Quote:
no non-ko scaffolds with territory scores can intersect at an integer score when 1 > t > 0. Regular go thermographs are subterranean thermographs for chilled go. All non-ko values at t = -1 are integers. So all non-ko thermographs with scaffolds that intersect when 0 > t > -1 have fractional masts. Suppose that the Black scaffold of a subterranean non-ko thermograph is a vertical line, s = 1. There are only three possible non-ko subterranean White scaffolds that intersect it or coincide with it. One is, obviously, the line, s = 1. Obviously the thermograph is the vertical mast starting at t = -1. Another is the line, t = s - 2. Again, the mast of the thermograph starts at t = -1. The third is the line, t = s - 1. Its mast starts at t = 0. But it is not the thermograph of a number, but of an infinitesimal. You can do the cases where the Black scaffold is the line, t = -s. [...] But when integers do have non-ko subterranean thermographs with masts that start at 0 > t ≥ -1, they start at t = -1.
You try to tell me as much as possible about negative thermographs without answering my question:)
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Integers do not always have subterranean thermographs with masts intersecting or coinciding at t = -1. For instance,
Why "for instance"? What integers other than 0 = { | } do not have subterranean thermographs with masts intersecting or coinciding at t = -1?
- Click Here To Show Diagram Code
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The position simplifies to 0 = { | }.
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0 = { | }, and has neither a Black scaffold or a White scaffold.
Why, ACCORDING TO THE DEFINITIONS, does 0 = { | } have neither a Black scaffold nor a White scaffold? Let me try.
Siegel (slightly modified):
II.3.16: "Let G be a short game. The Left stop L(G) and the Right stop R(G) are defined recursively by
L(G) = G if G is equal to a number;
max_G_L (R(G_L)) otherwise;
R(G) = G if G is equal to a number;
min_G_R (L(G_R)) otherwise."
II.5.1: "Let t >= -1. We define G cooled by t, denoted by Gt, as follows. If G is equal to an integer n, then simply Gt = n. Otherwise, put G't = { G_L,t - t | G_R,t + t }. Then Gt = G't, unless there is some t' < t such that G't', is infinitesimally close to a number x. In that case, fix the smallest such t' and put Gt = x."
II.5.2: "The temperature of G, denoted by t(G), is the smallest t >= -1 such that Gt is infinitesimally close to a number."
II.5.3: "The Left and Right scores of G, denoted by Lt(G) and Rt(G), are defined by Lt(G) = L(Gt) and Rt(G) = R(Gt) [...] The thermograph of G is the ordered pair (Lt (G), Rt (G))"
II.5.8: "Let G be a game. We define trajectories β_t(G) and ω_t(G) (the walls of G), for t >= -1, as follows. If G is equal to an integer n, then β_t (G) = ω_t (G) = n for all t. Otherwise, first define the scaffolds β't (G) and ω't (G) by β't (G) = max_G_L (ω_t (G_L) - t) and ω't (G) = min_G_R (β_t (G_R) + t)."
0 = { | } is equal to an integer so, by II.5.1, the cooled game is Gt = 0 for t >= -1. (*)
By II.5.2, the temperature is t(0) = -1.
By (*) and II.3.16, the Black stop is L(Gt) = L(0) = 0 and the White stop is R(Gt) = R(0) = 0. (**)
By II.5.3 and (**), the Black score is Lt(0) = L(Gt) = 0 and the White score is Rt(0) = R(Gt) = 0.
By II.5.8 for t >= -1, 0 = { | } is an integer so the walls are β_t (0) = ω_t (0) = 0 and there are no scaffolds.
- Click Here To Show Diagram Code
[go]$$B
$$ -----------
$$ | . X . . |
$$ | X X X X |
$$ | O O O O |
$$ | . O . O |
$$ -----------[/go]
The position simplifies to 1.
1 is equal to an integer so, by II.5.1, the cooled game is Gt = 1 for t >= -1. (*)
By II.5.2, the temperature is t(1) = -1.
By (*) and II.3.16, the Black stop is L(Gt) = L(1) = 1 and the White stop is R(Gt) = R(1) = 1. (**)
By II.5.3 and (**), the Black score is Lt(1) = L(Gt) = 1 and the White score is Rt(1) = R(Gt) = 1.
By II.5.8 for t >= -1, 1 is an integer so the walls are β_t (1) = ω_t (1) = 1 and there are no scaffolds.
- Click Here To Show Diagram Code
[go]$$B
$$ -----------
$$ | . X . X |
$$ | X X X X |
$$ | O O O O |
$$ | . O . . |
$$ -----------[/go]
The position simplifies to -1.
-1 is equal to an integer so, by II.5.1, the cooled game is Gt = -1 for t >= -1. (*)
By II.5.2, the temperature is t(-1) = -1.
By (*) and II.3.16, the Black stop is L(Gt) = L(-1) = -1 and the White stop is R(Gt) = R(-1) = -1. (**)
By II.5.3 and (**), the Black score is Lt(-1) = L(Gt) = -1 and the White score is Rt(-1) = R(Gt) = -1.
By II.5.8 for t >= -1, -1 is an integer so the walls are β_t (-1) = ω_t (-1) = -1 and there are no scaffolds.