Sente is a fact about the rest of the board

A Go player calls a move sente when the opponent has to answer it, and gote when they do not. The words are used as though they described the move: this corner is sente, that one is gote, and a player learns which are which.

They do not describe the move. Whether an answer is forced depends on what else is on the board, and the same local fight changes category when the rest of the board changes — which can be measured, by solving the whole board at a range of ambient temperatures and reading off what optimal play does.

The position, and why it has a follow-up #

{5 | {4 | 0}} is the smallest fight with the right shape. Left’s move ends the matter at 5. Right’s move goes to {4 | 0}, which is itself a fight — a switch worth 2 on average, with four points swinging on who takes it.

That inner fight is the follow-up. If Right plays and Left ignores it, Right takes it too and the position lands at 0 rather than 4, so ignoring costs Left four points. If Left answers, the exchange closes at 4.

So Right’s move carries a threat, and answering it is worth four points locally. Whether four points locally is worth a move depends entirely on what a move is worth elsewhere, which is the ambient temperature.

What a threat is, in the arithmetic #

The word threat is doing work in the paragraph above, and it is worth converting into a quantity, because the conversion is what makes the rest measurable.

Right’s move goes to {4 | 0}. The two things that can happen next are: Left answers, taking the follow-up to 4; or Left plays elsewhere, and Right takes the follow-up to 0. The difference between those two futures is four points.

Four points is the swing of the follow-up, and half of it — 2 — is the follow-up’s temperature. So “the threat is worth four” and “the follow-up’s temperature is two” are the same fact in the two units the subject keeps side by side, and the factor of two between them is the one every endgame book carries.

The ambient temperature is in the second unit: it is what a move elsewhere is worth, half of what changes hands there. So the comparison the sweep makes — follow-up temperature against ambient temperature — is a comparison of like with like, and a player comparing “four points of threat” against “a three-point move elsewhere” is comparing a swing against a temperature and will get it wrong by a factor of two.

That is a small thing and it is exactly the sort of small thing that makes a rule of thumb unreliable. The rule works when both sides are measured the same way, and the two ways are habitually mixed.

Where the answer stops being forced #

Reading down the sweep: at an ambient temperature of 1/2, 1, 3/2 and 2, optimal play answers. At 5/2 and above it does not.

The follow-up’s temperature is exactly 2.

That is the rule, stated as a prediction rather than as a definition: a move is answered exactly while the ambient temperature does not exceed the temperature of its follow-up. It is the formalisation of what a Go player means by “the follow-up is bigger than the biggest move elsewhere”, and it is checkable.

The first thing to check is whether the rule is about the shape of the fight or about the size of its numbers, and the cheapest test is to shrink the fight and run the same sweep. $\{2 \mid \{1 \mid 0\}\}$ is $\{5 \mid \{4 \mid 0\}\}$ with everything smaller: one move ends the matter at 2, the other leaves a switch worth a half.

Two positions is not a test either, so the rule is run against a spread of fights whose follow-ups are as unlike each other as the family allows.

The exception is worth having rather than hiding. {6 | {4 | {3 | 1}}} has a follow-up of temperature 1, and at an ambient of exactly 1 the rule predicts an answer and optimal play tenukis. At a tie the decision is settled by what lies one level further down, and the rule — which looks only one level down — has nothing to say about it.

What the local position does and does not report #

The local fight’s own temperature is 1. Its mean is 4. Neither number is the crossover, and it is worth being precise about why not.

Temperature is what the position is worth to move in. Right’s move gains one point on average, which is why the local position is not urgent.

The follow-up’s temperature is what the answer is worth. Four points swing on it, so the answer is urgent even when the move that provoked it was not.

Those two quantities can be arbitrarily far apart, which is what makes sente confusing to name. {10 | {9 | 1}} has a temperature of 1 and a follow-up of temperature 4 — the fight is worth almost nothing and the answer to it is worth a great deal.

The claim in that caption is a prediction about play, so it is settled by play. The same sweep is run on that fight, over ambients from 1 up to 5 — a range four times the position’s own temperature, which would be an absurd place to look if the local diagram were governing anything.

Why a flat wall means a forced answer #

The mechanism has a visual signature and it is worth pointing at, because once seen it makes the whole phenomenon obvious.

Right’s wall in the thermograph of {5 | {4 | 0}} is flat. A wall bends when the option it is built from is worth fighting over; it stays flat when the option’s own fight cancels the tax exactly. Flat means Right’s move buys nothing on the diagram: whatever Right gains by moving, Left’s answer takes back.

That is what “must be answered” looks like from the accounting side. The exchange is priced as a single unit rather than as two moves, and the theory’s temperature of 1 is the price of the unit.

The moment the ambient temperature exceeds the follow-up’s, the unit comes apart: Left would rather spend the move elsewhere and let Right have the follow-up, so the exchange stops being an exchange and Right’s move becomes a straightforward gain of four points.

The rule looks one level down, and that is where its exception lives #

The one failure in fifty is described above as an awkwardness at a tie. It is more useful than that: it says exactly what kind of rule this is, and what the rule would have to become to have no exceptions.

The prediction compares two numbers — the ambient temperature and the temperature of the follow-up — and the follow-up is one level down the local position’s tree. So the rule reads two levels of the fight and nothing further: the move, and the thing that answers it.

That is enough whenever the comparison is strict. When the two numbers are equal the rule has nothing left to consult, and the decision is made by whatever lies below — which on $\{6 \mid \{4 \mid \{3 \mid 1\}\}\}$ is a third level, since its follow-up has a follow-up.

So the natural repair is lexicographic. Compare the ambient against the follow-up’s temperature; on a tie, compare against the next one down; and so on until the two differ. A fight with a chain of $k$ follow-ups can require $k$ comparisons, and the rule as stated is the first term of that sequence.

The essay does not claim the repaired rule, because one tie is not evidence for a general procedure and the site does not promote a patch that fits a single row. What the row does establish is the shape of the incompleteness: not that the rule is approximately right, but that it is exactly right except on a measure-zero set of ambient temperatures, and that the set is where a coarser reading of the position has run out of information.

Which makes sente a property of a chain #

That reframes the essay’s title claim in a way worth stating, because the title is already a correction of a common error and the correction can be taken one step further.

Sente is not a property of the move — it depends on the ambient temperature. True, and the sweep measures it.

Nor is it a property of the pair — the move and the ambient — because at a tie the pair does not decide. What decides is the whole descending chain of follow-ups, compared against the ambient in order.

So the object that carries the answer is not the local fight and not its temperature but its sequence of nested stakes: what the move is worth, what the answer is worth, what the answer to the answer is worth, down to where the fight bottoms out in numbers. A Go player calling a corner “sente” is naming the first term of that sequence and assuming the rest never matters, which is right except at the ties.

And it explains why the thermograph is the right picture after all. A wall bends once per follow-up, so the chain of nested stakes is exactly the sequence of bends in the wall — a thermograph with two bends is a fight whose answer has an answer. The rule that reads one level down is the rule that reads the first bend; the lexicographic version reads them in order; and a position whose wall is straight has no chain, no follow-up, and cannot be sente at any ambient temperature whatever.

That last consequence is worth having as a test. A plain switch is never sente. Not usually, not at low temperatures — never, at any ambient, because there is nothing below it for an answer to be worth. It is the one claim on this page that a sweep can establish outright rather than bracket, because it is a claim about every row.

The contrast with the hero is the whole of the mechanism. {5 | {4 | 0}} and {4 | 0} differ by one level: the first has a fight where the second has a number, and that difference is what turns an ignorable move into a forced exchange. Right’s move opens the fight in both at low ambients, and only in the first does anything come back.

It also settles which boards the orthodox count is safe on. A board of plain switches has no follow-up anywhere, so no region on it can be sente at any ambient, so the orthodox account — means added, temperatures taken in turn from hottest down — can treat each region as a single move and be exact.

What this does to the endgame count #

Add one region with a follow-up and the account has to decide whether to treat the exchange as one move or two, and the right answer depends on the rest of the board. A count that treats a sente exchange as two independent moves double-counts the tempo; a count that treats it as one unit is wrong the moment the ambient temperature rises past the follow-up’s.

That is why Go’s own accounting has two counting conventions and a rule for choosing between them, and it is why the first time temperature told somebody something was an endgame where the orthodox account got the order wrong.

Playing it, rather than counting it #

The rule this site gives players is move in the hottest component, and it is worth asking what that rule does with a sente move, since it does not mention follow-ups at all.

Put {5 | {4 | 0}} on a board beside a switch worth 3 and another worth a half, and the sente fight is the coldest of the three at a temperature of 1 — so hottest-first leaves it alone while the three-point switch is still standing. That is exactly right, and the sweep above says why: at an ambient of 3 the exchange is not forced, so playing it early would be spending a move to buy nothing.

Hottest-first therefore gets the timing right by accident and for the right reason: it plays the big things first and comes back to the sente exchange when the board has cooled to where the exchange is worth making. The rule has a proved bound, and the bound covers the sente case without ever naming it.

The rule players actually use, and where it fails here #

A player sizing moves by how much changes hands — deiri counting — sees a sente move as a large move, because the swing between Left taking it and Right taking it is large. {5 | {4 | 0}} swings five points between the two openings and is worth one point of temperature, and a player working in the first unit will play it about five times too early.

Big is not the same as hot measures the cost of that over 240 lines. What this essay adds is the timing: the count is not merely wrong about the size of a sente move, it is wrong about when the move becomes worth making, and the crossover is a number the count never computes. The two sweeps on this page make the gap concrete — {10 | {9 | 1}} has stops at 10 and 9, one point between them, and is answered up to an ambient of 4; {4 | 0} has stops four points apart and is answered at no ambient at all. A rule that sizes a move by what changes hands puts those two in the opposite order to the one play puts them in.

Reverse sente, and the move nobody plays #

One category is missing from everything above and it is the one that most surprises a reader who plays Go.

If Right’s move at {5 | {4 | 0}} is sente, then Left’s move there is reverse sente: a move that prevents an exchange the opponent was going to get for free. Its value is not the local temperature either — it is the value of denying the opponent a sente move, which is a statement about a move that never gets played.

The theory handles it without a special case. The whole board is solved, the optimal move at each ambient temperature is reported, and whether that move is Left’s own or a denial of Right’s is a description rather than a computation. In the sweep above, the row where Right opens the local fight and the row where Right plays elsewhere are two different optimal answers, and the transition between them is the point at which Left’s reverse-sente move became worth making.

That is a real advantage of computing over naming. A vocabulary with four categories in it needs a rule for each, and rules for categories multiply; a minimax over the sum has one rule and the categories are read off its output.

The cost is that the output does not come with the vocabulary attached. Nothing in a solved board says “this was sente” — the words are a reader’s summary of what the solver did, and the summary is exactly what this essay is measuring the reliability of.

Where the model stops #

One switch is not a board. The sweep puts the local fight beside a single ambient component. A real board holds a dozen, and the relevant quantity is then the largest temperature among them — which is what “ambient temperature” means and is exactly what the single switch stands in for. The reduction is standard and it is a model.

The rule is one level deep. “Answered while the ambient does not exceed the follow-up’s temperature” looks at one follow-up. The one failing row in fifty is a position whose follow-up has a follow-up, and a rule that looked two levels down would presumably handle it and fail on something three levels down.

And sente is not a two-way classification. Real Go distinguishes reverse sente, double sente and gote, which are four categories rather than two, and they are about which player’s move carries the threat. Everything above is about one player’s move and one threat.

The same lesson, three fields along #

A local property that turns out to depend on the whole board is not new here, and the pattern is worth naming once.

Independence is a claim makes it about decomposition: whether a board splits into parts is a fact about the board rather than about the drawing, and a Kōnane row cut at a gap of any width is wrong some of the time.

Which part to move in makes it about moves: the value of a sum is the sum of the values and the move in a sum is not the move in any part, so no amount of local information settles it.

Where the fight stops makes it about summaries: the stops of a sum are not the sums of the stops, because alternation is a property of the board.

Sente is the fourth instance and the most familiar one, because it has a name players use daily. In every case the local quantity is exact and the mistake is in believing it is sufficient — and in every case the correction needs the rest of the board, which is precisely the thing local analysis was adopted to avoid consulting.

That is not an argument against local analysis. It is an argument for knowing which of a position’s properties survive being placed on a board: the value does, the temperature does, the mean does, and whether a move must be answered does not.

What the picture cannot show #

The sweep is a table of what optimal play did, and a table of decisions hides the thing that produced them.

Each row is a complete minimax over a two-component board, played to the end, and the interesting content is the line — the sequence of moves — rather than the verdict on the first two. A figure showing the lines would be five or six moves deep for every row and would not fit; what is drawn instead is the first two decisions and the final score.

The other invisible thing is the tie. The one row where the rule fails is at an ambient temperature exactly equal to the follow-up’s, and “exactly equal” is a measure-zero condition that a drawing cannot make look important. It is drawn as one red entry among fifty, which is proportionate to its frequency and not to its interest.

The convention, named #

Normal play, short games, and the scoring convention that makes the sweep mean anything.

Play continues until every component is a number, and the score is the sum of those numbers. That is the standard reading of a partizan sum as an endgame, and every “answers” and “plays elsewhere” above is the move that optimises that score. Under a different accounting — a game with a clock, or one where the last move carries a bonus — the same boards would have different best moves.

The second convention is that the ambient is a switch, so its own temperature is exactly half its swing and it has no follow-up of its own. An ambient with follow-ups would make the crossover depend on two things at once, and separating them is the reason the experiment is built this way.