Double sente is not a property of the position

Go players have a word for a local fight that seems to be sente for whoever plays in it first. It is double sente, and it is spoken as though it named a shape: this corner is double sente, so take it before anything else.

Sente is a fact about the rest of the board established that the simpler word is already relative — a move is answered while the ambient temperature does not exceed the temperature of the follow-up it leaves behind, and the crossover is a number the local position never reports. Double sente is that claim made twice over, once for each player, and the interesting question is what happens when the two claims are true at different times.

The answer is that a fight with a threat on both sides has three verdicts, not two, and which one applies is settled by a number that is nowhere in the fight.

The position, and its two follow-ups #

{ {5 | 3} | {2 | −4}} is the smallest shape with the property the word is about: both options are themselves fights, so each player’s move leaves something behind that the opponent may want to answer.

Left’s move goes to {5 | 3}, a switch with thermograph(·, 16).temperature equal to 1, mean value 4, and stops 5 and 3. Right’s move goes to {2 | −4}, a switch of temperature 3, mean value −1, and stops 2 and −4. Those two numbers, 1 and 3, are the whole of what follows.

The asymmetry is worth pausing on, because it runs the opposite way to intuition. {5 | 3} is the option that is worth more: its mean value is 4, against −1 for the other. {2 | −4} is the option that is hotter: three points of temperature against one. Big is not the same as hot is the essay that separates those two measurements, and here the separation decides which player’s move is forcing. The follow-up that leaves Left in a wonderful position is the one that lets Right ignore Left’s move soonest.

The fight itself reports neither number. Its own walls meet at a temperature of 2, its mean value is 2, and its stops are 3 and 2 — four quantities, all exact, all about the fight and none of them 1 or 3.

What the solver did, row by row #

Nothing above is a definition of double sente that a computation could act on, so the sweep does not use one. It builds a board out of the local fight and one ambient switch, plays the whole board out by exhaustive search from each player in turn, and records what optimal play chose.

Each row of the hero is two such searches. For the first, Right opens the fight and the question is whether Left’s optimal reply is in the fight or in the ambient switch. For the second, Left opens and the question is whether Right answers. The words “answers at once” and “plays elsewhere” are read off the solver’s move, never assigned by a rule.

The scores in that column are worth reading as a sequence: 3/2, 1, 1/2, 0, −1/2, −1, −1/2, −1, at ambients of 1/2 through 4 in steps of a half. They fall steadily while the exchange is forced, because a hotter ambient is worth more to whoever gets it, and then they stop falling: the score ticks back up to −1/2 at an ambient of 7/2, which is exactly the row at which Left stops answering. The non-monotone step is the crossover showing up in the score rather than in the verdict — the one place the two columns check each other.

The band edges are the follow-ups’ own temperatures #

Ten ambients were swept, at 1/4, 1/2, 3/4, 1, 3/2, 2, 5/2, 3, 7/2 and 4. The verdicts fall into three runs with no interleaving: double sente at 1/4 through 1, sente for Right only at 3/2 through 3, gote for both at 7/2 and 4.

The lower edge is 1, which is the temperature of {5 | 3} — the fight Left’s move leaves behind. The upper edge is 3, the temperature of {2 | −4}, which is what Right’s move leaves behind. So the rule is rung one’s rule applied twice:

Right must answer Left’s move while the ambient temperature does not exceed the temperature of G^L; Left must answer Right’s while it does not exceed the temperature of G^R. Both hold below the smaller of the two, neither holds above the larger, and in between exactly one holds — the one belonging to the player whose move leaves the hotter fight behind.

All ten rows agree with that, and the same held for every ambient swept over the four double-sente positions on this page and over the tightened grid below: forty-seven rows of forty-seven. Each of those figures checks the prediction itself and prints how many rows failed, which on every one of them is none.

That last caption is not modesty for its own sake. A sweep over ten ambients on a quarter-point grid establishes that the verdict changed somewhere between 1 and 3/2, and it does not establish that it changed at 1. The claim being tested is a prediction from the follow-up’s temperature, and what the sweep can do is fail to break it.

The same fight in three other shapes #

Three more positions were built to see whether the three bands survive rearrangement. The second is the reason this essay exists, and the third is built to a specification rather than found.

The coincidence in that figure is instructive and the generator refuses to let it pass silently. { {8 | 2} | {1 | −3}} has temperature 3 and mean value 2; its band edges are 3 and 2. A reader who met this position first would leave believing the thermograph reports the band. It does not, and the first position on this page is the demonstration: temperature 2, mean value 2, band edges 1 and 3.

Now the third shape.

{ {6 | 2} | {1 | −3}} is what a Go player means by double sente in the strong sense: a fight that is answered whichever way it opens, right up to the moment it is answered by nobody. The theory produces that behaviour, and it produces it as a coincidence of two numbers. The middle band has width |T(G^L) − T(G^R)|, and the band is empty precisely when the two follow-ups happen to be equally hot. Change one of Left’s options from {6 | 2} to {8 | 2} — the same fight one point larger on one side — and the band opens up to 5/2 through 3.

If the width really is |T(G^L) − T(G^R)| then a band of any size can be ordered in advance, so the fourth position is built to a specification rather than found: leave Right’s follow-up at {1 | −3} of temperature 2 and take Left’s up to {12 | 2}, of temperature 5. The formula predicts a middle band running from 2 to 5 — three points wide, half as wide again as the first position’s two.

Two things in that figure are worth more than the width. The fight’s own temperature is 3, which sits inside its own middle band — so at an ambient of 3 the local thermograph has closed and the verdict is still two rows away from changing. And the band’s position was ordered as well as its size: both edges moved when only Left’s follow-up did, which is what it means for the two edges to be two independent numbers rather than one interval the position reports.

So the strong form of the word does describe something real. What it does not describe is a kind of position. It describes an equality between two temperatures, which is a condition of measure zero on the numbers and has no visible signature in the shape of the fight.

What it does to the count, and to the play #

The practical consequence lands in two places, and the orthodox account is the first of them.

Put the fight on a board with three plain switches and run the orthodox account over it — each region summarised by its mean, the means added, the stakes taken in turn from hottest down. On a board of four regions whose means come to 8 and whose alternating stakes come to 1, the account predicts 9 and the recursion says Left moving first gets 10. The region the account is out by is the one with a fight inside it.

One point in ten is not a rounding error in a subject where counting at the end decides games. The account is exact on a board of plain switches, and it is out by one here because it prices a forced exchange as two independent moves — which is right above the crossover and wrong below it. A count that hard-codes either answer is wrong on half the board states, and the boundary between the halves is the band this essay measures.

The second place is move selection, where the news is better.

Rank the same fight against two others by temperature and the ordering does the work on its own: {3 | −3} is hottest at 3, the double-sente fight is one colder at 2, and {1 | 0} is colder again at a half. Hottest-first therefore leaves the double-sente fight alone while the board still holds a three-point switch — and at an ambient of 3, the hero says that fight is sente for Right only, so leaving it is right.

Playing the hottest never mentions follow-ups and never computes a crossover, and it gets the timing right anyway, with a proved bound on how much it can cost. That is the same accident rung one found for the one-sided case, and it is not really an accident: the rule waits for the board to cool, and the bands are defined by how cool the board is.

Nothing in that rule’s output says “double sente”. The verdict is a reader’s summary of what the solver did, and the summary is what is being audited here.

Where the model stops #

One switch is not a board. The ambient is modelled as a single switch, whose temperature is exactly half its swing and which has no follow-up of its own. A real board holds a dozen regions and the relevant quantity is the largest temperature among them. The reduction is standard, it is what makes the sweep a sweep over one variable, and it is a model.

The verdicts are read off a finite grid. Ten ambients at quarter and half steps bracket each crossover; they do not locate it. ambient(t) refuses an off-grid temperature outright rather than rounding it, so the grid can be tightened around either edge, and tightening it is the only way to say how well the edges are known.

That is as far as the instrument goes. A grid can shrink a bracket and cannot close it, and the claim being tested — that the edges are exactly the two follow-ups’ temperatures — is a claim about a real number that a sweep can only fail to break.

The rule looks one level down. “Answered while the ambient does not exceed the follow-up’s temperature” reads the temperature of G^L and G^R and nothing deeper. Rung one found one row in fifty where a follow-up with a follow-up of its own broke that at an exact tie; the four positions here all have plain switches as follow-ups, so none of them can exhibit the failure.

And the three bands need three verdicts to exist. { {6 | 2} | {1 | −3}} has only two, and the generator that draws the table throws when every row gives the same verdict — a table with one band in it would not be evidence of anything.

Who named it, and what the name was for #

The word is Go’s, not the theory’s, and it is centuries older than any of this. Japanese Go literature has sente and gote as basic vocabulary, and double sente as the awkward category that every endgame primer warns about and none of them defines sharply.

The reason the warnings exist is that the category is unstable in play, and strong players know it empirically: a double-sente point is one that has to be taken early, before the board cools, and a player who leaves it too long discovers it was never double sente at all. That is exactly the upper band edge, described from inside a game rather than from outside it.

The apparatus that makes it a measurement is Conway’s, dating from around 1970 and published in 1976: temperature, the thermograph, and the mean value of a position. Its application to Go is Elwyn Berlekamp’s, from the 1980s and 1990s, with David Wolfe — Mathematical Go is the book, and the first time the theory told somebody something was a constructed Go endgame in which the analysis beat a professional who had the position in advance.

What temperature theory contributes to the old word is not a replacement but a boundary. The Go player’s instinct — take the double-sente point first — is correct, and the theory says how long “first” lasts and what it is measured against. The instinct was never wrong; it was underspecified in one variable, and the variable is the rest of the board.

What the picture cannot show #

The thermograph of { {5 | 3} | {2 | −4}} is a complete and exact description of the local position, and it cannot report the band. That is the essay’s argument turned into a statement about the drawing, and it is worth making exactly.

The walls, exactly, from thermograph(G, 20): Left’s runs (0, 3) → (1, 3) → (2, 2) and Right’s runs (0, 2) → (2, 2). The one bend at t = 1 is half the answer, and it is there because a follow-up colder than the position survives into the drawing. The follow-up hotter than the position does not, because a thermograph stops at the height where its walls meet and everything above that height is a fact about a position nobody would move in.

There is a second thing the drawing hides, and it is the one that would mislead a careful reader rather than an incautious one. { {6 | 2} | {1 | −3}} — the position with no middle band — has both follow-ups at temperature 2, below its own temperature of 5/2, so its thermograph carries two bends and both stand at 2. The diagram reports the band edge in precisely the case where there is no band. Where the band is widest, the diagram shows one edge; where the band is empty, it shows both.

The third invisible thing is the line of play. Each cell of the sweep is an exhaustive search over a two-component board played to the end, and what is printed is the first decision and the verdict. The sequence of moves that produced them would be six deep in every row and would not fit on a page.

The convention, named #

Normal play throughout: the player unable to move loses, and misère play is a different game with the same rules and none of this arithmetic. Two further conventions are doing real work.

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 at once” and “plays elsewhere” above is the move that optimises that score under exhaustive search. Under a different accounting — a clock, a bonus for the last move — the same boards would have different best moves and the bands would move with them.

The ambient is a single switch, so its own temperature is exactly half its swing and it has no follow-up. An ambient with follow-ups of its own would make the crossover depend on two things at once, and separating them is the reason the experiment is built this way. Temperatures do not add is the essay on why a sum of switches cannot simply be summarised by one number, and it is why the reduction to a single switch is a model rather than a fact.

One convention is Go’s rather than the theory’s, and it is the one that makes any of this apply to a real board: a Go endgame has to be treated as a sum of independent regions, and the rule that makes Go finite is what stops a fight from being reopened for ever. Establishing which real positions decompose exactly enough is Go knowledge, not mathematics.

Where the ladder goes next #

This is the second rung of the sente ladder, and the first — sente is a fact about the rest of the board — supplies everything the sweep here reuses. Four further rungs are visible from this one.

Reverse sente, and what denying a move is worth. If Right’s move at { {5 | 3} | {2 | −4}} is sente over an ambient band, then Left’s move there is reverse sente over the same band: it prevents an exchange Right was going to get for nothing. Its value is not the local temperature and not the follow-up’s; it is the value of a move that never gets played, and the sweep already contains it as the difference between two rows it does not currently subtract.

A fight that cannot be answered locally. A ko is the shape that breaks the whole apparatus, because the answer to a ko capture is not a move in the ko — it is a threat somewhere else, and the exchange is not local at any level. Nothing on this page survives that, and the theory’s treatment of it is a genuinely different construction.

Orthodox accounting, corrected. The one point the account is out by has a name in Go — it is the difference between the two counting conventions — and the correction is computable from the bands rather than chosen by judgement.

And sente in an environment made of nothing else. An environment made of coupons replaces the single ambient switch with a whole graded stack, and the sente exchange appears there as two consecutive moves in the game with no coupon between them. That is this essay’s crossover arriving from a construction that never mentions sente, which is the best kind of confirmation there is.