The answer that starts another fight

Sente is a fact about the rest of the board established the rule this anchor rests on. A local move is sente when the opponent answers it locally rather than playing elsewhere, and whether they do depends on how hot the rest of the board is: the move is answered exactly while the ambient temperature does not exceed the temperature of the follow-up, the position the answer would be made in.

That essay checked the prediction on five local positions, at ten ambient temperatures each, and reported one of the five disagreeing. The disagreeing row has been on this site ever since and nothing has explained it.

It has an explanation and it is exact.

The row that disagreed #

The five positions the rung below swept were {5|{4|0}}, {2|{1|0}}, {10|{9|1}}, {6|{4|{3|1}}} and {4|0}. The last has no follow-up at all — Right’s move is into a number, so there is nothing to answer — and three of the remaining four agree with the prediction exactly.

The fourth is {6|{4|{3|1}}}. Its follow-up is {4 | {3 | 1}}, whose temperature is 1, and it is answered up to an ambient of ½ and ignored at 1. That row is one line of a table above, so it is worth running on its own and on a finer grid, because a single line of a summary is not a place to start an argument.

The difference between that position and the other three is one clause. In {5|{4|0}} the follow-up is {4 | 0}, whose options are the numbers 4 and 0: whoever moves in it takes the whole stake and the fight is over. In {6|{4|{3|1}}} the follow-up is {4 | {3 | 1}}, whose Right option is itself a switch: taking it leaves a smaller fight behind.

One is an answer that ends the exchange and the other is an answer that opens the next one. That is the clause, and it is worth exactly a factor of two.

Thirty-eight, fifteen and two #

Testing that on five positions would prove nothing, so the sweep is built rather than chosen: seventy-four local positions of the form {a | F}, with F constructed to have a follow-up of a stated depth, and with the crossover found by play — the whole board solved at each ambient temperature on a quarter-point grid, and the largest temperature at which Left’s optimal reply is still the local one reported.

• Nineteen have a follow-up that is a number. There is nothing to answer and no crossover; they are counted and set aside.
• Thirty-eight have an answer that settles the fight, and on all thirty-eight the crossover is the follow-up’s own temperature.
• Fifteen have an answer that starts another fight, and on all fifteen the crossover is exactly half the follow-up’s temperature.
• Two have a third level of fight below that, and on both the crossover is still half — not a quarter.

All three counts are assertions rather than reports. A position in any class whose crossover missed would stop the build, because each of the three sentences above is a “without exception” and a figure that averaged over an exception would be describing something else.

Why half #

The factor of two is not a fitted constant, and the reason is the same reason a switch’s temperature is half its gap.

The arithmetic is worth doing on the smallest case, because it is three lines and it makes the factor inevitable rather than empirical.

Consider Right’s move into the follow-up F. Left is deciding between answering locally and taking the ambient exchange. If F is a plain switch between two numbers, answering it takes the whole of the stake: the gain is the full gap, and the comparison against the ambient is a comparison of the gap against twice the ambient — which is what “the follow-up’s temperature” already is, since the temperature of a switch is half its gap.

If F has a follow-up of its own, answering it does not take the whole stake. It takes the stake and hands back a smaller fight, and the opponent gets the next move in that fight. Half of what the answer gains is given back immediately, so the answer is worth half as much, so it stops being worth making at half the ambient.

Put numbers on it. {4 | 0} has a gap of four and a temperature of two, and answering it moves the local count by the whole four; against an ambient exchange worth 2t the answer is worth making while 4 ≥ 2t, which is t ≤ 2, which is the temperature. {4 | {3 | 1}} has stops at 4 and 2 and a temperature of one, and answering it moves the count by two and hands Right a fight worth one back — a net of one against 2t, so t ≤ ½. The published rule reads the temperature and gets 1; the play gets ½.

That pair is the cleanest statement of what the classification is about. Nothing changed in the expression {4 | {3 | 1}} between the two figures; what changed is whether the fight below it is the one being answered or the one being handed back. The class is a property of the pair — the move and the answer to it — and never of a position on its own, which is why no amount of reading a single thermograph settles it.

That is why the third level does not halve it again. The give-back is a single exchange: the answer gains, the opponent takes the residue, and what happens below the residue is settled between two players who have both already moved. The correction is one factor of a half, applied once, and it applies whether the fight below runs one level deeper or three.

The two classes, said in the rules rather than in the values #

The classification is stated above in terms of the follow-up’s options, which is a statement about a value. On a board it is a statement about the shape, and the translation is worth making because it is what a player would use.

An answer settles the fight when, after it, the local region is worth a number — nobody wants to move there again at any temperature. An answer starts another fight when the region still has a move in it that somebody wants. On a Go board that is the difference between a connection that finishes a corner and a connection that leaves a hane behind; in Domineering it is the difference between a placement that fills a region and one that splits it.

The test is not how many moves are left in the region. A region with four moves left in it that are all forced, or all worth nothing, settles the fight; a region with one move left in it that is worth two points does not. What matters is whether the residue has a temperature, and a residue with a temperature is one that will be fought over again.

That gives the practical form of the rule, and it is one line longer than the one it replaces: a local move is answered while the ambient is below the follow-up’s temperature, or below half of it if the answer leaves something behind.

What this repairs, and what it does not #

The published rule is not wrong. It is a rule for positions whose answers settle the fight, and it is exact on those, and the sweep here confirms it on thirty-eight of them rather than the three the rung below had.

What the rule was missing is the class it does not cover, and the class is not exotic. Every Go endgame worth analysing is in it. A corner sequence in which Black descends and White must connect and Black then has a further push is a fight three deep, and it is the ordinary case rather than a constructed one; a shape whose answer ends the matter is unusual on a real board and common in a textbook, because a textbook wants the example to be finite.

So the practical statement is the opposite of the way the rule reads. A reader applying answered while the ambient is below the follow-up’s temperature to a real shape will over-estimate the range in which it is sente, by a factor of two, on exactly the shapes real games are made of.

The correction also applies to each side of a fight independently. A position with a follow-up on both sides has two crossovers, one for each player, and each of them is the full temperature or half of it according to what that side’s answer does — so a fight can be in one class for Left and the other for Right, and the band between the two crossovers moves by a factor of two at either end.

Where the halving is visible #

The crossover is found by play here, and it is worth asking whether it can be read off the diagram, since that is what a thermograph is for.

Partly. The follow-up’s temperature is the height at which its two walls meet, and that is the number the published rule uses. The halving corresponds to a bend in one of those walls below the meeting point: a wall bends exactly when the option it is built from is itself a fight, which is the same condition that separates the two classes above.

What the diagram does not obviously give is the factor. The bend’s height is not half the temperature in general — in {4 | {3 | 1}} the right wall bends at 1 and the temperature is 1, so the bend is at the meeting point rather than half way down it — and the crossover is ½. So the halving is a fact about what answering is worth, not a height that can be pointed at on the picture.

That is a real gap and it is where the anchor goes next. The measurement says the number is half; nothing here says which feature of the diagram half is a function of, and a reader wanting to compute a crossover from walls rather than from play still has to know which of the two classes the position is in.

Where it leaves the coupon reading #

There is a second way this site measures where a fight stops being worth entering, and the two ought to agree.

An environment made of coupons puts a graded stack of known stakes beside a position and watches which coupon is on top when somebody first plays in the game instead of taking one. That stopping coupon is a crossover of the same kind, measured a different way, and its docstring records that it tracks the hottest temperature anywhere in the tree rather than the position’s own.

The halving is consistent with that and sharpens it. A position whose answer starts another fight has something hotter below its follow-up, so the coupon reading and the ambient reading are both looking past the follow-up; what this page adds is how far past — exactly one factor of a half — and that the factor does not compound.

Whether the two measurements agree numerically on the same pool is a check nobody has run, and it is the sort of check that either confirms both instruments or finds a bug in one. It is one rebuild away and it has not been made.

What a factor and a subtraction would look like from here #

The correction found here is a factor — the crossover sits at half the follow-up’s temperature — and it is worth asking what would distinguish that from the other natural shape, a subtraction, because the two agree on exactly the pool this sweep has.

A factor of a half sends a temperature $t$ to $t/2$. A subtraction of a half sends it to $t - \\tfrac12$. The two agree precisely when $t = 1$, and they diverge in opposite directions on either side: at $t = 2$ the factor gives 1 and the subtraction gives $1\\tfrac12$; at $t = \\tfrac12$ the factor gives $\\tfrac14$ and the subtraction gives nought.

So the pool decides which reading a sweep reports, and it decides it entirely. A pool whose follow-ups mostly have temperature one cannot separate them, however many positions are in it and however carefully each one is played out — the two hypotheses make the same prediction on every member.

That is the check this page does not run, and it is cheap to state as a requirement on a future one: a pool for this question needs follow-up temperatures spread over a range, not a count of positions. Fifty-five positions whose answers are all about the same size are fifty-five confirmations of a coincidence.

It also says what to distrust in the reading above. The halving is real — the crossover genuinely is not at the follow-up’s own temperature — and the form of the correction is the part resting on the pool rather than on the measurement. A reader carrying one number away should carry the direction and the rough size, and should expect the shape of the law to be settled somewhere with a wider spread of answers in it.

What the sweep does not say #

Three limits.

The pool is constructed. These seventy-four positions were built to have follow-ups of stated depths, because a sweep of naturally occurring positions would have almost none of the third class and the finding is about the classes. Constructed pools are the right instrument for a claim of the form this always happens and the wrong one for a claim about how often it happens, and no claim of the second kind is made here.

The grid is a quarter-point grid. The crossover is reported as the largest gridded ambient at which the move is answered, so a true crossover strictly between two grid points would be reported at the lower one. Every crossover found is a quarter-point and every predicted value is a quarter-point, so the grid is not hiding anything on this pool; on a pool with eighths in it the grid would have to be finer.

Nothing here is a proof. The arithmetic in the third section is an argument about one shape and the sweep is fifty-three positions with a crossover; between them they make the factor of a half very hard to doubt and neither of them establishes it. A proof would go by showing that the incentive of Right’s move into a follow-up with a follow-up is exactly half the incentive of the same move into the corresponding plain switch, and that is a statement about incentives rather than about temperatures.

And the ambient is a single switch. ambient(t) is {t | −t}, one fight of temperature t standing for the rest of the board. A real board is many fights of many temperatures, and the sharper theory replaces the single number with a graded stack of them. Whether the halving survives that replacement is the same question one rung up and it is not answered here.

The convention, named #

Normal play. The ambient board is the switch {t | −t}, whose temperature is t; the crossover is read off optimal play over the sum of the follow-up and the ambient, computed by minimax with no strategy assumed; and the follow-up’s temperature is the height at which its own two walls meet, computed by the thermograph.

The depth of a follow-up is counted along Right options only, because it is Right’s move that is under test and Left’s answer that is being priced. A follow-up whose Left options are fights and whose Right options are numbers counts as settling the fight, which is the right convention here and would be the wrong one for the mirror question.

Where the ladder goes next #

The sente anchor has four rungs to here: what sente is, that double sente is a band rather than a property, what the reverse costs, and now that the rule the first three rest on has two cases. The five above it are one measurement being corrected four times, and the sequence is worth knowing before any of the numbers are trusted.

What the halving is a function of answers this page’s own question first: the two classes are not a fact about forms but a feature of one wall — whether the follow-up’s right wall rises straight up before it leans. Fifty-five positions, no exception, and the crossover becomes something a reader can see on a diagram.

Then the halving itself gives way. A subtraction not a factor builds a pool that is deliberately deep and finds the crossover to be the follow-up’s temperature less a half on eighteen of twenty, with the factor of a half agreeing only where that temperature is one — which nearly every position in the earlier pool had. And half of the smaller temperature shows that both readings are regions of one law: over 128 fights with answers ranging from a number to a temperature of six, the correction is half the answer’s temperature, saturating at half the fight’s own.

The last two rungs are about how far down a position the answer depends. The two numbers at the top settles it by construction rather than by proof: group positions by their two top temperatures and the crossover is single-valued on every group however far apart the third temperature is, and the formula survives a fourth level of fight. The premises an induction would need then states plainly what is still missing — the law holds at five levels and survives translation, heating and cooling, and none of that is the inductive step.

So the reading a player should carry from this anchor is the last one and not this one. A fight’s crossover is its own temperature less half its answer’s, and the factor of a half this page reports is that law seen through a pool whose answers were all the same size.

Two neighbours are worth the trip. What a move nobody makes is worth is the reverse question — what denying an exchange is worth — priced by the same sweep. And a rule with a guarantee is where playing by temperature is given a bound, and where a temperature that is twice the true one is exactly the input that makes the bound useless.