In https://www.lifein19x19.com/viewtopic.p ... 71#p278371 Gérard TAILLE asks:



Usually, current theory presumes no kos now or later. Some theory assumes an ideal environment while other theory is a bit more flexible allowing any set of simple gotes without follow-ups in the environment. Combinatorial game theory about sentestrat, orthodoxy etc. has had rough approximations about +-T/2, which can be large.

My (partly Bill's) theory in Endgame 4 - Global Move Order and Endgame 5 - Mathematics for the late endgame is exact for some classes of positions. During the early endgame, my theory uses rather tight approximations 2D, where D is the ideal environment's drop. In practice, D is about 1/2 or 1 so my error remains +-1 or +-2 compared to perfect play (which you call god play).

However, such generalised theory is for one local endgame in an environment. You want to study two local endgames in an environment. Such generalised theory does not exist yet.

More specifically, some theorems presume a specific kind of local endgame, such as an arbitrary local gote with one or two simple follow-ups, or a local endgame that is an arbitrary local gote, ambiguous or local sente. Already each of some of their proofs took me up to three weeks and cover up to 11 printing pages in compressed annotation.

Depending on the theory for the early endgame, the conditions differ and include, e.g., one or two of these:


Always start in the environment.

Mgote ? T

Msente ? T1

Gote-sente-difference ? T


My late endgame theory is more sophisticated.