What the halving is a function of

A move is sente when the opponent answers it, and whether they answer depends on the rest of the board. Sente is a fact about the rest of the board establishes that, and the answer that starts another fight found the rule has two cases:

The move is answered while the ambient temperature does not exceed the follow-up’s temperature — on the 38 positions whose answer settles the fight. And while it does not exceed half of that — on the 17 whose answer starts another.

That essay closed by naming the problem with its own result:

The halving is a number and it is not yet a height on a diagram, so a reader with two thermographs still cannot compute a crossover without knowing which class the position is in. Finding the feature of the walls that the factor is a function of would make the crossover readable rather than playable.

The feature is one wall’s foot.

What a right wall does at the bottom #

A thermograph’s right wall starts at the Right stop — the value the position settles at if Right moves first and both play out. As the tax rises, the wall traces where the position’s value goes.

Whether it leans immediately depends on what is under it.

If Right’s option is a number, there is nothing to hold the wall up: taxing the position makes Right’s move worth less at once, and the wall leans in from the moment it leaves the axis.

If Right’s option is itself a fight, that fight has its own temperature, and while the tax is below it Right’s move is still worth making at full value. The wall goes straight up for exactly that height before it starts leaning.

So a vertical stretch at the foot of the right wall means one thing: the answer starts another fight, and the height of the stretch is that fight’s temperature.

That is a fact about how a thermograph is built rather than an observation about these positions. A wall is the trace of a stop as the tax rises, and a stop moves only when the move producing it stops being worth making; a fight below holds it still for exactly as long as it is worth more than the tax. So the vertical foot is not evidence that the answer starts a fight — it is what starting a fight looks like when the diagram is drawn.

The rule #

Sort the 74 positions of the depth census by that feature alone.

Nineteen of the seventy-four have no crossover at all and are set aside; of the remaining fifty-five:

Thirty-eight have a leaning foot and every one of them is answered up to the follow-up’s full temperature.

Seventeen have a vertical foot and every one of them is answered up to half of it.

No exception, and the census asserts it: a position whose wall foot says one factor and whose play gives the other would stop the build. It also refuses to build if the pool contains only one kind of foot, which is the failure mode a rule like this invites — a classification that has never had to discriminate is a classification with no evidence behind it.

Nineteen more positions have no crossover at all, because their follow-up reduced to a number — there is no fight to answer and nothing to compute. They are set aside rather than counted as failures, and the section below is about why setting them aside is legitimate rather than convenient.

Why this is progress #

The rung below sorted the same positions by depth of form — whether the follow-up’s options are numbers or fights. That is a true classification and it is a fact about how the position is written.

Sorting by the wall is a fact about the diagram, and the difference matters for three reasons.

It is visible. A reader who has drawn the two thermographs the situation calls for — the local fight’s and its follow-up’s — can see the foot without doing any further work. Reading a form’s depth means looking at the braces, and the whole complaint against brace notation is that it makes structure hard to see.

It survives rewriting. A form’s depth changes when the position is written differently; a wall does not, because a thermograph is a function of the value. Two positions with the same value have the same walls and the same crossover, and the depth reading has to be applied to a canonical form to be well defined at all.

And it says what the number is. The vertical stretch is not merely present or absent; it has a height, and the height is the sub-fight’s temperature. So the diagram carries the whole of the third quantity the calculation needs, and a reader holding two thermographs is holding three temperatures.

What the reader actually does #

The procedure, written out, is four steps and no play.

Draw the follow-up’s thermograph. Read its temperature — the height at which its two walls meet. Look at the foot of its right wall: if it rises straight, measure how far, and if it leans, there is nothing to measure. The crossover is the follow-up’s temperature, halved if the foot was vertical.

That is the whole of it, and it replaces a minimax over a sum with a glance at a diagram. The saving is not small: the census finds each crossover by playing the position against forty-one different ambients and reading where the answer stops, which is forty-one full evaluations of a sum for a number the wall gives away.

There is one caution and it is the caution on every thermographic reading. The follow-up’s thermograph is not the local fight’s. The position being played in is one game; the thing whose temperature governs the crossover is its Right option, one move down. A reader who reads the foot of the local position’s right wall is reading the wrong wall, and on many of these positions the two look similar enough to confuse.

The nineteen with no crossover #

Nineteen of the seventy-four positions never have their move answered at any ambient, and they are worth a paragraph because they are the degenerate case the rule has to handle.

In each of them the follow-up reduces to a number. {6 | {4 | {9 | 1}}} looks like a three-level fight and its Right option is worth 5 — the inner braces collapse under the reduction, and what is left has no temperature at all.

A position whose follow-up is a number offers the opponent nothing to answer — which is what a number does to a fight in its simplest form: taking the ambient is better at every temperature above nought, and there is no crossover to find. The wall reading handles it without a special case, because a number’s thermograph is a vertical line and a vertical line has no right wall to read a foot off — it is the mast, and the whole diagram is the mast.

That is a reminder worth keeping about the whole apparatus. A follow-up that looks like a fight in braces may not be one, and every reading on this page is of the canonical form rather than of what was written down. Two hundred and fifty-six ways of writing twenty-two things is the general statement of that hazard, and here it accounts for a quarter of the pool.

Why a half and not something else #

The rule gives a factor of one or a factor of a half, and nothing in between. That is worth an explanation, because a feature with a height in it might have been expected to produce a factor depending on the height.

It does not, and the reason is the shape of the exchange. When the answer settles the fight, the opponent’s decision is answer or take the ambient, and the two are compared directly — so the crossover is where they are equal, which is the follow-up’s temperature.

When the answer starts another fight, the opponent’s answer also hands back a move. The exchange is now two moves rather than one, and the value of a two-move exchange against a one-move alternative is compared at half the rate. The crossover halves because the exchange doubles in length, not because of anything about the sizes.

So the height of the vertical foot decides nothing except that it is there. It says the answer starts a fight; the fight’s size does not enter. The heights in the census are 1, 5 and 6, and the factor is a half in all three cases.

That is worth setting beside what happens to the same quantity elsewhere on the site. How cold a sum of hot games can be finds a whole permitted range and asks where in it a sum lands; here there is no range, only two answers. The difference is that a sum of two hot games is a genuinely continuous problem and a crossover is a comparison between two exchanges of definite length, and a length is a whole number.

That is a slightly disappointing shape for a result and it is worth saying so. The census makes the classification readable and does not make it graded, and a reader hoping the diagram would produce a whole schedule of factors will not find one here.

What it buys a player #

The practical claim is that a Go or Domineering player who has learned to draw thermographs can now settle a question they otherwise have to feel their way through, and it is worth stating carefully because the claim is narrow.

What was hard was not the temperature. Answer while the board is cooler than the follow-up is a rule anybody can apply once they know the follow-up’s temperature, and the temperature is the easy reading on a thermograph — the height where the walls meet.

What was hard was knowing which rule. The two-case structure meant a player had to know whether the answer settles the fight, and that is a question about what happens two moves later, which is exactly what a diagram was supposed to save them from. So the rule as the rung below left it had a hidden lookahead in it.

The wall foot removes the lookahead. The diagram already contains the information — it was drawn from the follow-up’s options, so the sub-fight is in there — and what this page does is say where to look.

The residue is that a third diagram is still lurking. The height of the vertical foot is the sub-fight’s temperature and reading it means the follow-up’s thermograph has been drawn accurately near the axis, which in practice means knowing the sub-fight. So the lookahead is not gone; it has moved from a decision a player makes to a diagram a player draws, which is a better place for it but is not nowhere.

From a fact about forms to a fact about a picture #

The move this page makes is small to state and worth naming as a kind, because it is the move that turns a classification into something usable.

A class defined by forms is a class a reader cannot recognise. Positions whose Right option is itself a switch with a number on the left is a condition on a written expression, so applying it means having the canonical form in hand — which means having run the recursion, which is the expensive thing the crossover rule was supposed to save.

A class defined by a feature of the diagram is different in kind. The thermograph is drawn from the position and the feature is visible on it: does the right wall rise straight before it leans? A reader with the diagram has the answer by looking, and the diagram is a thing this subject draws anyway.

So the same partition, re-described, changes from a lookup into a glance, and nothing about the mathematics moved. That is the whole content of this page, and it is worth separating from the finding that the partition is exact.

It also predicts where the next improvement will come from, and the ladder above confirms it. A feature of a wall is one step from a number read off the wall, and once the classes are known to be about a wall’s shape the natural question is what quantity that shape is a function of. The answer turns out to be a temperature, and the two classes turn out to be two regions of one formula — which could not have been guessed while the classes were still described as shapes of expression.

What the census does not say #

Four limits.

Seventy-four positions from one construction. The pool is built by nesting switches at two and three levels, which is what the rung below built and what makes the two classes both populated. A position from a real board — a Go endgame, a Domineering corner — has a follow-up whose wall could have a foot the construction does not produce.

Two classes and no third. Every position here has a factor of one or a half. Nothing establishes that no third factor exists; the census would notice one, since a position matching neither is reported as other, and none has turned up.

The wall reading and the depth reading agree on this pool. They are two descriptions of the same partition here, and the claim is that the wall version is the readable one, not that it is a different one. A position where they came apart would be very interesting and there is none in the census.

And the halving argument is a sketch. A two-move exchange is compared at half the rate is the mechanism above and it is stated in words. Turning it into a derivation means writing the two comparisons out against the ambient explicitly, which is a piece of thermographic algebra rather than a computation, and this page has not done it.

The convention, named #

Normal play, and the temperature is the standard one: a tax on every move by both players, with the temperature the height at which neither wants to move. A number is given temperature −1.

The ambient is modelled by a single switch of the stated temperature, which is the site’s standing convention for the rest of the board is worth this much, and the crossover is found by playing the sum out exactly at each ambient on a grid of quarter-steps.

A thermograph’s right wall is the trace of the Right stop of the cooled position as the tax rises. Its foot is the segment leaving the axis; it is vertical when the value does not move as the tax rises from nought, which happens exactly while the game below is worth more than the tax.

The follow-up is the position’s Right option, taken in canonical form.

Where the ladder goes next #

The sente anchor has five rungs to here, and the four above take the halving this page has just located on a wall and dissolve it into something larger.

A subtraction not a factor builds a deliberately deep pool and finds the correction to be the follow-up’s temperature less a half rather than half of it — with the two readings agreeing only where that temperature is one, which is what nearly every position measured before had. So the classes this page identifies are real and the arithmetic attached to them was fitted to a narrow pool.

Half of the smaller temperature then shows both readings to be regions of one law. Over 128 fights whose answers run from a number up to a temperature of six, the correction is half the answer’s temperature, saturating at half the fight’s own — linear below the cap and constant above it, which is exactly the shape that looks like a factor on one pool and a subtraction on another.

The last two rungs settle how deep the answer reaches. The two numbers at the top groups positions by their two highest temperatures and finds the crossover single-valued on every group, however far apart the third is — so the dependence stops at two numbers, established by construction rather than by proof, and the formula survives a fourth level of fight. The premises an induction would need then states precisely what is still missing: the law holds at five levels and survives translation, heating and cooling, and none of that is the inductive step.

What this page contributes to that sequence stands unchanged. The two classes are a feature of one wall, and the wall is what a reader can see — which is why the ladder above could go looking for a formula at all rather than for a rule about forms.