Below zero

Temperature is introduced as the size of what is at stake — the difference between moving first in a position and moving second, in the currency the position is played for. That is a quantity with an obvious top, since a position can be worth fighting over by any amount, and no obvious bottom at all. Nothing at stake ought to be zero and there ought to be nothing below it.

There is something below it. The scale stops at $-1$, and what sits on the floor is exactly the numbers.

What the floor is, and why it is not zero #

The temperature of a position is the height at which the two walls of its thermograph meet. Below that height the two walls are apart, which says the mover has something to gain; at and above it they coincide, which says the position has been taxed until nobody wants it.

For a number, the two walls never diverge. Cool a number by any positive amount and it is unchanged — it is already frozen, there is nothing to freeze — so the walls sit on top of each other from the ground up and the height at which they “meet” is the ground itself.

That gives $0$, and $0$ is the wrong answer, because it is also the answer for $\ast$ and for $\uparrow$ and for every other position that has no fight in it but is not a number. The convention that separates them is to put the numbers at $-1$:

$$t(x) = -1 \quad \text{for every number } x$$

and it is a convention. Nothing computes it. What makes it the right convention is the job it does, which is to keep the ordering strict: a number is strictly colder than $\ast$, strictly colder than $\uparrow$, strictly colder than anything with an infinitesimal in it. Without the clause the coldest class holds two very different kinds of object and the word “cold” stops distinguishing them.

The classification, done twice #

A convention baked into the code is a bad thing to measure with. If the only way to know that a value has temperature $-1$ is that the thermograph routine special-cases numbers, then “the floor is $-1$ and the numbers are on it” is a fact about the program.

So the census classifies each value by a second and completely independent computation: whether its two stops agree.

The left and right stops are what each player gets by moving first and fighting on until somebody is left facing a number. If they are equal, the fight is worth nothing and the position sits infinitesimally close to that number. If they differ, there is a gap, and the gap is what the temperature measures.

Three classes come out of it: a number outright; the stops agree and the value is not a number; the stops differ. And the two computations agree on all 1,474 values — every value in the first class is at $-1$, every value in the second is at exactly $0$, every value in the third is strictly above $0$, with no exceptions.

That is what makes the picture a measurement. The convention is a convention; the structure it sits on is not.

Up is the case to hold on to, because it is the one a reader expects the floor to catch and the floor does not. Its two walls stand on top of each other from the ground upward, exactly as a number’s do, so the height at which they meet is nought; and the clause that would drop it to $-1$ does not fire, because $\uparrow$ is not a number. Nothing whatever is at stake in it and it is nevertheless not nothing — Left wins it moving second — which is the entire distinction the floor exists to record. The thermograph of $\uparrow$ is a single line, which is what a diagram built to measure a gap looks like when there is no gap.

The agreement between the two classifications is easier to audit one day down, where the whole pool can be counted by hand.

That the proportion falls so far in one day is the first warning that the histogram above the floor is a fact about the construction rather than about games. Each day builds far more positions with something to fight over than positions without, so the cold end thins as the pool grows, and any share read off one day is a share read off that day.

The rung at zero #

Three hundred and thirty-seven of the 1,474 values sit at exactly zero. They are the class this site keeps meeting under other names.

A value at temperature zero has both stops at the same number $x$, which says precisely that $G - x$ is infinitesimal: smaller than every positive number and larger than every negative one. So $G$ is a number plus something no number can see.

That description covers $\ast$, $\uparrow$, $\downarrow$, $\ast 2$, $\uparrow\!\ast$, the tinies and the minies, and everything in the all-small class; and it also covers those things with a number added — $1\!\ast$, $-\tfrac12 + \uparrow$, and so on. Being at temperature zero is not the same as being infinitesimal; it is being infinitesimally close to some number, which is a much bigger class.

And it is exactly the class the reduced canonical form replaces by a number. That reduction’s first clause is “if the two stops agree, take the number they agree on”, and its input condition is temperature zero, stated in the other vocabulary.

So the rung at zero is the boundary between two kinds of position and it is a boundary with a name in three places: temperature zero, stops that agree, and infinitesimally close to a number. All three are the same set.

Only fifteen numbers #

The count that gives the scale its shape is at the other end of the same row. Of the 1,474 values born by day three, fifteen are numbers.

One per cent. The construction produces the dyadic rationals slowly — a day adds only the simplest number in each gap — and everything else it produces is a switch, an infinitesimal or a fight of some kind. So the temperature scale over this pool is overwhelmingly populated above zero, with a floor almost nobody is standing on.

That is a fact about the construction rather than about games. The values real rulesets produce are 465 numbers out of 1,193, which is nearly two in five, because a game’s endgames are numbers and every game has endgames. The two pools are differently shaped at the cold end and the difference is large.

Cooling below zero is heating #

If the scale runs below zero, the operator that moves along it ought to as well, and it does.

Cooling by $t$ charges a tax of $t$ on every move; heating by $t$ pays a bonus of $t$. Written as one operator with a signed parameter, cooling by $-t$ is heating by $t$, and there is one scale rather than two.

The table says something the $-1$ convention only gestures at. A number is unchanged by cooling because there is nothing to freeze, and it is unchanged by heating because heating leaves numbers alone by an explicit clause in the definition. It is a fixed point of the operator in both directions, and no other value in the pool is.

So the floor is not arbitrary in the way a chosen constant is arbitrary. It marks the class of positions on which the whole family of operators does nothing at all, and the only choice made is where on the number line to put that class. Putting it at $-1$ rather than at $-\tfrac12$ or at $-\infty$ is what makes the interval between “a number” and “a number plus an infinitesimal” the same size as the interval between two adjacent integer temperatures, which is a convenience and nothing more.

The floor in a game rather than in the construction #

Two games on this site are made entirely of positions on the floor, and they are the sharpest illustration of what being there means.

Shove and Push are partizan strips of coins, and every one of their 728 positions up to six squares is a number. So the whole of both games sits at temperature $-1$: there is never a move worth making, both players would rather the other had to move, and the entire analysis is arithmetic with no search in it anywhere.

That is what a game looks like from $-1$, and it explains why the floor is worth marking rather than merging into zero. A game whose positions are all at temperature zero would look completely different: Clobber is such a game, its values are infinitesimals, and deciding one of its positions requires counting ups rather than adding numbers. Both games are “cold” in the loose sense and they need entirely different machinery.

What the picture cannot show #

A thermograph draws a position’s temperature and cannot draw the reason for the floor, because the floor is below the axis the drawing starts at. Every thermograph on this site begins at height zero and runs upward; the $-1$ lives in the caption.

The second thing it cannot show is the difference between the members of the zero class. $\ast$, $\uparrow$ and $\uparrow\!\ast$ all have identical thermographs — two walls on top of each other at the ground, running straight up — and they are three different games with three different outcomes. The thermograph is exactly the wrong instrument for them: it is built to measure a gap, and there is no gap. Deciding between them is atomic weight’s job and it is a completely different computation.

And the third is size. A temperature of $2$ is not a claim that the position is worth a lot, and big is not the same as hot is the essay about that. The hottest value born by day three is $\{2 \mid -2\}$ with temperature $2$, and it is worth $0$ on average.

The shape of the scale above the floor #

The two ends of the distribution are the interesting parts and the middle is worth a sentence too.

Above zero the values are spread over seven temperatures — quarters, halves, three-quarters and so on up to $2$ — with by far the largest group at $\tfrac12$: 497 of the 1,474, a third of the whole pool. The next largest are $\tfrac34$ with 288 and $1$ with 160.

A temperature of $\tfrac12$ is what a switch between two numbers one apart produces. $\{1 \mid 0\}$ has mean $\tfrac12$ and temperature $\tfrac12$; so does $\{0 \mid -1\}$; so do a great many values whose options are day-two values one apart. The construction produces those in quantity because the day-two values themselves are packed into a short interval, and the temperature of a switch is half the width of the gap between its stops.

So the histogram is not a fact about heat. It is a fact about how tightly the day-two values are packed, which decides how wide the gaps between them can be, which decides the temperatures available on the next day. The scale above zero is inherited from the scale below it.

What the convention costs #

There is a small price for the $-1$ clause and it should be stated. It makes the temperature of a sum harder to talk about, because $t(G+H)$ is not a function of $t(G)$ and $t(H)$ in any case, and the special value at the bottom means the failure has an extra shape: two numbers add to a number and $-1 + -1$ is not $-1$ in any sense the arithmetic supports.

The clean statement is that temperature is not a homomorphism, full stop, and two hot fights that add to a cold number is the essay about that. The convention neither helps nor hurts; it just means the reader should not read $-1$ as a number that can be added.

The other price is that “temperature” and “urgency” come apart at the bottom. A position at $-1$ and a position at $0$ are equally not worth moving in, in the sense a player cares about. What separates them is what happens when they are put in company: a number is inert, and a position at temperature zero can decide a sum of infinitesimals. The floor is a statement about company rather than about urgency, and the word does not carry that.

Who chose it, and what the alternatives were #

The $-1$ is Conway’s, and On Numbers and Games introduces it exactly as a convention with a job — the numbers have to be somewhere and putting them at $0$ would confuse them with the infinitesimals. Winning Ways states it the same way and adds the phrase that stuck: a number is infinitely cold, in the sense that no amount of taxing changes it, so any finite floor is arbitrary and $-1$ is the convenient one.

Two other choices were available and both are worse for a reason worth seeing.

Putting numbers at $-\infty$ is faithful to “no tax touches them” and destroys arithmetic: a temperature is a real number that gets compared, added to and subtracted from all over the Go endgame literature, and a value that is not a real number breaks every one of those.

Putting numbers at $0$ alongside the infinitesimals loses the distinction the word is carrying. A player holding $\ast$ and a player holding $\tfrac12$ are in different situations — one of them has a move that matters in company and the other does not — and a scale that reports the same number for both has stopped measuring the thing it was built for.

What $-1$ buys is that the two classes are one unit apart, so the phrase “cooling by one” takes a position at temperature $1$ or below all the way to a number, and chilling — the operator the Go and Domineering literature actually uses — is exactly cooling by one. The floor was chosen to make the standard operator land on it.

The clearest place to watch that happen is not in the construction at all. Run the same tax over five Domineering boards and the two ends of the scale sort themselves out on a board a reader can look at.

Three of the five end up smaller than every positive number, which is what the operator is for: chilling turns a board of fights into a board of infinitesimals and leaves the banked material where it was. The boards it moved were the ones at temperature $1$ and $0$; the boards it did not move were the ones on the floor. A tax of one is exactly the distance from the floor to the top of the class the operator is meant to collapse, and that is the whole of what the convention buys.

That last point is worth stating as the general shape of the choice rather than as a coincidence about one operator. A convention at the bottom of a scale is chosen by what it makes the arithmetic above it say, not by what it says about the objects at the bottom. Nothing about a number determines that its temperature should be $-1$; every reading of how much is at stake in a number gives nought or nothing at all. What determines it is that the subject’s most-used operator subtracts one from every temperature, and a floor one unit below the infinitesimals is the floor that operator lands on exactly.

So a reader meeting $-1$ for the first time is right to find it unmotivated by the definition and wrong to conclude it is arbitrary. It is a fixed point chosen after the fact, of exactly the kind that makes a notation worth using, and the test of it is not whether $-1$ describes a number but whether the sentences it lets people write come out true.

Where the ladder goes next #

cold opens here with the floor of the scale, and the six rungs above it all take the first question this page leaves: what the distribution above the floor actually looks like, and whether it describes anything a player meets.

How hot a day gets explains the top of the histogram exactly. A day of construction buys one degree, the record-holder is unique each time — $\ast$, then $\{1 \mid -1\}$, then $\{2 \mid -2\}$ — and the record is a fact about how far the integers get rather than about anything the fights do. The shape underneath stays untidy: it peaks at a half, leans upward out of its own peak, and has a hole at one and three quarters where nothing is born.

The four rungs after that answer the question this page does not think to ask — whether a census over values describes a game at all — and the answer is no, in a definite direction. How hot a real position is reweights by position rather than by value: a tenth of the subject is hot by value and a twentieth by position, two thirds of positions are worth numbers outright, and ten of seventeen rulesets never produce a hot position at all. What a game actually produces narrows it again from a catalogue to a played game and finds 16 per cent hot against the catalogue’s 53 — a factor of three that every catalogue census on this site, including this anchor’s, was quietly carrying.

Then two corrections in the other direction. One fight makes a board a fight tallies at the board rather than the piece and gets 32 per cent, and the obstacle was the catalogue removes the limit that made the early game unmeasurable and finds a board hot four times in five three moves in, cooling as it breaks up. So a real game is hot at the start and cold at the end, which is the reverse of the impression a value census gives.

Eight squares and no hotter closes the anchor with a ceiling the construction has no analogue for: no Domineering position is hotter than three halves, the attaining region has eight squares, and the ceiling holds at nine and ten where the obvious extrapolation predicted more.

The second and third directions this page names are elsewhere. The 337 values at exactly nought need the atomic weight calculus, which is the scale’s replacement below its own floor. And the operator that reaches from the cold end into the hot one is the operator that puts the star back, whose whole content is what it does to the class this page’s floor is made of.