Initial Position
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[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . . . X . X X .
$$ | O O . O X . O . O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
Main Sequences
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[go]$$B gote option, H = -12
$$ ---------------------------
$$ | O C B B C C 2 3 X . X X .
$$ | O O C O B C O 1 O X X X .
$$ | . O O O C O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
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[go]$$B sente option, S = -19
$$ ---------------------------
$$ | O C B B C 4 1 3 X 5 X X .
$$ | O O C O B 6 O 2 O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
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[go]$$W White starts, R = -23 1/3
$$ ---------------------------
$$ | O C B B C C C 1 X . X X .
$$ | O O C O B C O C O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
Gote option's gote move value MGOTE = (H - R) / 2 = (-12 - (-23 1/3)) / 2 = (11 1/3) / 2 = 5 2/3.
Sente option's tentative sente move value MSENTE = S - R = -19 - (-23 1/3) = 4 1/3.
Follow-up
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[go]$$B start of the sente sequence
$$ ---------------------------
$$ | O . X X . . 1 3 X . X X .
$$ | O O . O X . O 2 O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
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[go]$$B intermediate follow-up position
$$ ---------------------------
$$ | O . X X . . X X X . X X .
$$ | O O . O X . O O O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
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[go]$$B Black continues locally, -9
$$ ---------------------------
$$ | O . X X . 1 X X X . X X .
$$ | O O . O X . O O O X X X .
$$ | . O O O . O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
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[go]$$W White continues locally, -19
$$ ---------------------------
$$ | O C B B C 1 X X X 2 X X .
$$ | O O C O B 3 O O O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
Sente option's follow-up move value F = (-9 - (-19)) / 2 = 5.
Temperature Region
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[go]$$B Temperature T = 5 1/2
$$ ---------------------------
$$ | O . X X . . . . X . X X .
$$ | O O . O X . O . O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O C O O O . X O X . X .
$$ | . O B B B O X X O X . X .
$$ | . O B B . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
AssumptionsThe assumptions are fulfilled: we have
H ≥ S > R <=> -12 ≥ -19 > -23 1/3 (meaningful options: the gote option might be considered because its result is at least the result of the sente option; Black does not pass because the result of playing is larger than the result of letting White play first locally; similarly, White does not pass),
F < T <=> 5 < 5 1/2 (high temperature),
MSENTE < F <=> 4 1/3 < 5 (sente option due to its increasing move values).
Therefore, the tentative sente move value of the sente option is its sente move value.
TheoremAt high temperature, the theorem for a local endgame with gote and sente options is:
"If F < T, usually start
- in the environment if MGOTE ≤ T,
- locally if MGOTE ≥ T (the creator chooses the gote option)."
Application of the TheoremThe example has a local endgame with gote and sente options, and the contextual and explicit assumptions are fulfilled so the theorem applies.
We have MGOTE ≥ T <=> 5 2/3 ≥ 5 1/2.
Black as the creator starts.
Applying the theorem means choosing the case "locally if MGOTE ≥ T (the creator chooses the gote option)".
Hence, usually Black starts locally choosing the gote option.
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[go]$$B Black's correct starting sequence
$$ ---------------------------
$$ | O . X X . . 2 3 X . X X .
$$ | O O . O X . O 1 O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O . O O O . X O X . X .
$$ | . O X X X O X X O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
Reading and Counting
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[go]$$B Black starts locally, count -23, Black continues in the environment
$$ ---------------------------
$$ | O C B B C C 2 3 X . X X .
$$ | O O C O B C O 1 O X X X .
$$ | . O O O C O O X O X . X .
$$ | . O C O O O . X O X . X .
$$ | . O B B B O X X O X . X .
$$ | . O B B 4 O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
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[go]$$B Black starts in the environment, count -23 1/3, Black continues in the environment
$$ ---------------------------
$$ | O C B B C C C 2 X . X X .
$$ | O O C O B C O C O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X 1 O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]
By the method of reading and counting, Black achieves the larger, better result by starting locally.
Judgement on the ExampleThe example has these remarkable characteristics:
- The theorem suggests Black's local start. The method of reading and counting suggests Black's local start. Both methods agree. This is no coincidence but related, in particular, to theorems 128 and 129 in [22] about comparing counts for a local endgame with gote and sente options.
- The values F and T are close.
- The values MSENTE and F are close.
- The values MGOTE and T are close.
For these reasons, the example is particularly well designed.
InventorsBill Spight has first
- identified the type of local endgame with gote and sente options,
- studied its MGOTE,
- studied its MSENTE,
- proved a theorem for a local endgame with gote and sente options at low temperature during the late endgame,
- derived a proof of a theorem for a local endgame with gote and sente options at low temperature during the early endgame,
- identified the assumption H ≥ S > R (for Black's start),
- compared F and T to identify low temperature,
- identified the assumption MSENTE < F for the sente options itself being a local sente.
He has been too scared to study high temperature by theorems though.
Robert Jasiek has first
- distinguished tentative from confirmed values,
- proved theorems for a local endgame with gote and sente options at high temperature during the late endgame,
- derived the proof of the theorem for a local endgame with gote and sente options at high temperature during the early endgame.
Therefore, everybody here should have been able to study the example according to Bill's study here about seven years ago except for the theorem, whose application has been possible for everybody at least since I have stated it here. Application of the theorem is the easy part. The time-consuming part is before and the difficulty is not getting confused with what sequences and value comparisons to use or avoid.
EDITs: proofs for late / early endgame; minor corrections.