RobertJasiek wrote:
Presuppositions
Suppose the local endgames each with one follow-up M0|F0 and M1|F1 (the sente move values M0, M1 and follow-up move values F0, F1) of the creator with M0 ≥ M1 (*0) and M0 + 2F0 ≥ M1 + 2F1 (*1) in a (possibly empty) environment E without kos now or later.
Theorem 3
The preventer's start in M0|F0 is at least as good as in M1|F1.
Discussion
We can write the local endgames as {2*F0 + K | 0 + K || 2*M0 + K} and {2*F1 + L | 0 + L || 2*M1 + L}. We do not specify whether either local endgame is a local gote, ambiguous or local sente because we do not specify M0 ? F0 and M1 ? F1, that is, whether the move values of either local endgame decrease, are constant or increase. Since we can write the local endgames as before, M0, M1 are the sente move values. However, we should state the presuppositions (note to myself: and annotate the font) more carefully: M0, M1 are the TENTATIVE sente move values. Thereby, the theorem also applies to one or two local gotes, whose gote move values we need not know. It was my mistake to just pretend that local endgames' moves values would be sente move values. Instead, I must declare to use the TENTATIVE sente move values. Does this solve everything?
Some 50 years ago, years before
On Numbers and Games came out, I realized that a complete theory of the endgame could be constructed using only local scores, as traditional evaluations and move values were only heuristics. To that end I came up with some idiosyncratic names for score differences.
For instance, although I did not use slash notation, for the game, G = {a|b||c|d}, where a > b > c > d, I called b - c the double sente value (which was the traditional name), but I called a - d the double gote value (definitely not the traditional name). I called a - c the forward gote value and b - d the backward gote value. For the game, H = {e|f||g}, where e > f > g, I called f - g the sente value, regardless of whether H was a local sente or not. The concept of local sente is only a heuristic. I called e - g the gote value, since there was no forward gote value or backward gote value unless there was an odd number of dame along with g, and in that case they were equal, anyway. For the game, I = {h|i}, where h > i, I called h - i the gote value, which was the traditional name. I did consider values such as b - (c+d)/2 for G, but only as approximations.
I am not sure, but it seems to me that you are taking the heuristics as basic. The heuristics are quite useful. Professor Berlekamp recommended starting the analysis of a game by deriving its thermograph. But it's the final scores that are basic.