sente options in an environment of simple gotes without follow-ups. The temperature T is high with F < T, where F is the [sente option's] follow-up move value [...] the tentative gote move value MGOTE [...]
Proposition 132
[early endgame, creator's perspective and start, high temperature]
If F < T, it is a good approximation that the creator starts
- in the environment if T ≥ MGOTE,
- locally with the gote option if T ≤ MGOTE.
[... Proof filling one printing page applying proposition 66, whose proof covers half a page, and theorem 121, whose proof refers corollary 67 and proposition 120 for the late endgame, whose proof covers more than one page ...]
Proposition 133
[early endgame, preventer's perspective and start, high temperature]
If F < T, it is a good approximation that the preventer starts
- in the environment if T ≥ MGOTE,
- locally if T ≤ MGOTE.
Proof [filling almost one and a half printing pages applying proposition 66 and theorem 125 for the late endgame, whose proof also refers to proposition 124, whose proof covers one page...]
Theorem 134 [early endgame, high temperature]
If F < T, it is a good approximation that the starting player starts
- in the environment if T ≥ MGOTE,
- locally if T ≤ MGOTE (the creator chooses the gote option).
Proof
Combine propositions 132 + 133 and their proofs.□" [22]
No surprise you have doubts before reading the proofs;)
Citation of an extract from the proof of proposition 132:
"MSENTE < F is a necessary condition for the sente option itself being a local sente. For the average (MSENTE + F) / 2 of MSENTE and F, we have MSENTE < (MSENTE + F) / 2 < F. We have assumed a high temperature, that is, F < T. The tentative conditions suggest T ≤ (MSENTE + F) / 2 as a necessary condition for starting locally with the sente option. In summary, for starting locally with the sente option, we get the conditions MSENTE < (MSENTE + F) / 2 < F < T ≤ (MSENTE + F) / 2.
Since this is a contradiction to the assumption in the tentative conditions of the existing case 'starting locally with the sente option', it does not exist." [22]
Simply speaking, in conclusion, proved by contradiction to the assumption of a high temperature!
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Gérard TAILLE
