A ko is won somewhere else

The rung below ends by naming what its figures throw away, and it is the same thing every account of a ko fight throws away when it draws the ko on its own:

What is missing is the ko threat: a move elsewhere on the board that forces an answer, buying the time to come back and recapture. Ko threats are what make ko fights interesting, they connect the ko to every other part of the board, and they are precisely the thing this model has thrown away by drawing the fight in isolation.

This rung puts them back, and what comes out is not a refinement of the rung below. It is the reason a ko fight is the standard counterexample to the one assumption every technique on this site is built on.

What a threat is, exactly #

A ko threat is a play somewhere else that the opponent must answer or lose something bigger than the ko. In the model that is one clause:

A player may play a threat instead of capturing. It changes no ko point, it costs one of a stated finite stock, and it lifts the prohibition — because after the threat and its forced answer, the point captured is no longer the point captured last move.

The third half of that clause is the part that makes it a ko threat rather than a pass. The whole purpose of playing one is to come back, and coming back is legal precisely because a move has intervened. Nothing else changes: the same graph, the same retrograde labelling, the same normal-play convention.

The stock is finite because a board is finite. A player has some number of moves elsewhere that are worth more than the ko and the opponent has some other number, and those two counts are facts about parts of the board that this figure does not draw.

The rule is what makes the threats matter #

Before reading the grid it is worth running the experiment that says the two mechanisms are one.

Switch the repetition rule off. Now recapture is always available, so nobody is ever forced to play a threat, so no threat is ever spent — and the fight is drawn from almost everywhere, whatever either side holds.

So the prohibition and the threats are not two separate features of a ko fight. The rule is what converts a free recapture into a purchase, and the currency is counted on the rest of the board. That is a sharper statement of what the rung below found: the ko rule buys termination, and it buys it by making the fight consume a resource that runs out.

It also explains, in one line, why the fight terminates at all under the rule. Each round trip through the ko costs each side one threat, the stocks are finite, and a game with a strictly decreasing quantity in it ends — which is the condition the whole recursion rests on, arriving from an unexpected direction.

The threshold, searched for rather than assumed #

Read the grid and the answers fall into a pattern. Whoever is further ahead on threats wins the fight; where the two stocks are level in the right sense, whoever moves loses.

The obvious guess for “level in the right sense” is equal, and it is wrong. On a single ko point held by Black, the balance the fight turns on is Black one ahead — a=b+1 is the diagonal where whoever moves loses, and Black needs a strict extra threat merely to hold.

The reason is not a rule about ko, it is bookkeeping. The player holding the point has no capture available and must therefore spend first; the opponent captures for free. So the holder is one purchase behind before anything happens, and the fight’s threshold is offset by exactly the holdings.

Nothing in the generator was told any of this. The threshold is searched for — every candidate offset is scored against the labelling and the best one reported — precisely because asserting an offset of one was right for a single ko and wrong for two.

Every decided cell of both grids fits its threshold, with no exceptions. That is what makes the sentence the fight goes to whoever has more threats a computed statement rather than a piece of folklore quoted at a figure.

Reading one line of play #

The grid is a table of verdicts and the mechanism it hides is a sequence, so it is worth walking one.

Black holds the ko point. White is to move, with two threats to Black’s two. White captures — the point is now White’s, and it was free, because nothing had been captured on the previous move. Black may not recapture immediately, so Black plays a threat; White must answer it, and the answer costs White nothing but the move; and now the prohibition has lapsed. Black recaptures. White may not recapture immediately, so White plays a threat, Black answers, and White recaptures.

Count what has been spent. One full turn of the wheel costs each side one threat, and returns the position to where it was with the holdings swapped. Two turns and the stocks are empty; whoever is holding the point when they run out is holding it for good; and the arithmetic that decides who that is is the arithmetic of the two counts and the initial holding, which is exactly the threshold the sweep reports.

The step worth noticing is the second one, where Black plays a threat rather than passing. A pass would not lift the prohibition — a pass changes nothing, and the point captured is still the point captured last move. The threat works because the opponent is obliged to answer, and it is that obligation, not the move itself, that puts a move between the capture and the recapture. A pass is not a move is the general statement of the difference, and a ko fight is where it is most visible: the same tempo spent two ways, and only one of them buys anything.

What this does to independence #

Here is the consequence, and it is the reason this rung belongs to the applied field rather than to a field about Go.

Every technique on this site starts from a position broken into parts that do not interact. A value is a property of a part; parts add; the total says who wins. Splitting a position is a claim about the position, and the claim is that a move in one part changes nothing in another.

A ko fight with threats in it fails that claim in the most direct way available. The fight’s outcome is a function of a number counted on the rest of the board, so the ko does not have an outcome until the rest of the board has been counted; and a move elsewhere that removes a threat changes the ko’s outcome without touching the ko.

That is not a weak coupling. It is not that the ko is approximately independent and the threats are a correction term. The grid has cells where Black wins whoever moves and cells where White does, and moving between them is a matter of one threat spent somewhere the picture does not show.

The nearest thing on this site is sente, where a move that must be answered stops being answered once the rest of the board is busy enough — and the crossover sits at a temperature computed from the whole position rather than from the fight. That is the same shape one field over: a property that reads like a fact about a local position and is a fact about its company.

It is worth being exact about which kind of failure this is, because the sums field distinguishes two and they behave differently. One is a split that is nearly right: the parts interact a little, the sum is close, and the error can be measured and is usually small. The other is a split that is not a split at all, because the components share a thing. A ko fight is the second. There is no version of the picture that is approximately correct, because the missing quantity is not a small correction to the answer — it is the answer.

The same distinction decides a chess ending on the rung about kings: two pawn files priced separately are exact when the kings cannot move and name the wrong winner once a king has somewhere else to be. A shared piece and a shared stock of threats are the same defect in two games, and neither of them is a matter of degree.

The stock is spent by both sides at once #

One property of the fight is unlike anything in a game of pure position, and it is the reason a Go player counts before starting rather than during.

A threat is spent to continue, not to win. Playing one gains nothing on the board — the ko is where it was, the threat has been answered, the move has bought a single recapture. So the fight consumes both stocks in step, and the side that runs out first loses the point, however good the position looked when the fight began.

That makes a ko fight the rare position where a player can know the outcome in advance and be unable to change it. Counting threats is not an evaluation heuristic standing in for a search; it is the answer, exactly, and the sweep says so on every decided cell of every grid on this page. A search that read the ko alone would have to be wrong, because the information the answer depends on is not in what it is searching.

There is a second consequence for the practice of the game. Since the fight consumes threats and gains nothing, a player who is behind on threats should not start the fight at all, and a player who is ahead should create more before starting it. That is a statement about the order of play on the whole board, generated by a local position, and it is the same kind of statement the hottest-first rule makes about temperature — one part of the board dictating when another should be touched.

The triple ko is not rescued #

The rung below finds the position Go’s own rule sets name and score as no result: three ko points under the simple rule, nineteen labels the propagation never reaches. It falls out of the move rules, and nothing in the module was told about it.

Threats do not fix it, and they do not fix it for a reason worth having.

The mechanism is the one from the second section, read backwards. Threats matter when the rule forces a purchase. With three ko points the simple rule never forces one, because a player denied one point may take another — so the cycle can be traversed for ever with the stocks untouched, and a stock nobody spends decides nothing.

That is a clean statement of what a no result is in this vocabulary: not a fight nobody can win, but a fight nobody can be made to pay for.

Superko settles it, at the price the rung below prices: a position stops being a configuration and becomes a configuration plus a history, and three points go from twenty-five states to eighty-six. Under superko the threats matter again on all three point counts, because a repetition rule strong enough to forbid the cycle is a rule strong enough to force the purchase.

Why a Go player already knows this and the theory does not #

The folklore is exact and old: whoever has more ko threats wins the ko, and a player counts threats before starting a fight rather than during one. What the grid adds is not the rule; it is the offset, and the fact that the offset is a function of the holdings rather than a constant.

What the theory has to say about it is more interesting for being awkward. The rule a player uses is a comparison of two numbers, and neither number is a value in this site’s sense. A threat is not a game with a value that adds; it is an option that expires, a resource, a count. And a count is exactly the kind of object the disjunctive theory has no place for — the whole apparatus turns positions into values and adds them, and there is nowhere in it to put “and Black has three of something”.

So a ko fight is a position that a strong player evaluates by a method the theory does not contain, and the method is right. That is worth stating plainly rather than apologetically. The theory buys arithmetic and pays for it in coverage, and a ko fight is one of the places the coverage stops.

The stock is a model of something that is not a stock #

One clause of the model deserves more scrutiny than the others, because it is where the simplification bites.

Real ko threats are not interchangeable. Each is a move somewhere that costs the opponent something if unanswered and costs the threatener something if answered, and their sizes vary; a threat may be worth less than the ko, in which case playing it loses more than it gains; and answering a threat may itself create a new threat for the other side. What a player counts, when they count threats, is threats larger than the ko — a number that depends on what the ko is worth, which depends on the fight.

None of that is here. The model has a stock of identical, free, always-answerable threats, and the number is handed to the solver.

What survives the simplification is the structural claim: there is a resource, it is counted elsewhere, and the fight’s outcome is a function of it. What does not survive is any quantitative statement about a real board, and no figure here makes one.

What the picture cannot show #

The threats are not drawn, because they are not there. Every figure on this page draws a ko fight and a number, and the number is the whole of what the rest of the board contributes. A reader wanting to know where threats come from, how many a position has, or whether one is large enough to use is asking about a Go board, and this model has none.

Nor is there any territory in it. Go is scored, and a scoring game is a different object from the normal-play games this site is built on. The ko fight here is won by whoever leaves the opponent unable to move, which is a model of the fight and not of the game it sits inside — the rung above this one is about how much that substitution costs.

And the stocks are static. A real fight spends threats on both sides while the board changes and new threats appear; here the two numbers only fall. That makes the model cleaner than the game and, in one specific direction, easier: a fight in which threats can be created is a fight in which the count is not a count.

The convention, named #

Normal play throughout: a player with no capture and no threat left loses.

That is a substitution and it is the same one the rung below makes, but the threat clause gives it a sharper edge. A player who has run out of threats has not lost anything in Go — they have merely run out of the ability to contest one point, and the game continues. Here the game is the fight, so running out is losing, and every verdict on the page inherits that.

What survives the substitution is exactly the comparison. Whether the fight goes to Black or to White is a question about which side runs out of purchases first, and that question does not depend on how the rest of the board is scored. What does not survive is any claim about who wins the game of Go, and none of the figures makes one.

The surprise: the rule was doing two jobs and only one of them is famous #

The ko rule is always explained as the thing that stops the game going round in circles, and the rung below shows that reading is exactly right — the rule is the finiteness hypothesis, and removing it removes the values.

The grid says the same clause does a second thing, and it is the thing a player actually experiences. The prohibition is what makes a ko fight cost something, and what it costs is drawn from the rest of the board. Without the rule, a ko is free and endless. With it, a ko is a purchase, and a purchase has a price paid in a currency stored elsewhere.

Those two jobs are the same edge deleted from the same graph, and the second is invisible in a picture of the graph. Termination and coupling arrived together, from one clause, and a reader who has understood why the rule makes the game finite has not yet understood why it makes the game non-local.

The general shape is worth carrying past Go. A rule that forces a player to spend something is a rule that ties the position to wherever the something is kept. Any game with a resource — a bank of waiting moves, a limited number of passes, a hand of cards — has this property, and the decomposition every technique here relies on is not available in it. The ko rule is the smallest instance anybody has found, and it took a game people have played for millennia to supply it.

Where the ladder goes next #

go now has the rule that makes the game finite and the resource that rule creates, and both essays evaluate a fight by asking who moves last.

Go is scored. The rung above is what that costs: territory scoring and area scoring are two conventions in daily use, they are not variants of one rule, and on a counted fraction of endgames they name different winners without either player having played differently.