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 Post subject: Re: Numerical evaluation theory of thickness
Post #21 Posted: Thu Aug 11, 2022 12:10 pm 
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What concepts are derivable from this sort of theory that isn't covered by sente/gote etc. ?

Well, shielding is certainly one. Aji too (though the theory isn't strong enough to estimate it well, only informally tell you principles). Weak points as a key source of value of moves that gets split up as more moves are played. Each move is like splitting the area around into more independent endgames. If you take a vital point of life, it may be difficult for the opponent to make eyes in each independent region, so you may be "winning" in each, and profit arises from this.

My better guesses for komi tune to the known k=7 for d=2. So suppose it takes d+1 moves to capture the corner.

Perhaps m+n=(6d-4)/(4d-2). And m-n= 1-2*(-d-1)

I think this m-n value is the closest I've come to JF's units of thickness/atsui

Then
k=[4d-2 + d(6d-4)errf( (1-2*(-d-1)/(4-2/(2d-1))]/2

Which predicts

d=1
k=1.5
note that this is unreliable as (4d-2),(6d-4) break down for d<2. However, the answer does seem reasonable as any stone you place on the board that has 2 liberties lives.
d=2
k=7
d=3
k=15.7
d=4
k=27.8

what goes wrong for the complete graph? well, the smallest group size, (4d-2) increases with degree and the number of neighbours, and it is larger than the whole board, so we can no longer talk about the "influence" such a group has on (6d-4) etc. because that is outside the board and so on.

This tends to something a bit more than

(3d^2+2d-2)/2

which has roots at (-1 +/- sqrt(7))/3 . not quite golden ratio.

Funny that d=0 gives k=-1 . There is only one intersection and play there is suicide.
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 Post subject: Re: Numerical evaluation theory of thickness
Post #22 Posted: Mon Aug 15, 2022 1:24 pm 
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When points seem obvious, attention turns to how best to present and which words to use. Some ideas here are already an obvious improvement in how to think (at least for me), but much is fuzzy.

Today, I am pondering on two seemingly unrelated topics.

How does 2d affect the board concepts the opening/middlegame compared to well understood corridors in endgame theory?
How do physics/chemistry analogies help/apply/confuse?

e.g. Entropy, temperature etc

In physics, entropy maximisation, stationary action etc talk in terms of some optimisation. In Go, we have optimal play (lets say for score), and Go is discrete. Can this be thought of as some approximation to a problem of optimising one continuous function. I think it should be possible (in fact it is by definition in the game tree), but I haven't found a way merely as a function of the board graph, which is what I am focusing my research on. Perhaps its best to simply think of both players applying a "strength" of pull on the board. Equal strength should lead to equal score up to randomness. However, Go is not exactly stable (though not exactly unstable either except in semeai), and moves having positive value meaning that a single move usually doesn't affect the score that much. So on average, equal strength leads to equal score. Without even averaging, perfect play leads to perfect score deterministically as it is a combinatorial game (perfect information, finite, two player).

This is like a problem in non-equilibrium thermodynamics. There are many possible low temperature states that the game tends to over time (heat death), with completely different scores. Both B win and W win are stable possibilities (or else there is nothing to fight for). The board is the medium of interchange between the two players who are sources of energy like a star? The colder player draws the opponent into traps. When B plays a move, there is an interaction between B and the board that absorbs B's will (maximise their points) and moves oscillate between B and W. But which intersections maximise B's points and how does that change after a move? Are there waves spiralling outwards from a fight? Or could say that B has infinity temperature, W has -infinity temperature, and this is how they absorb the value of the board over time. Though this is probably not a good idea as the game is symmetric between B and W, and this tends to be misleading when compared to standard thermodynamics. Go is quite random in the sense that slightly different starting positions lead to vastly different outcomes on the board (and board positions are the only physical thing we have access to as observers), basically chaotic as it is so unstable (at least in perfect play), but principles are common to each timeline.

Temperature is a common term in Go endgame theory, but as I have studied some thermodynamics, I don't really want to use it here without understanding if it really matches the thermodynamics concept, or if it is actually a slightly different concept. Otherwise bad analogies will be made. In physics, temperature is defined as 1/T=dS/dE (entropy,energy)

Stephen Wolfram seems to say that energy is akin to computation activity. Physically, high energy tends to mean you don't know how the materials involved will spread and react (or where they came from, going backwards in time). And yet it also implies that it is stored and insulated from externals at the moment, not yet forming bonds and attachments that lower the energy. Or perhaps it is to due with more physical fluctuations than time fluctuations. It seems that by friction, hot systems tend to lose their energy to the environment until they reach a much lower local energy state. In the case of a ball rolling in a parabola, it falls to the bottom and remains there at the extreme low of energy local to this system. But by conservation of energy, this energy needs to go somewhere, and that may mean small (mechanical or even quantum) oscillations likely remain. Statistical physics uses temperature to balance the energy exchange of different (mostly) independent systems. High temperature systems are likely to give their energy to low temperature systems. If so much energy is stored invisibly (say in air heat, or the mass itself), this suggests that these are comparatively low temperature for their energy. Invisible suggests that our conscious brains don't interact with them directly. This may be because they are small and similar temperature to ourselves. Note that higher temperature systems change (entropy) less compared to lower temperature systems.

If moves are particles, their high or lower value (or energy change?) depends on the stability of the position and how many other unsettled intersections are nearby that it affects. Is the move itself stable enough claim the whole potential or only a portion?


Weak point theory in 2d
From the point of view of territory and eyespace maximally bonding to a secure group (pass-alive ideally), what about potential territory and insecure groups?

Given an insecure group, there are a sequence of possible directions to attack it. The opponent will attack from their best direction until the group dies or the moves aren't valuable enough. The best direction tends to be connected with other attacks into a potential moyo, or perhaps it is on the side/corner with better control of mutual eyespace (Territory before eyespace before potential is a reasonable rule of thumb). This is especially useful in securing (at least for now) both an attack and supporting weak attacking stones around if the defender must leaning from the centre on them as this helps the attacker reinforce their connection to side eyespace.

Likewise the defender will likely play on a specific set of these weak points themselves in response if they want to save their group. Aliveness and killing each take a finite number of moves, and at vital points, these often overlap. Normally, any attacking move is a valuable defensive option. Both are close to the heavier (larger, closer to boundary of life and death) weak group in question. Heavy is proportional to size of group plus potential (times appropriate reducing factors for aliveness to prevent double counting) and exponential in distance from life and death boundary. At the boundary it is highest. One move away is 1/2, two moves is 1/4, etc, as usual. However, not all chains of the group may be at the same point relative to the life and death boundary. Those closer to the outside might have sacrifice techniques that threaten to rescue them by leaning on the weakness of the attacker's seal.

Often, stretching a little to cramp the opponent in the short term, even if they get better shape than normally locally by spending two moves (in the long term) may profit, because those two moves might not be valuable direction for them (e.g. next to your living group even if they break your potential). For example, a 3-3 invasion sometimes continues with a one space jump instead of the knight's move, just to block the 4-4 wall's potential on that side. The corner is thinner, but taking more of the development potential also means that the wall is more attackable from the other side. This can make a difference if that side may be more valuable than the corner.

Stretching is the attacker's responsibility. Normally the side with eyespace has more influence locally. If you are attacking, you often have less eyespace since you have to come close to the opponent, and the opponent may pincer to sacrifice. Your eyespace and much potential territory comes from the potential kill or at least follow up attacks.

At the same time, if your attacking stone is too weak (which probably means the defender is too strong, so consider adding nearby support first), then the defender may threaten that attacking stone perhaps attaching or at least occupying or cutting it from the next follow up, or even pincering and counter-attacking. Then at the least, the attacker may have to respond and such a play may be gote, or not gain as much as simple estimations of value will predict. Don't overcommit to weak attacking stones (they are light) as even if they aren't easy to capture, one stone isn't many points, and their potential is already reduced by the defender's strength locally. This means delay adding a move to defend them. Ideally threaten to sacrifice them by profiting from other attacking moves first (using it as a threat without even losing a move reinforcing it), and only develop with them if the opponent remains weak or threaten to capture it on too large a scale. Remember even then that their main value comes not from making territory locally but from threatening the opponent's weak points. Even if the opponent doesn't have many weak points, playing higher to reduce the opponent's more solid potential may be better than a lower territory that remains flimsy.
A higher thinner move can make miai of a larger territory and just connecting. A tighter move is better to make secure territory/eyespace locally and prevent any incursions by the opponent, and is more appropriate for the defender of territory/eyespace.

They say don't play close to thickness, but where should you play? The above tries to explain that playing close to weak points (your own or the opponent's) is good but only if it works and is strong enough to support follow ups (even if the opponent responds). Bad shape arises when you play near weak points but in the wrong way, either inducing a defence, or spending too many moves defending (perhaps without probing the opponent's weak points and hence making the opponent's attack light). The worst extreme is playing in no man's land where both sides have strong stones and all areas are interchangeable. Hence each can be sacrificed with profit from tenuki or leaning on it with one's own local strength. So securing more intersections locally is less valuable. Sometimes there is a wall with few weaknesses (connected to a living group) that is threatening a large territory but you also have some stones there with some control of eyespace. This is not quite no man's land then, though you may still prefer to play lightly and consider sacrificing. An area which is may potentially be large B territory or W territory (without semeai, just both have large walls) is not that rare, and may arise from each taking territory on the other side and pushing the opponent towards such a less valuable area. This isn't quite no man's land, but is unlikely to grow very big. It is like: you take low, then I take high. You play high, then I invade.


Balance, miai, evaluation of bonds
Weak groups and potential territories often find themselves in balance with each other or close to in perfect play. The less clever explanation is that this is because highest temperature moves are played first, so that remaining moves are often close to similar in size if they are large. The clever explanation is that this is planned since having big moves in miai is good insurance for the second player. In perfect play, the first player shouldn't expect to get much profit except at similar cost. Biggest moves are those concerning weak groups or large territories. For example, if a group A is weak and doesn't have much potential (i.e. the opponent is strong nearby), then the value differences locally may be dominated by the attacks and corresponding defensive responses. If you can be certain that play will occur on such areas, then even if the local areas isn't so big (perhaps they are already close to miai), premptive control of the attacker's potential (often by 3-3 invasion or approach), or threaten a bigger moyo with the escape route. So in good play, the defender (who normally has more control of their group) should try and is able to balance the major attacking points by supporting their weak groups when choosing direction of play before they are 1 or 2 moves away from being forced to respond. We can call this "playing close to a potential fight", or just that the defender should try to make more options for themselves, even at the cost of territory. Of course the attacker tries to prevent this but can normally only choose the largest attacking direction in combinations with leaning on other weak groups of the defender. This is why they say playing Black is harder than playing White. The attacker maximises profit with a splitting attack on the defender's nearby two weak groups. (driving tesujis are the same point even if not a long term attack, but only one weak single cutting stones that were already heavy and quickly grow).

The least valuable region next to a wall such as the adjacent points is most likely not going to be competed for, and hence is most likely not going to be competed for as it isn't valuable for eyespace. So it can be thought of as part of the chain since its life and death is so closely linked, even towards the end of the game in dame filling. This is another way we actualise influence of strong shapes.

Similarly, if the opponent builds a moyo, it is often a good idea to build a moyo adjacent. This fight for the centre should make it more awkward for them to expand their potential (as you can just answer with a large territory with also threatens their boundary stones) and also to invade (as your strength or centre control from attacking will help support invasions of your own, or reductions, or give you a larger centre potential that works with reducing moves). In general, centre influence gives you more tuning "control" of how the game flows. This is in part because there is more balance between the value of the opponent's territory and their potential, so you have more choices of strategy which is better if you know how to use it, and if it compensates for your loss of territory. Normally the opponent's territory is much more valuable for you to fight for especially as it is a source of mutual eyespace. In part it is also that if they lose their territory/eyespace, then they may find it difficult to get compensation if you control the centre so they may remain weak without much potential (of course, your potential is likely reduced too).

A lives vs A dies vs A dies and the local area is the opponent's potential.

rather than territory. But perhaps there remains space to fight for locally. Then, the miai balance may be to play looser or higher to make A's potential territory miai with A's connection back to security.

If instead it is the opponent that is weaker locally, then there may be many value differences locally. There may be

A lives vs A is attacked vs ... vs A sacrificed vs A dies
A kills the neighbour opponent chain vs A leans on the opponent to build vs A has small centre potential

Then, the stronger side may need to balance many things. The largest term may be preventing the opponent's attack at all. That is the balance. This may mean you occupy that point even if it is close to your own group. This can be called removing aji, thick play, ready to attack any opponent group nearby. Then it may simply be endgame and territory in one sector may be miai with territory in another.

Evaluation of bonds (territory with group) (enemy groups)(friendly groups)

Here we think of binding board intersections to chains and moves. This is "territory", "development", "potential", "aji", etc. Binding to territory is stronger if the chains are more alive. It spreads further if the opponent finds it difficult to live locally. Moves are worth how much they change territory. They are worth most when affecting a large area and when at the boundary of life and death (perhaps for multiple fighting groups). The total binding strength can be thought of as the sum total of this control/influence. The marginal can be thought of as the value of the largest move there. i.e. what happens when a player makes a move that vacuums up value?

A weak group should normally expect a positive impact overall unless the negative impact dominates, namely losing the possibility of using that intersection as a liberty or eyespace. But merely from attack, a group should expect a positive impact compared to that group not being on the board, even if its value comes from being able to sacrifice it for some profit elsewhere. You can't get worse than being dead in Go, so if a group isn't worth saving, its owner simply won't play there and neither will the opponent. If an attacking fight occurs, the group must have positive enough value to be worth saving. However, the direction of impact may also be predictable. It will likely be forced to live in a relatively low value area where it isn't so hard to make eyespace, while the opponent gets influence around. We can say that a weak group may have a shell of positive influence (positive factor that is, so more akin to itself) on the most nearby intersections, but a negative influence on the places the opponent can secure by attacking it.

From this point of view, the analysis of A above can be though of as:
If nearby to A, the opponent is strong, this cancels out the negative influence of the weak group already. Each move may be fully positive, in proportion to how alive the group is rather necessarily about territory and potential.

If nearby to A, the opponent is weak, then A may be alive enough to get a large proportion of the positive influence locally with any move. The main additional value is preventing the opponent's sacrifice strategy and double threats by minimising their shell of negative influence.

When two friendly weak groups are nearby, the attacker may find value in cutting even if such a move is an overplay relative to each. We can say that the negative influence overlaps too much with the other group. In the most severe case, one weak group (the smaller) will have to be sacrificed for the other to live. That is a successful double attack. An overconcentration would be if the defender must play a slow move to connect the weak groups to prevent such a threat. If optimal this means there is no active way to defend. i.e. defending one's own weak point and claiming the extra potential by leaning on the attacker's weak points isn't worth the cost.

When attacking weak points where there is a cut, it may be that the defender already has reasonable eyespace on all sides. Then the attacker should consider balancing the direct cut with more control of the eyespace region. The cut probably should be the focus, as attacking can get the extra forcing moves to profit, but the largest threats arise from more immediate threats, so remaining within 1 or 2 moves of being able to cut is important.

I'm not sure if there is some kind of inverse square law of gravity/charge that says when friendly/enemy groups are attracted to the same territory/eyespace.

An ideal theory would be able to prove good enough approximations that in order to refute a move, all you would have to do is show the refuting move, count and compare. For example showing that 4-4 8-3 enclosure is too vulnerable to the 3-3 for this to be efficient.
In terms of automation, I think that at the least, each large weak group should have slots for the order the opponent is expected to attack in with a field of expected influence in that scenario. This guides maximin to which direction to play in globally to compensate local imbalances. If the group is so weak that a narrow 1d path to life needs to be found, then many slots may be needed. It may need miai at each stage between living with another eye or escaping towards a potential other eye.

I realise that this does end up fairly related to GnuGo, so perhaps I didn't give them enough credit. But this is also a guide for how humans should think when they play and evaluate in a review or use tewari.


What is a fight?
What are standard fighting possibilities? Why is there a windmill spiral wave?
A fight is probably one where all attacking possibilities are heavy. During a fight, the temperature may increase, with threats getting larger since they get closer to the boundary but also may concern the life and death of a larger area as the cutting stones involved become larger groups. However, in optimal play, the end result will be balanced overall (extra value for the side that spends more moves, that is close to the average temperature during the fight).
Evaluations and decisions may be as difficult as a semeai, which is basically a fight where there is no room for retreat and life and death is completely dependent on killing the opponent (even if full evaluation may allow sacrifice).

The choices are often about status and judgement. Which cut to take?

Just live vs Capture the weakest boundary stone vs Cut the weakest boundary stone first vs Cut into empty space with a stronger cutting stone.

Many modern 4-4 joseki have these choices repeatedly appearing. If the opponent is too strong nearby that you don't have much potential (i.e. little buffer room before you have to live small), you should try to live faster. If the opponent invades the 3-3, perhaps play the double hane which connects all your stones but takes gote and reinforces the opponent's side control on both sides. This can be slack, i.e. not attacking at the maximum and hence making the opponent lighter. However, your centre gets stronger and the opponent's isn't alive yet. It can be used to sacrifice an area where you are already weak.
Other 4-4, 3-3 invasion joseki flow into the outside, giving way in the corner. The most complicated ones refuse to let the corner out into the sides but leave two cuts on the outside (and two unsettled boundaries). The judgement is often that the attacker should choose the most valuable point and the defender should respond accordingly.
Some 4-4 joseki have the attacker getting eyespace on both sides by attacking the corner and sealing it. This is common to reduce the opponent's potential when there isn't a clear direction to build the 4-4's own (perhaps because building leads to balancing it against the corner's development which may be too large since the corner is alive). However, they become thin this way and the corner can live and cut the outside with local pressure on both sides from the centre which can be used to sacrificed on just live easily and eliminate the attacker's potential.


Other
In a cross fuseki, the value of a 3-3 point stands out as approach may be met with the small knight's move which costs you on the other side which was also your potential. However playing 3-3 directly loses you centre potential too, but you expect some return from being able to attack both sides.

Since eyespace has more influence than a weak group (which may have negative influence), then if when it comes to choosing which side to block for a 3-3 invasion to a 4-4, consider that the threat of a large territory also has influence but not as much as the factor a living group has on its potential. Normally don't block in a way that pushes the opponent into your territory because that may make it too small for too little gain. Be careful that the local situation may remain tense as weak points remain, especially if there is a cutting point. This can give one side the extra influence of several moves in a row locally.

The search for the "divine move" (in Hikaru no Go) perhaps asks for a fair result on the board. But biased in practice due to player's differing experiences.

What is "pressure" in Go? It is in proportion to strength perhaps. But increasing pressure locally is perhaps about asking opponent to fight more locally or else lose out locally. Like the 3-3 invasion, one space jump variations.


To summarise the key lesson of the lengthy post,

The opponent's influence near your group is related to how easy it is for each side to live there, and may be positive if you are weak. However their total influence should be less than if you didn't have a group there at all. This should inform my estimate of komi, though I don't know how.

I think it must be local balance equations but how exactly?

power
If power (Dahl 1957) is to do with A getting B to do something B would otherwise not do, perhaps weak points interact when one with high temperature (or at least larger follow ups, perhaps in a semeai) affects the local main line elsewhere (nearby, or perhaps a ladder breaker).

Note that the threat of follow ups is more powerful than the execution holds perhaps under this definition. The execution loses the aji of different threats.


tightness on weak points
by playing tighter on a weak point, you play closer but your own stones may be in the short term weaker. However, the opponent's weak point may remain weak if their whole group is weak unless they can sacrifice or cut through your weak point.

full tightness may capture their stones.

Slightly less than full tightness may be ok when capture doesn't work or isn't worth it, but often also helps the opponent reinforce their weak point. Consider the classic variation of the 3-3 invasion to the 4-4 with the hane but not double hane. The 3-3 lives in sente, but in return, the 4-4 still has some endgame on it and solidly poking into the side, which may be good if this damages stones their. However, if there is no centre/side support for the influence, this can lose control of shape and eyespace since the 4-4 still has a cutting point and the weak points of the 3-3 are much smaller (the corner can't be captured). Hence, the 3-3 invader may be able to take sente to reduce the side first and sometimes even force the 4-4 to live small.

This goes downwards towards just solid connected shape (as is more common today in 4-4 joseki) which asks the opponent to balance weak point defence with reducing the outside influence, so weak points will likely remain, but the influence of the outside may not be as extensive, but it will be stronger where it exists. And the 4-4 gets sente.

__

When both have a weak point, consider the direction in which you can build a moyo. Try to make miai of a bigger moyo and playing closer to weak points.

If the opp has no weak point, avoid playing nearby, but defend if you must. If the opp has a weak point, try to defend actively by attacking to help connect to your weak groups or make moves smaller or first shoring up weak points if you expect to be able to make territory from attacking.

Don't overconcentrate by ignoring boundary weak points. If the opp is solid enough that you don't expect to be able to invade, then especially at the boundary of development, try to lean on them.

To say a move "doesn't work" normally refers to boundary plays that stretch too far, so that tactically the end result isn't good. It is normally a "fight" which increases the temperature, but the end result isn't positive for the starter, or perhaps only equivalent to a small gote.


how to play bad go.

ignore weak points. take the opponent's eyespace when they are safe in the centre and attacking your weak group. surround strong groups. play forcing moves that aren't actually forcing. only see that your moves are connected and your opponent's aren't locally, without appreciating double attacks.


heuristics for eyespace. How many stones are required to secure the centre?
IIRC Erdos's (not very good) heuristic for the maker-breaker game is that a set that requires n more elements selected to reach completion is worth 2^-n to the maker. The value of a move is accordingly the sum of various powers of 2.

https://en.wikipedia.org/wiki/Maker-Bre ... _for_Maker
In Go, d2=2, k=4, |F|~|X|, so this condition is impossible. Curiously when k=2 (i.e. in the corners), then this means Maker wins. In that case, in higher dimensions, perhaps Maker can't win, and hence my komi prediction was wildly inaccurate and should be much lower?

Another way to think about it is that if n moves are required to make an eye and each move threatens n eyes. Then initially the value of a point is 1/2^n. The first move has value n/2^n. If subsequent moves are of this sort, and 8n-4 moves are required in total, then?

NB in go, an advantage for the breaker is that eventually the maker may threaten eyes that the breaker may have already broken. An advantage for the maker is that to break an eye, the breaker also needs to be alive.

Hence it is much easier to make eyes that this suggests.

Perhaps sum over different possibilities? Must be less than 1/2 chance in total on a neutral board as an eyespace point needs to be empty and not dame (i.e. controlled by one side). Then we integrate over all possible. It might not be finite.

F(f(theta)) from 0 to 2pi. Where F is the integral of f(theta)dtheta. f is distance to eyespace point and should be continuous-like (i.e. step size at most one). Finally we want sum over 2^-(F) over all possibilities. Say f must be integer. And that f is constant for a range of pi/2f.

If a range of dtheta can be arbitrarily plugged into any integer value of f. Then contribution to F is accordingly 1,2,3,... . This can be factored out and the contribution is 1/2+1/4+1/8+ ... . This tells us to double the integral for every dtheta which will blow up. This is approximately continuous for dtheta small and f big already.

This all probably made no Go sense. The best heuristic so far is probably just 1/2^(-n) as originally or 1/16.


Flow of go
common fighting shapes in go are huxiangqianzhi. Mutual restriction. weak groups adjacent, fighting for boundary.

Go is about harmony, judgement. In empty space, the decision is about which weak point to take against valuable areas, and what to do when there is no big weak point.

If mutually weak, that "attracts moves by ?" unless repelled by difficulty of local life. every stone repels the opponent according to the cost of living, but its potential attracts them.

e.g. normally don't attach to strong wall as what are you trying to control? even if you aren't trying to control next to the wall, opp can counter easily and even if they can't take what you are trying to control, there will be a higher cost since your boundary stone is weak.

nb most valuable ladders arise from joseki that surround opp from centre but also try to block off a side.

myt: why is fight tense? every move is killing, needs to count liberties. playing a weak point loses at the next weak point if the opp must respond but may gain more beyond that with threats.


attraction of weak points
well value is linear so the value of a move is roughly sum of value on each 4 neighbours. We can say that moves are attracted to weak points. However, to what degree? How much does it change if the move is further away? Well, if it is about taking out the last eye or shorting a liberty, far away has zero effect. If it is about making partial surround, then perhaps less moves are required to kill and value is in proportion to that. If it is about supporting profit from attacking moves, then be careful not to overconcentrate, but the value will be in proportion to that area with little to no connection to the value of the weak group, beyond perhaps making moves forcing earlier (if they were forcing anyway, there might be no impact).

I doubt there is any particular inverse square law, though there might be if we consider this for all weak points around even summing the smallest no man's land intersections. It's just that weak points dominate the calculation, like black holes sucking up moves.


non-equilibrium thermodynamics. It is clear that this is deeply relevant to go, and probably any game, but I don't understand the theory well enough. Go is fairly general, so go concepts are likely to be relevant to other games (with some stretching, metaphors, etc.)

What is entropy in this case? The depth of the tree? my point about temperature being like extremes for W and B makes some sense. What are the appropriate extensive variables? The must be functions of stone positions but is just saying stone positions enough for a theory?

critical points correspond to my weak points notion, the most unstable areas where each move can affect a lot.
what is free energy then (if F+TS is conserved?). what are thermodynamic forces? probably linearish locally and strongest near weak points. If optimal play is considered the lowest energy state then, does the second law become minimising differences to this? probably free energy is like local temperature. and temperature in go doesn't quite correspond to thermodynamic temperature.

If a partition function can be found, much can be done.

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When Venus transits, we can align our clocks to one event. By measuring the angle to flat Earth at two places far apart on Earth, we can compute the distance to Venus and the Sun.

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