The hotter residue survives

A number and a fight took a hot position apart into three pieces. Charge the position exactly its own temperature — cool it by the height at which the fight stops being worth having — and what is left is its mean with something attached. Subtract the mean and the something is an infinitesimal, the residue: a star for most hot positions, an up or a down or a nimber of two for the rest, and never nothing. Over all 1,122 hot values born by day three the residue was a star 942 times. So a hot position is its mean, plus a fight, plus a remainder no number reports, and the remainder is what decides close games.

The essay closed by asking what happens when several hot positions are added, and it listed two of the three answers in advance: the means add, and the temperatures do not. The third was the residues, and the natural guess — they add as infinitesimals do, so that a board’s residue is the sum of its regions’ — would make the three-piece decomposition a genuine arithmetic of boards. The guess is wrong, and wrong in a way that is exact rather than approximate.

Seven thousand eight hundred and fifty-four sums #

The pool is every hot value born by day two, seven of them, against every hot value born by day three, 1,122 of them. For each of the 7,854 sums the residue is computed from scratch: the sum is put in canonical form, its thermograph gives its temperature and mean, it is cooled by that temperature, and the mean is subtracted. The parts’ residues are computed the same way, and the three are compared.

The pairs split cleanly by one question: are the two parts equally hot?

On the 4,712 pairs whose parts have different temperatures, the sum is exactly as hot as the hotter part, and the sum’s residue is the hotter part’s residue — every time, and never with any trace of the colder part’s. On the 3,142 pairs whose parts are equally hot, three things happen. On 880 the sum stays at that temperature and the two residues add, as the guess said they would. On 2,237 the sum is colder than either part, and its residue belongs to whatever fight is left. And on 25 the two fights cancel completely and the sum is a number, with no residue at all.

So the guess holds on 880 sums in 7,854 — about one in nine — and it holds there for a reason that makes it a special case rather than the rule.

Cooling a sum cools every part by the same tax #

The rule has a two-line explanation, and both lines are facts about cooling rather than about residues.

The first is that cooling is additive: cooling a sum by some tax gives the same game as cooling each part by that tax and adding. Cooling adds and heating does not checked this on 1,768 pairs at two taxes and found it exact every time; it is a theorem about the operator. The second is that a position cooled by more than its own temperature is exactly its mean. Past its temperature there is nothing left to fight about; the cooled position is the number the fight was worth, with no infinitesimal attached.

Put the two together. The residue of a sum is the sum cooled by its temperature, less its mean, and by the first fact that is each part cooled by the sum’s temperature, less each part’s mean, added. When the parts differ in temperature the sum is as hot as the hotter part, so the tax is the hotter part’s own temperature. The hotter part, cooled by exactly its own temperature, leaves its residue. The colder part, cooled by more than its own, leaves its mean and nothing else, and less its mean that is nought. The sum’s residue is the hotter residue plus nought.

The first worked sum shows it happening. $\{3 \mid -3\}$ has temperature 3 and residue ∗. $\{2 \mid \{1 \mid -1\}\}$ has mean 1, temperature 1 and residue ↑ — one of the 122 day-three values whose residue is an up. Their sum is hot to temperature 3. Cooled by 3, $\{3 \mid -3\}$ becomes ∗ and $\{2 \mid \{1 \mid -1\}\}$ becomes the number 1. The sum’s residue is ∗, and the ↑ has gone. The second, $\{4 \mid 0\} + \{3 \mid 1\}$, is the same thing with a star in place of the up: temperatures 2 and 1, both residues ∗, and the sum’s residue ∗ — one star, not the two that would cancel.

The third is the other case. $\{2 \mid \{1 \mid -1\}\}$ and $\{2 \mid 0\}$ are both hot to temperature 1, the sum stays hot to temperature 1, and each part is cooled by exactly its own temperature. Both residues survive and add: ↑ and ∗ make ↑∗.

The picture is visible in the thermographs themselves.

A thermograph’s walls meet at the temperature and stand as a single vertical mast above it. Cooling by a tax is reading the thermograph at that height. At height 3 the thermograph of $\{2 \mid \{1 \mid -1\}\}$ is two units up its mast, where it says only “the mean, 1”. Whatever infinitesimal lay at the foot of that mast is below the line being read. The sum’s thermograph closes at the hotter part’s temperature, and it is read there.

The temperature obeys the same two facts #

The two facts decide more than the residue, and it is worth seeing that they decide the sum’s temperature too, because that is where the residue’s rule comes from.

Take parts of temperatures $t_G < t_H$ and cool the sum by any tax between them. By additivity the result is $G$ cooled plus $H$ cooled. The colder part, cooled past its own temperature, is its mean — a number. The hotter part, cooled by less than its own temperature, is still a fight. A number plus a fight is a fight, so the sum is still hot at every tax up to $t_H$, and its temperature is at least $t_H$. It cannot be more, because the temperature of a sum is at most the largest temperature in it. So when two parts differ in temperature, the sum is exactly as hot as the hotter one — on all 4,712 pairs here and all 2,791 of the sample, and by the argument always.

That is the case how cold a sum of hot games can be found to be the usual one: 864 of its 1,035 sums sat exactly at the maximum temperature. What looked there like a tendency of sums is a rule for every sum of parts that are not equally hot, and all the room below the maximum — the frozen sums, and the eleven that landed in between — belongs to parts that are. The residue simply rides along with the temperature. The sum’s fight is the hotter part’s fight, shifted by the colder part’s mean, and so its residue is the hotter part’s residue.

Seen that way the residue behaves like a maximum rather than a sum, and so does the temperature, and for the same reason: cooling reads every part at one height, and at the height that matters the colder parts have already stopped being fights.

What the sum forgets #

The rule does not care what the colder residue is.

Among the 4,712 pairs of different temperatures, the colder part’s residue was a star 4,572 times, an up 58, a down 58, ∗2 sixteen times, and ↑∗ and ↓∗ four each. Every one of them is missing from the sum’s residue. The ups and downs come in equal numbers because every pool here is closed under negation — the negative of a value born by day three is born by day three — so each pair with an up has a mirror pair with a down. The stars are the least of it — a star is what any plain switch leaves, and two stars cancel anyway — but the ups and downs are exactly the residues a number and a fight singled out as the ones that decide close games, and they are dropped just as completely. Nothing about the kind of infinitesimal protects it. The temperature of the other part is the whole of the rule.

Equal temperatures: add, cool or cancel #

The 3,142 pairs of equal temperature are where residues can add, and most of them do not.

When the parts are equally hot, the tax that cools the sum is each part’s own temperature — if the sum is still that hot. On 880 pairs it is, and both residues survive and add. On 2,237 it is not: two fights of the same temperature partly cancel, the sum is colder than either, and cooling it by its own lower temperature leaves both parts still hot. The residue of the sum is then the residue of a fight neither part had on its own. $\{1 \mid 0\} + \{1 \mid \downarrow\}$ is the smallest example: two fights hot to a half, and a sum with no temperature to speak of whose residue is neither part’s star.

And on 25 pairs the cancellation is total. $\{1 \mid 0\} + \{2 \mid 1\}$ is two switches of temperature one half with means a half and three halves; each is its mean plus $\pm\tfrac12$, and $\pm\tfrac12 + \pm\tfrac12$ is nought, so the sum is the number 2. That is the extreme of the effect how cold a sum of hot games can be measured from the side of temperatures, where 160 of 1,035 sums were frozen outright. Seen from the residues, it is the reason equal temperatures are so rarely additive: two fights of one temperature are the fights most likely to answer each other.

The same rule on day-three pairs #

Seven day-two values are a narrow base, and the rule should be tested where neither part is small.

Four thousand pairs drawn with a fixed seed from the 1,122 hot day-three values give the same shape. On all 2,791 pairs of different temperatures the sum is as hot as the hotter part and keeps its residue alone. Of the 1,209 equal pairs, 427 add their residues, 777 cool into something else and five cancel to a number. The colder residues dropped are more varied here — 190 downs and 181 ups among them, and 59 nimbers of two — and every one is dropped.

Forgotten by the residue, not by the game #

It would be easy to read all this as saying that a colder region’s infinitesimal does not matter once something hotter is on the board. It says nothing of the kind.

Take the hot fight $\{3 \mid -3\}$ and put beside it each of three colder parts from the essay before this one: $\{2 \mid 0\}$, $\{2 \mid \{1 \mid -1\}\}$ and $\{2 \mid \{1 \mid -1\}, \downarrow\}$. All three have mean 1 and temperature 1, and their residues are ∗, ↑ and ↓∗. All three boards have residue ∗, the hot fight’s. Now add $\{2 \mid -4\}$, which is $\{3 \mid -3\}$ less one — it cancels the hot fight exactly and takes away the colder part’s mean. What is left of each board is its colder part less one: $\{1 \mid -1\}$, $\{1 \mid \{0 \mid -2\}\}$ and $\{1 \mid \{0 \mid -2\}, -1{\downarrow}\}$. The first is a win for whoever moves, the second a win for Left whoever moves, the third a win for whoever moves again. Three boards with one residue, and a single companion that separates the middle one from the other two.

The middle board can be played out in two lines. With Left to move in $\{1 \mid \{0 \mid -2\}\}$, Left takes the 1 and is a point ahead with nothing left to play. With Right to move, Right’s only move is to $\{0 \mid -2\}$, Left answers by moving to 0, and Right, facing an empty board, has lost. The ↑ in the colder part is exactly the tempo that lets Left answer last. In the first board, whose colder part carries a star instead, Right’s move leads to $-1$ and Right wins; in the third, the down’s extra option $-1{\downarrow}$ lets Right escape the same way. None of that is visible in the board’s residue, which is the hot fight’s star in all three.

So the residue of a board is not a summary of the board’s infinitesimal content. It is the infinitesimal content of the board’s hottest fight, and the hottest fight is the one that will be played first. Once it is played, the next-hottest region’s residue becomes the board’s, and so on down. The decomposition mean plus fight plus residue is exact for each region and lossy for the board, and how many ups — the measure of infinitesimals that does add, on the positions where it is defined — is the tool for the part the residue forgets.

What a player should carry instead #

The practical reading is short. A player summarising an endgame region by region — mean, temperature, residue for each — has a correct summary of every region and an incorrect way of combining them if the residues are added. The board’s mean is the sum of the means. The board’s temperature, when the regions differ in temperature, is the hottest region’s. And the board’s residue is the hottest region’s residue. Nothing about the colder regions’ residues is in any of the three board-level numbers, and all of it is still in the board.

So the summary that survives addition is the list: every region’s three numbers, kept separately. The board-level residue is recoverable from the list — take the hottest region’s — and the list is not recoverable from the board-level residue. That is the same shape as the relation between a thermograph and a value, one level down. It is also why the colder residues come back into play: when the hottest fight is taken and the region settles, the next-hottest region becomes the hottest, and the board’s residue changes to that region’s. In an endgame played hottest first, the board’s residue is not one number but a sequence, read off the list one region at a time as the fights are resolved.

The rule for three regions follows from the same argument — cooling at the hottest region’s temperature freezes both colder ones into their means — and the sample checks it directly: random triples of hot day-three values whose three temperatures are all different keep the hottest region’s residue alone, as the last line of the sample’s figure records.

The two facts, and the convention under them #

Every residue here is computed the same way: put the value in canonical form, compute its thermograph up to a tax of sixteen, cool it by its own temperature by the recursion that defines cooling, and subtract its mean. A value is hot when it is not a number and its temperature is positive. The day-two and day-three pools are the canonical values of all forms with options from the previous day, and the day-three sample is four thousand pairs drawn with a linear congruential generator from a fixed seed. The rule’s two ingredients — that cooling adds, and that cooling past the temperature leaves the mean — are properties of the cooling operator as defined, and in particular of its convention that a position cooled by more than its temperature is its mean rather than its mean plus something. That convention is the standard one, and it is precisely what removes the colder residue. Under normal play, which is the only convention the thermograph is defined for.

What the census cannot show #

Pairs, and a sample of triples. The census counts pairs exhaustively and triples only as a sample; a board of four or more regions is covered by the argument and not by a count. The equal-temperature cases are described, not explained. Why 2,237 of 3,142 equal pairs cancel into something colder, and what the residue of the colder fight is in terms of the two parts, is not worked out here; the counts say only that it is usually neither part’s. And the pools are small values. Day three reaches temperatures of a few units and residues among a handful of infinitesimals; a residue that was something stranger — a tiny, a far star — would be dropped by the same argument, but none occurs in the pools.

Still open: what the colder fights leave when they cancel #

The rule for different temperatures is settled, by argument and by count. What remains is the equal case’s commonest outcome: two fights of one temperature whose sum is colder. Its residue belongs to the fight that survives the cancellation, and the next thing to measure is whether that fight — and so its residue — is a function of the two parts’ thermographs, or depends on more than a thermograph shows. The residues as a sequence found that copies of one position produce residues in patterns of period one, two or four; equal-temperature pairs of different positions are the general case of which copies are the special one, and the question is whether the patterns survive when the two fights are not the same fight.