Two hot fights that add to a cold number

Two identical fights on a board. Each is worth $\{2 \mid 0\}$ — whoever moves in it gains two, whoever is left with it gains nothing — so each has a mean of $1$ and a temperature of $1$. Both players want to move in either.

Put them together and the total is worth exactly $2$: a plain number, with no temperature at all. Neither player wants to move in it, and the fight has vanished.

Every position carries two numbers rather than one. The mean is what it settles to once the fighting is over. The temperature is how much moving there is worth — how large a tax on moving a player would pay rather than give the move up.

The mean is well behaved under addition and the temperature is not, and the asymmetry is worth stating as sharply as possible.

$$\operatorname{mean}(G + H) \;=\; \operatorname{mean}(G) + \operatorname{mean}(H) \qquad\text{always.}$$

$$\operatorname{temp}(G + H) \;\leq\; \max\bigl(\operatorname{temp}(G),\, \operatorname{temp}(H)\bigr) \qquad\text{and that is all.}$$

An inequality, in one direction, with no formula. The sum’s temperature can be anything from nothing at all up to the hottest part’s, and which it is depends on the components in a way no arithmetic on their temperatures can capture.

For the simplest positions both numbers are read straight off the braces. A switch $\{a \mid b\}$ has mean $(a+b)/2$ and temperature $(a-b)/2$ — the midpoint of the two options and half the distance between them — so $\{2 \mid 0\}$, $\{4 \mid 0\}$, $\{1 \mid -1\}$ and $\{3 \mid -3\}$ carry means of 1, 2, 0 and 0 and temperatures of 1, 2, 1 and 3. Any two of those can be added, and the mean of the result follows from the list while the temperature does not.

Why the means add #

The mean is the value a position settles to, and settling is what happens when the fighting stops. Formally it is the value of the position after cooling — charging a tax on every move — at any tax above the temperature, at which point the position has become a number.

Cooling a sum by $t$ cools each part by $t$, so a tax high enough to freeze both parts freezes the sum, and the frozen sum is the sum of the frozen parts. Numbers add in the ordinary way. That is the whole argument, and it explains why the mean is the well-behaved half: it is a statement about the position after all the interesting behaviour has been taxed out of it.

The temperature is a statement about exactly that interesting behaviour, which is why it does not survive.

Why the temperature does not #

Two identical switches cancel each other’s urgency, and it is worth walking through.

Left moves in the first, gaining two. Right moves in the second, gaining two the other way. Nobody has gained anything relative to the other, and there is nothing left to fight over. Because there are exactly two fights and they are the same size, the whole thing is a fair exchange — so the sum is a number, and a number has no temperature.

The condition for that is not “the temperatures are equal”. It is much more delicate, and the first thing to establish is what happens when the two sizes are plainly different — because if the sum’s temperature were some blend of the parts’, an unequal pair is where the blending would show.

One point of difference is a thin margin, so the same question is worth asking with the gap opened as far as the small position can be shrunk. If a small fight contributes anything at all to the total, a fight worth six times less than its partner is where the contribution would have to be smallest and still visible.

The exchange that makes it happen #

The disappearance of the temperature in the first figure is worth following move by move, because it is the mechanism behind everything else on this page.

Two fights, each worth two to whoever gets there. Left moves in the first and is two ahead. Right, rather than answering in the same fight — there is nothing left to answer — moves in the second and is two ahead there. Net: nothing, and the board is settled at $1 + 1 = 2$.

What made the exchange fair is that the two fights were the same size. If the first were worth six and the second two, Left would take the six, Right would take the two, and Left would come out four ahead of the settled position — which is a real advantage to whoever moves first, so the sum is still hot. That is a claim about a specific pair, so it is worth drawing rather than asserting.

That is the whole story of why the sum’s temperature depends on the components and not on their temperatures. It depends on whether the components can be paired off — and pairing off is a fact about the sizes together, not about any one of them. The endgame account is that exchange carried out on paper for a whole board: add up what each region settles to, then add the largest thing at stake, subtract the next, and so on down the list.

What it costs a player #

This is not a technicality about diagrams. It is the reason playing the hottest component first is a rule of thumb rather than a theorem.

A player looking at a board with several fights on it wants a single number saying how much is at stake overall, so as to compare this board with a different one. Temperature does not supply it. The sum’s temperature is bounded above by the hottest part’s, and can be far below — so a board of six evenly matched fights may have no temperature at all, while a board with one big fight and five trivial ones has the big fight’s.

The practical use of the number is a ranking rather than a total. Set four components beside each other and order them by what is at stake, and that order is exactly right for choosing the next move — which is what playing the hottest component is — while saying nothing at all about the value of the whole board. The board is the sum, and the sum’s temperature is not on any list of the parts’.

Worse, the number a player would naively compute — the sum of the temperatures — is not merely inaccurate but wrong in a consistent direction. It is always at least the true value and usually far above it, so a player who adds temperatures systematically overestimates how much is at stake and treats settled boards as fights. The error is not noise around the right answer; it is a bias, and a bias is the harder kind of mistake to notice from experience.

The consolation is that the mean does supply the number a player wants for comparing boards, and the mean adds. Somebody evaluating a whole position adds the means and gets an exact answer about where the position settles; the temperatures are then used one at a time, to decide where to move next, and are never added.

That division of labour is the practical content of the theory of temperature, and it is exactly aligned with which of the two quantities is additive.

The number that does add, and what it is worth #

Since the mean is the well-behaved half, it is worth being clear about what a player gets from it — and about what the addition is actually asserting.

$\operatorname{mean}(G+H) = \operatorname{mean}(G) + \operatorname{mean}(H)$ says the settled positions add. It does not say the values add in any simplified sense; the values add because addition of games is defined so that they do, and the mean is a number extracted from the value afterwards. What the theorem contributes is that extracting the number commutes with the addition.

That is worth more than it sounds. It means a board can be scored by scoring each region and adding the scores — the ordinary thing every Go player does — and the answer is exact rather than approximate. It fails only where a component has no mean, which for short games it never does.

It is tempting to extend the licence to the other numbers along the horizontal axis, since they are values too and the mean is the well-behaved reading. The stops — what each player gets by moving first and fighting on with no tax — are the obvious candidates, and they do not add either.

So the horizontal axis is not uniformly safe. What composes is the mast, and it composes because the mean is what survives cooling; the stops are readings taken at zero tax, where the two components are still able to answer each other.

One of them is a homomorphism and the other is not a function #

The asymmetry can be stated in a way that says exactly how deep it goes, and the statement is shorter than the two sections above it.

The mean is a homomorphism. It carries the sum of games to the sum of numbers, it carries the negative of a game to the negative of its mean — since mirroring a position mirrors its thermograph — and it carries the zero game to nought. So it is a structure-preserving map from the group of games onto the dyadic rationals, and everything a homomorphism licenses is available: a board can be scored region by region, a region can be replaced by another of the same mean without changing the total, and the accounting is exact rather than approximate.

The temperature is not a function of the means, or of the temperatures, or of anything smaller than the positions. That is stronger than saying it fails to add. A quantity can fail to add and still be determined — the maximum of two numbers does not add and is perfectly well determined by them — whereas here two pairs of positions can carry the same two temperatures and the same two means and give sums of different temperature. There is no arithmetic on summaries, of any kind, that produces the answer, which is why the bound is an inequality rather than a formula with an error term.

So the two numbers a position carries are not two readings of one kind of thing. One is an image under a map that respects the whole structure; the other is a feature of the position that no such map preserves.

The shape of the bound has a name #

The inequality is worth looking at for its form as well as its content, because the form is one mathematics has a word for.

$$\operatorname{temp}(G + H) \;\leq\; \max\bigl(\operatorname{temp}(G),\, \operatorname{temp}(H)\bigr)$$

is not the triangle inequality. The triangle inequality would put a sum on the right, and a sum is what a length or a size obeys. What is on the right is a maximum, and that is the ultrametric inequality — the strong triangle inequality, the one that defines a non-archimedean valuation rather than an ordinary one.

That is worth noticing because it says the naive expectation was not merely quantitatively wrong but the wrong kind of quantity. A player adding up temperatures is treating them as sizes, which accumulate; the bound says they behave like orders of magnitude in a valuation, where the largest term swallows the rest and nothing accumulates at all. The paragraph above about the error being a bias rather than noise is that mistake seen from the outside — adding things that were never meant to be added.

It also predicts where the interesting question is. Ultrametric quantities have a characteristic behaviour: the inequality is tight, and the value is exactly the maximum, except where the two arguments have the same size, which is the one case in which cancellation is possible. That is precisely the pattern the figures above display — the sum stays as hot as its hottest part in every unequal pair drawn here, and collapses only in the pair of equal switches.

Whether that is a law or an accident of these figures is not something this rung establishes, and it is exactly what the rung asking how cold a sum can get goes on to measure. What belongs here is the observation that the question has a shape worth asking about: an inequality whose right-hand side is a maximum invites a completely different follow-up from one whose right-hand side is a sum.

The bound, and why it holds #

The inequality is worth an argument rather than an assertion.

Suppose the hottest component has temperature $t$. Charge a tax of $t$ on every move — cool the sum by $t$. Each component, cooled by $t$, has become a number, because $t$ is at least each one’s temperature. So the whole cooled sum is a sum of numbers, which is a number.

A position that has become a number after cooling by $t$ has temperature at most $t$. So the sum’s temperature is at most the largest of the parts’, which is the bound.

Nothing in that argument goes the other way. It gives no lower bound at all, and the first figure shows why there cannot be one: two components of temperature one summing to temperature nothing.

What the solver computed, and how #

Every wall in every thermograph here is computed as an exact piecewise-linear function by the site’s own recursion, not sampled. The temperature is where the two walls meet, found by intersecting the pieces, and the mean is where the mast stands.

The sum’s thermograph is not assembled from the parts’. It is computed from the summed game — build $G + H$ with the ordinary disjunctive sum, canonicalise, and run the same thermograph routine on the result. That is essential to the argument: a figure that built the sum’s diagram out of the parts’ diagrams would be assuming the very thing under discussion.

The generator then checks two claims before drawing anything. The means must add, to within floating-point tolerance, and the sum’s temperature must not exceed the largest of the parts’. Either failing stops the build with the offending numbers printed. So the two statements in the second section of this essay are not quotations from a textbook; they are conditions the figures on this page satisfy or do not exist.

The site’s gate also confirms that the bound is not vacuous — that at least one of the drawn pairs has a sum strictly cooler than its hottest part. Without that check, a bug that made every sum come out at the maximum would pass, and the essay’s central example would silently become false.

Where the model stops #

Small positions. Every component here has a birthday of two or three, and the thermographs are drawn to a height of a dozen or so. Real Go endgames have components with fights inside fights, and their thermographs have several vertices.

The bound is stated for two summands. It holds for any number by the same argument, and the figures show two, because three thermographs on one frame is already at the limit of what can be read.

Temperature is not the only thing that fails to add. The move itself does not add either: the best move in a sum is not the best move in any part, and there is no rule that reads it off the parts’ values. This essay is one instance of a general pattern, and it happens to be the instance where the failure has a clean bound.

A thermograph assumes the position is short. A loopy component has no thermograph in this sense, and a sum containing one is outside everything on this page.

The bound is an upper bound and the essay has no lower one to offer. That is a real gap rather than an editorial choice. Knowing that a board’s temperature is at most three is worth something to a player only if there is also a reason to think it is not zero — and there is no general statement of that kind here. In practice the sum’s temperature is close to the maximum whenever the components are of visibly different sizes, and collapses when they pair off; but “visibly different” is not a definition and the essay declines to pretend it is one.

Cooling is being used, not derived. The argument for the bound is a sentence about what a tax does to a sum, and it depends on cooling distributing over addition. That is a theorem, it is stated in the essay on cooling, and it is assumed rather than proved here.

Who found it, and when #

Thermographs and the theory of temperature are Conway’s, developed in On Numbers and Games (1976) and worked out in detail with Berlekamp and Guy in Winning Ways (1982). The bound on the temperature of a sum is standard there, as is the additivity of the mean.

There is a reason the failure is not usually presented as a failure. In the literature the additivity of the mean is a theorem with a name and the non-additivity of the temperature is a remark, because from inside the theory it is obvious: the mean is what survives cooling and the temperature is the height at which cooling finishes, and there is no reason a height should add. From outside, where the two numbers arrive together as “the two things a position carries”, the asymmetry is the surprising part — and it is the part a player has to internalise, since adding up temperatures is exactly the mistake the vocabulary invites.

The motivation was Go. Berlekamp’s programme of applying the theory to Go endgames turns entirely on the two quantities behaving differently: the means are added up to score the board, and the temperatures are compared one at a time to choose the move. Strong Go players had been doing both for centuries under the names miai counting and sente, without the vocabulary — which is one of the more satisfying convergences in the subject, and also a caution, since the players’ rules of thumb turn out to be provably close to optimal and provably not optimal.

Where the ladder goes next #

This is the third rung on the thermograph ladder, after how to read one and the orthodox accounting an endgame gets from it. This rung asks what happens to the diagram under the one operation the whole subject is built on, and finds half of it surviving.

The next rung is the natural follow-up to a quantity that is bounded rather than determined: how tight is the bound? The sum’s temperature can be anything from zero up to the maximum, and what decides where in that range it falls is the relationship between the components’ thermographs, not their temperatures — which is a statement about the shape of two diagrams and is exactly the sort of thing a figure can make obvious and a formula cannot.