In the thread Go maths literature,
https://www.lifein19x19.com/viewtopic.php?f=15&t=19571dhu163 asks for my endgame theory and its concepts in Endgame 5 - Mathematics applicable also during the opening and middle game. Let me give such references as follows.
General concepts: count and white-count, gote count, sente count, tentative gote count, (a player's) tentative sente count, score *, move value, gote move value, (a player's) sente move value, tentative gote move value, (a player's) tentative sente move value, gain, net profit, (concepts of) alternating sum, temperature and value of starting in an environment, high temperature, low temperature, gote-sente-difference, local region (or local endgame), environment, ideal environment, follow-up and follower, early endgame (in contrast to late endgame), larger local endgame, parity, standard area komi *, tail, traversal, value perspective
* = hardly useful during the opening and middle game.
Types of local regions (or local endgames) and related concepts: without follow-up, one follow-up, follow-ups of both players, with gote and sente options, local gote, (a player's) local sente, (a player's or doubly) ambiguous [local region / endgame], creator, preventer
Methods: applying a principle, comparing counts, comparing move values, comparing net profits, comparing the opponent's branches *, comparing two sequences, comparing three sequences, making a hypothesis *, reading and counting
Theorems / propositions (and their numbers):
- equality of move value and gains in a simple gote (10, 19)
- playing simple gotes in order of decreasing-or-constant move values (11) [Everybody assumed this the most frequently applied principle but I have proved that this is correct in general indeed. This 'first take the larger cake' is a well-known principle in number theory or the mathematical theory of alternating sums and so basic that I could not find a proof elsewhere so had to prove it presumably again by myself. No problem as it was a 20 minutes exercise, which every maths student should be able to do quickly, or should not study mathematics:)]
- equivalence of comparing a) tentative sente and gote counts, b) tentative gote move value and follow-up move value, c) tentative sente move value and follow-up move value, and d) tentative sente move value and tentative gote move value (12~16, 23~24)
- equivalences of counts and white-counts, and of move values using Black's or White's value perspectives (17~18, 25~26)
- non-existence of a local double sente (20)
- relation of the count of a local gote, ordinary ko or ko threat to the move value (27)
- the net profit of any alternating or non-alternating sequence transforms the initial count to the resulting count (29) [This is absolutely basic, of the utmost importance and characterises net profit as well-defined.]
- net profit T/2 of starting in an ideal environment (50) [Bill Spight had known this, of course.]
- maximum net profit T of starting in a simple environment (51)
- timing of a local gote (55~59) [Everybody had assumed the 'obvious' but I proved it and found bounds for premature or delayed timing.]
- timings for playing in a local endgame / region with one follow-ups during the early or late endgame, at low or high temperature, with the creator or preventer starting (62~63, 68~69, 73, 84~85, 89)
- difference of net profits of Black versus White starting alternating play (72)
- equivalence of different methods (83, 109)
- still tenuki from a local sente during the early endgame at high temperature (90)
- timings for playing in a local gote with follow-ups of both players at high, low or medium temperatures (93, 95, 99, 103, 107, 110) [In particular, at low temperature play locally!]
- timings for playing in a local endgame / region with gote and sente options during the early endgame at low or high temperature (130~131, 134)
- order of playing in several local endgames / regions (137~139)
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RobertJasiek
