You can find on the link https://senseis.xmp.net/?MiaiValuesList%2F000To099#toc2 a list of small miai values but the smallest one listed (value 1/6) is wrong => the smallest in this list is then the simple ko (value 1/3).
Is it the smallest one?
No, Bill Spight discovered the following one (value 1/5)

Click Here To Show Diagram Code

[go]$$Wc White komaster
$$ . . . . . . . . . |
$$ . . . . . . . . . |
$$ . . . . . O O O O |
$$ . . . X X X X O . |
$$ . . X X O O O . . |
$$ . . X . X . . . . |
$$ ------------------[/go]

Is it the smallest one?
My answer is no. In fact I discovered you can reach a value as small as you want (strictly speaking the only limit is due to the size of the board).
For example can you build the value 1/16 ?

You don't want to hear -1 for the pass token...

So please show yours! (After waiting more for others' attempts.)

I do not understand what you mean Robert. Is my previous post unclear or ambiguous?

Oh, the pass token, actually -1/2, is no longer on the SL miai value list, but there is still its discussion on https://senseis.xmp.net/?MiaiValuesList%2FDiscussion

Ok, I get it, you have only considered on-board miai values and not considered environmental coupons of arbitrary, fixed size +-T.

I'll just drop this here. Hopefully this is enough to show why abstract value calculations should have nothing to do with situational factors such as who is ko master.

Area scoring, black is ko master, the marked stones are connected to a living black group, no dame left elsewhere.

Click Here To Show Diagram Code

[go]$$
$$ | . Y Y O . . . . . , . . . . . , . . . |
$$ | X O O O O O O O O O O O O O O O O O O |
$$ | . X X O X X O X X O X X O X X . X X O |
$$ | X X . X X . X X . X X . X X . X X O . |
$$ ---------------------------------------[/go]

What examples like yours show is that value calculations can become arbitrarily complicated and impractical. However, kos do depend on environments and their ko threats. Komaster is an abstract model of an environment with ko threats. More specifically, it is a simplified model of one player being able to win the ko due to enough large enough ko threats, due to which one enables such local evaluation regardless of the exact kinds and shapes of ko threats.

There are also models without komaster (or komonster) for environments when neither player has enough large enough ko threats but typically has simple (gote) plays elsewhere. This does not mean that such models would also model well environments in which one player has enough large enough ko threats.

Ko evaluation can be abstract or practical, general or for a specific example position, exact or approximative.

If ko evaluation is abtract and, as you suggest, avoids environmental considerations, it still has to relate and compare different possible local outcomes. In the simplest case, there are exactly two possible outcomes: Black wins and dissolves the ko or White wins and dissolves the ko. Either outcome is reached by some number of local excess plays. So an approximative, naive local evaluation assumes that each such local play has the same, constant gain and we would calculate this gain, a move value and an initial count accordingly, like we do for an ordinary ko (and like my naive first evaluation attempt did for a corner stage ko). As approximation, such may be good enough in practice. For theoretical consistency of values, such can fail (as we have seen for a corner stage ko).

As the example https://senseis.xmp.net/?BQMRJ000 shows, a ko can have more than two possible outcomes. Therefore, comparing two particular outcomes may not be good enough. The environment and its ko threats matter! In abstract evaluation, komaster is a possible model and does make sense but we also must not just consider the two possible wins due to enough large enough ko threats calculating some sort of weighted average; we must also consider outcomes achieved without ko fights.

Different outcomes and choosing them do depend on the environment! Unlike placid kos, hyperactive kos are not sufficiently characterised by only one move value, only one count and only one gain! (Even a simple sente is not sufficiently characterised by only one gain - the gain of Black's first play and the gain of White's first play are unequal.)

The environment can, and often does, contain ko threats. For a local ko to be evaluated, one must also taken into account ko threats of environments. If one player has enough large enough ko threats, a simple model for such is having a komaster.

Local ko evaluation is much more meaningful if environments are considered. Abstract local ko evaluation is very meaningful if also cases of environments with one player having enough large enough ko threats are considered. The name for some simple such cases is "komaster". Hence, abstract value calculations should consider situational factors, such as who is komaster.

This does not mean that abstract models would always be the most practically useful. If your example occurred in an actual game, komaster evaluation would be a great way of wasting thinking time.

Komaster evaluation of hyperactive (and therefore difficult) kos, although the model is relatively simple, tends to be very time-consuming because there might not be enough shortcuts to algebra or its visual representation by thermographs. Therefore, hyperactive kos or positions with them as follow-ups are best evaluated between games. Then one can apply calculated values for typical environmental cases like one can apply learned josekis or life-and-death shapes.

My position in a territory scoring environmement is the following (I used the bottom left corner of the board to simulate the use of ko threats, and I used the bottom right corner to simulate tenuki moves).
With the beautiful position discovered by NordicGoDojo we have now examples in both area scoring and territory scoring.

Nice but - How large a go board do you need for White to profit from this ko? 1 point must be more than 16 threats. In practice, White should be komonster or have a huge excess of ko threats with otherwise at most dame remaining.

As mentionned in my first post the only limit is the size of the board but that is not the problem because it is only an example that has to be analysed in a pure theoritical approach (I am not able to build N tenuki moves with value 1/N).
Obviously it is the same for NordicGoDojo's position in area scoring.
No, 1 point must be more than 16 tenuki moves (black has no ko threats)

Outside CGT definitions, I am used to call every pass or non-pass move elsewhere a threat because it enables recapture. But, ok, for the stricter sense of threat having a follow-up move or defensive reply, (ordinary) tenuki is the right word here or more specifically a play in a simple gote without follow-up.

Having thought a bit more about these things, I wonder: what would you suggest as value calculation independent from the environment / situational factors?

Possibility 1:

Assumption of all moves of Black's and White's alternating sequences to have the same gain. Such occurs in a simple gote. However, already for a simple sente, this would wrong. E.g. a reverse sente can have a 1 point reverse sente gain but a large gain of each move of the sente sequences. We must not increase the 1 point by forming an average gain and using it for each move. With already a local sente being problematic, we should not expect kos to be better.

Possibility 2:

We might consider two values as either player's best case: Black controls the whole local endgame region; White controls the whole local endgame region. Now, what would these values tell us and what would be missing in comparison to having counts and move values?

Other Possibilities?

Consider a situation such as below.

Click Here To Show Diagram Code

[go]$$
$$---------------------------------------
$$ | . X . . . . W . .
$$ | X X . . a b X O .
$$ | . X . . . . X O .
$$ | X X X X X X O O O[/go]

As far as abstract value calculation is concerned, ‘a’ is always superior to ‘b’; this is due to the higher value that black gets per played stone. This does not preclude the fact that, in the event that there are no other endgame moves left, black ‘b’ in fact gets one more point compared to ‘a’. In this case, black’s achieved value per stone is lower, but black ends up with more territory locally thanks to her being able to play an extra stone locally. This latter possibility is something that should not be forgotten, but should not be used in value calculations.

Next, consider the following.

Click Here To Show Diagram Code

[go]$$
$$---------------------------------------
$$ | . X . . d b W c .
$$ | X X . . X a X O O
$$ | . X . . . X O . .
$$ | X X X X X O O O O[/go]

By abstract value calculation, we assume black plays ‘a’ locally, and later in the game black ‘b’ is black’s privilege, forcing white ‘c’. (Else, black doesn’t play anything and white gets to play ‘a’.)

If this is all there is on the board, after black ‘a’ white will play ‘b’, followed by black ‘d’ and white ‘c’.

To try to get extra value, instead of ‘a’ black may play ‘b’ if she has enough ko threats. If black wins the ko, white ‘c’ and black ‘a’ may result. However, as far as the local shape is concerned, the ‘b’ ko is far from fair (because white almost has nothing to lose), and black should generally expect to make a loss from it; it will be an effective tactic in a tiny percentage of games.

Now, we might add considerations such as ‘ko master’ or ‘ko monster’, and have black ‘b’, white ‘c’, and black tenuki happen, so that black can get to play an endgame move elsewhere while still expecting to get to connect at ‘a’. This will maximise black’s profit due to her getting to play a larger number of valuable endgame moves than white.

I cannot see how all of these possibilities could be unified into an endgame theory that is still useful for players; considerations such as the ‘b’ ko should belong to the magisterium of reading, not counting.

You use the phrase "abstract value calculation" as if there was only one such thing but there are different models involving abstract value calculation. Some of them include last move considerations, some include a komaster concept and different abstract environments can be assumed.

Some abstract value calculations consider that a move A is superior to a move B in some situations but inferior in others.

If a local endgame has two move options A and B, they need not always be gote. One has to check, in particular, whether one of them is sente. In your first example, Black A - White nobi needs to be checked whether A is gote or sente.

Not all abstract value calculations only consider per move values. Therefore, if a move A is superior in one model assessing a larger per move value, it does not imply that move A needs to be superior in all models of value evaluation.

It is well known that getting the last move matters and some very sophisticated models focus on taking this into account. Such analysis can be useful for positions near the game end. Other models are designed to ignore the detail of the fight of getting the last move. For some classes of positions, such models nevertheless enable finding correct moves to including the last move of the game. In other classes of positions, such models can be wrong by about 1 point because they neglect the last move fight. The part of combinatorial game theory also considering infinitesimals is designed to also consider the last move fight. Per move value endgame theory is designed preferably for application earlier than the latest endgame, simplifies and does so by ignoring the last move fight. Nevertheless, for some classes of simple positions, it can be exact.

We do not have a unified endgame value theory that would be the most appropriate and correct for all positions. As you point out, some positions profit more from tactical reading etc. than from endgame evaluation. However, you should not characterise tactical reading as value-less. Simplified tactical reading works with symbolic results, such as "dead" versus "alive" but more accurate tactical reading is, e.g., the method of reading and counting, which coonsiders counts as the results of variations and decides by min-max on the counts. E.g., instead of "dead" we might assess "-30 points" and instead of "alive" we might assess "+2 points" as the resulting counts.

Ko fights can often be evaluated approximately by the method of reading and counting. If, however, a position is too complex to explore relevant variations sufficiently, kos must be evaluated by value-driven models. Hyperactive kos in local problem books should be discussed as to their different behaviours and outcomes depending on different environments and ko threat situations when the problem shape occurs in whole board positions. Such discussion can study different example ko threat environments or use abstract values and concepts (such as komaster). Problems having hyperactive kos as follow-ups can get out of hand in problem books if one would want to demonstrate various example ko threat positions; abstract value characterisations can be more appropriate to discuss larger numbers of problems.

Value models and tactical reading are not enemies. One should consider both tools and apply whichever is appropriate. Often one can combined both and other tools.

That you cannot see how all of these possibilities could be unified into an endgame theory that is still useful for players is no surprise. There is no unified go theory yet so nobody knows yet. For centuries, all that can be done is to further develop the various theories, models and skills, where some partial unification should be sought.

I think I understand what you mean NordicGoDojo, ko consideration like "ko master" or "ko monster" should not belong to counting and I agree entirely with you.
But you seem to oppose "reading" and "counting" and on this point I cannot agree with you. Robert said "Value models and tactical reading are not enemies" but my view goes much further.
For me "value calculation" is completely integrated in "reading" technics.
Think about it. Take your position and imagine a real (complex ?) environment on the board.

Click Here To Show Diagram Code

[go]$$
$$---------------------------------------
$$ | . X . . . . W . .
$$ | X X . . a b X O .
$$ | . X . . . c X O .
$$ | X X X X X X O O O[/go]

White has just play the circle stone and now it is black to play. Black can play either locally (at "a", "b" or other point like "c" and even other points which may appear playable for a beginner while very weak for an advanced player) or play somewhere in the environment.
Assume you are very strong at reading (as a computer is). How will you proceed?
For a theoritical point of view it is easy : you choose a first sequence form the initial position till the end of the game and then, beginning with the leaf of this sequence, you look to sone improvement by white or by black, changing your sequence to a better one, and continuing until no more improvements are still possible. At the end you have found an optimal sequence.
The point is the following : this procedure works perfectly but may be extremely long especially if you choose your first sequence at random. On the other hand if you are lucky and you choose an already quite good first sequence then the procedure will be drastically shorter. Here comes for your help the move value calculation. You choose your first move according to the result of the value move calculation and you continue your sequence in the same manner by only considering the value move calculation etc till the end of the sequence.
As you see the idea is to choose the first sequence with a good chance it is almost the best one. That way the following reading will be far quicker.
If your are lazy you can stop reading after having found this first sequence but a strong player must continue the process.

To summarize:
In the reading process you have two phases
1) Choosing carefully a first sequence (with the help of move value calculation or any other method)
2) Looking for improvment till the optimal sequence is reached

I consider "move value calculation" is an efficient a way to improve your reading itself.

Click Here To Show Diagram Code

[go]$$B Black to play
$$----------------------
$$ | O a O O O X . X . |
$$ | O X X X X X X X X |
$$ | O b O O X O O O O |
$$ | O X X X X O . O . |
$$ | O O O O O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$----------------------[/go]

Do you know that even in case of obvious simple gote areas the environment might contradict the move value calculation?
In the above position on a 9x9 board and assuming AGA rule, can you believe that it is possible to build an environment (on the four bottom lines of the board) such that the move at "b" is the only best move!

What do you mean by this? What kind of counting? Why?

For me the local position you analysed may be any kind of position including all types of ko you have in mind but I do not like ko assumption in the environment. I prefer to always say that neither black nor white is komaster.
You can easily find examples in which it is better to assume black (for example) is komaster but in practical games black is rarely really komaster; black may be able to win a ko but in that case white will often gain a (small?) compensation.
In addition it looks not easy and maybe unuseful to have to choose between a calculation assuming black is komaster or not.

IOW I prefer simply to always calculate the move value assuming no ko threats and to handle ko threats only in the second phase of reading (improvment till the optimal sequence is reached)

BTW do not forget that you can always add a ko threat to your local position if you think it is really relevant!

It is unnecessary to enhance a local endgame by a ko threat in the shape but it is sufficient to use something like the komaster rule; he may while the opponent may not recapture the ko.

What does the move value of assuming no ko threats / no komaster tell you for actual play in an environment with ko threats?

There are practical games with one player having clearly more ko threats. This affects strategy throughout the game. It is not rare. The model of this player being the komaster is a simplification of the actual game.

The difficulty is not assuming some komaster. The difficulty is application of thermography regardless of whether there is a komaster.

Yes Robert but that is not my point.
What is rare is that the ko loser has no compensation at all for losing the ko (I mean no secondary or tertiary ko threats).

@Gerard: Yes, it is rare is that the koloser has no compensation at all for losing the ko. However, why do you have a problem with this and, e.g., komaster evaluation? Such a model does consider compensation as gains T of each play elsewhere. Only komonster evaluation allows T to drop to 0 for no compensation.

@NordicGoDojo: Another remark on possibly unifying endgame evaluation theories and consideration of the last move: If we could already unify all endgame evaluation theories and solve the game, very likely it would be impractical to apply such a unified theory during a game. Combinatorial game theory started with exactness also considering the last move of a game. Bill Spight's endgame theory and my endgame theory involving move values, or the combinatorial game theory's orthodoxy and thermography are endgame evaluation theories that, in the general case, are approximations and ignore, in particular, consideration of the last move. Unless very late during the endgame, it is an advantage and great simplification to ignore, in the general case, the fight of getting the last move.