How hot a day gets

Below zero took the temperature scale down to its floor and closed by naming something it had not explained:

The 1,474 values are spread over nine temperatures with a mode at $\tfrac12$ and a maximum at $2$, and the shape of that histogram is a fact about the construction that nobody has explained.

Two questions are folded together there, and they have very different answers. The maximum has one and it is exact. The shape has one and it is a mess with a hole in it.

Day one tops out at nought. Day two tops out at one. Day three tops out at two. A day of construction buys exactly one degree, and the value attaining the maximum is unique each time.

The three record-holders #

They are $\ast$, $\{1 \mid -1\}$ and $\{2 \mid -2\}$, and once they are set out the pattern is not mysterious.

Day nought has one value, $0$, with no temperature at all because it is a number. Day one has $\pm 1$ and $\ast$; the integers are numbers and the star is not, and the star’s two walls meet at height nought. So the hottest value of day one is the star, at temperature nought — the smallest temperature a position can have without being a number.

Day two’s hottest is the switch between the two extreme integers of day one: $\{1 \mid -1\}$, whose walls start at $1$ and $-1$ and close at height $1$. Day three’s is the switch between the two extreme integers of day two: $\{2 \mid -2\}$, closing at height $2$.

The mechanism is now visible and it has nothing to do with fights. A switch $\{a \mid b\}$ between two numbers has temperature $\tfrac{a - b}{2}$, so making the hottest possible switch means finding the two furthest-apart numbers available, and the two furthest-apart numbers on day $n$ are the integers $\pm n$. Half of $2n$ is $n$.

So the temperature record is a fact about how far the integers get, and not about anything the fights do. A day adds one to the largest integer, as the number tree shows, and half of two is one.

Uniqueness, which was not guaranteed #

The value attaining the maximum is unique on every day, and that is a separate claim from the maximum itself. It is asserted in the code rather than reported: a day whose maximum is reached twice would break the reading, and the figure refuses to draw rather than the caption glossing over it.

The reason it is unique is worth a sentence, because it is the reason the record is so brittle. To reach temperature $n$ on day $n+1$, a position must have its left wall starting at $n$ and its right wall at $-n$ — which means Left has an option worth $n$ and Right an option worth $-n$, the two extreme values of the previous day. And a position with only those two options is $\{n \mid -n\}$ exactly. Adding any other option lowers a wall somewhere and cools the position, and there is nothing else to add that does not.

The three records are not merely a sequence of numbers, and the cleanest way to see how tightly they are tied together is to charge each of them the degree a day buys.

That chain is the same fact as the one-degree-a-day rule, arriving from the operator’s side rather than from the construction’s. A tax of one takes a switch $\{a \mid b\}$ to $\{a-1 \mid b+1\}$ for as long as there is a gap left to narrow, so it walks the record-holders down their own ladder a rung at a time and stops at nought — and it stops there because $\ast$ has no gap to narrow, which is why day one’s record is the last one on the chain and day nought’s value is not a record at all.

The integers are what supply the gap. The number tree adds one to the largest integer each day at either edge while everything else fills in between, and it is the march at the edges, not the filling in, that the temperature record is made of.

The hole at one and three quarters #

Day three’s temperatures are $\tfrac14$, $\tfrac12$, $\tfrac34$, $1$, $\tfrac54$, $\tfrac32$ and $2$ — with nought below them and $-1$ for the numbers.

There is no $\tfrac74$. There is exactly one value at $2$.

That gap is not an artefact of how the figure buckets its bars: every temperature on day three is a multiple of a quarter, which is asserted before anything is drawn, so the list above is complete and the missing entry is genuinely missing. Nothing born by day three has a temperature of one and three quarters, and one value has a temperature of two.

The explanation turns out to explain the whole scale rather than the one hole in it, and it is the best thing on this page.

The positive temperatures of day three are exactly the half-gaps between the numbers born by day two. Those numbers are $-2, -1, -\tfrac12, 0, \tfrac12, 1, 2$; the distances between pairs of them are $\tfrac12, 1, \tfrac32, 2, \tfrac52, 3$ and $4$; and half of those is $\tfrac14, \tfrac12, \tfrac34, 1, \tfrac54, \tfrac32$ and $2$ — which is the list above, entry for entry, with nothing over and nothing missing.

One half of that is easy. A switch between two numbers has temperature half the distance between them, so every half-gap is realised by an actual position. The other half is not obvious at all: a day-three value need not be a switch between numbers — most of them are not — and there is no evident reason why a position with a follow-up, or a star, or five options should land on one of seven prescribed heights rather than somewhere in between. Every one of the 1,122 hot values of the day does.

That is what puts the hole at $\tfrac74$: it would need two day-two numbers $\tfrac72$ apart, and there is no such pair. The position that would have that temperature is $\{2 \mid -\tfrac32\}$ — computed, its temperature is $\tfrac74$ and its mean is $\tfrac14$ — and it is a day-four value, because $-\tfrac32$ is born on day three rather than day two.

So the gap is an accident of the order the numbers arrive in, and it will not survive into day four. What will survive is the shape it produces: the top of the scale is not smooth. The values thin out, stop, and then produce a single record-holder standing alone above a gap.

What the record costs to hold #

There is a way of reading the three record-holders that turns the pattern into something a player can use, and it is worth taking because it explains why real games look nothing like this.

$\{2 \mid -2\}$ is a position in which the two players are fighting over four points, and it has no follow-ups: whoever moves takes the whole four and the game is over. It is, in other words, the crudest possible fight — all of the stake settled in one move, nothing left afterwards.

That is what makes it the record-holder, and it is also what makes it unlike anything on a board. A Go endgame region, a Domineering corner or an Amazons pocket is hot and has follow-ups: taking the point leaves a smaller point behind, so the immediate stake is only part of what is at issue. Every such position has a wall that bends, and a bent wall is a wall that has given something back.

So the construction’s hottest values are hot in the least interesting way available, and the interesting hot positions are all strictly cooler than the record for their day. That is not a coincidence: a follow-up is an option, an option pulls a wall in, and pulling a wall in lowers the temperature.

Which gives a small ordering worth carrying. Among positions built from the same material, the ones with something left afterwards are colder than the ones with nothing, and the difference is not a subtlety — it is the whole of the gap between the day’s record and the hottest position with a follow-up in it. A reader who thinks of a hot position as a complicated one has it exactly inverted: complication is what cools a fight, and the hottest thing a day can make is the simplest fight its numbers allow.

The mode, and which side it leans #

Below the tail the distribution is dense and lopsided. Counting the hot values of day three: one hundred and sixteen sit at a quarter, 497 at a half, 288 at three quarters, 160 at one, forty-four at one and a quarter, sixteen at one and a half, and one at two.

The mode is at a half with 497 — a third of the whole day. And the two neighbours of the mode are not close to equal: 116 below it and 288 above, so the distribution leans upward out of its own peak before collapsing.

That asymmetry is the part of the shape a reader is least likely to guess. The obvious model — that the temperatures thin out symmetrically from a central value — is wrong on the low side, and the reason is that the low side has a floor. A temperature of $\tfrac14$ is the smallest positive one available, and there is nothing below it but nought, which is a completely different kind of position.

The lean is already present one day earlier, where the whole distribution fits in four bars and can be counted by hand: seven numbers, eight values at nought, six at a half and one at one. Six against one on either side of the mode is the same asymmetry with two orders of magnitude fewer values behind it, which is worth knowing before reading any explanation of it as a fact about large pools.

And the floor is not a rounding artefact of where the bars were drawn. Every temperature on day three is a multiple of a quarter, so a quarter is not merely the smallest one observed — it is the smallest one available, and the band below it is empty by construction rather than by shortage.

The 337 at nought, which are not one thing #

The largest single group is neither hot nor a number: 337 values sit at exactly temperature nought.

A value there has both walls meeting at the foot, which means both its stops are the same number and it is worth that number plus something the temperature scale cannot see. That “something” is an infinitesimal, and the class is enormous and completely unordered by anything the diagram shows — where the stops run out is the essay about exactly this class, and it finds all four possible relations between a position and its own stop inside it.

So the histogram’s largest bar is a bar about the theory’s own blind spot. Nearly a quarter of the day’s values are gathered there because the temperature scale has stopped measuring, not because they are alike, and telling them apart is the whole subject of atomic weight.

Reading the whole shape at once #

Putting the three sections together gives a description of the day that is worth stating as a whole, because no single bar of the histogram says it.

Of the 1,474 values born by day three: fifteen are numbers and have no temperature; 337 are worth a number plus something invisible to the scale; 613 have a temperature of a half or less; 288 sit at three quarters and 160 at one; and precisely sixty-one are hotter than one, of which seventeen are hotter than one and a quarter and one is hotter than one and a half.

The collection is overwhelmingly cold. Nearly a quarter of it is at nought, more than half of the hot values are at a half or below, and the hot tail is a rounding error — sixty-one values in fourteen hundred, which is four per cent. A reader who has met the subject through its thermographs and its Go endgames will have the proportions almost exactly backwards, because the interesting positions are the hot ones and the hot ones are rare.

That has a practical corollary. A solver walking a game tree spends most of its time in positions whose temperature is nought or a quarter, where the temperature theory has nothing to say and the atomic weight has everything. The theory a reader meets first is the theory that applies least often.

What this predicts about day four, and why it cannot be checked #

The record is exact and the mechanism is clean, so the prediction is easy: day four’s hottest value is $\{3 \mid -3\}$ at temperature $3$, attained uniquely.

That is a prediction this site cannot test. The recursion this site cannot run is the account of why: day three needed the 1,474 values to be built out of every antichain of the twenty-two day-two values, and day four asks the same of 1,474 values whose order contains an antichain of at least forty-five. Two to the forty-fifth option sets a side is not a computation that gets easier with patience.

The shape is much less predictable. If the mode moves outward by a quarter each day it would sit at $\tfrac34$ on day four; if it moves in proportion to the maximum it would sit at $1$. The two guesses diverge immediately and nothing here separates them.

A last caution about reading any of the bars above as a description of the day. A temperature is one number extracted from a value, and the day arranged by what its values are rather than by how hot they are is a far wider object: two values sitting in the same bar may be nothing like each other, and two a bar apart may differ by a quarter and by nothing else. Every histogram on this page is that projection, and the projection is the point of it and also its whole limitation.

What the census does not say #

Three limits, and the first is the one that matters most.

The maximum is a maximum over a day, and a day is a bound rather than a description. A position on a real board with forty pieces on it is routinely worth a value born on day two, so the record-holders above are not the hottest positions anybody meets — they are the hottest values the first three days of the construction produce. A Domineering board can have a temperature well above two while being worth a value with a small birthday, because temperature and birthday are independent quantities and the birthday bounds neither the width nor the size of what it names.

The distribution is over values, not over positions. A value that a hundred different positions are worth is counted once here. Weighting the histogram by how often each value actually arises in a game would give a different shape, and this site has the gamut census that could supply the weights and has not applied them here.

And the gap is a fact about day three. Nothing says a temperature of $\tfrac74$ is unreachable in general — only that no value born by day three has it. A day-four value almost certainly does.

The convention, named #

Normal play, and the temperature is the standard one: a tax of $t$ charged on every move by both players, with the temperature the height at which neither player wants to move any more. Every number above is the height at which two exact piecewise-linear walls meet, computed by the recursion rather than sampled.

A number is given temperature $-1$ throughout rather than nought, which is a convention with a job: it puts every number strictly below every position carrying a star, which is where a reader’s intuition wants them. Below zero is the essay about that choice and what it costs.

Where the ladder goes next #

The cold anchor has two rungs to here: what the bottom of the scale is, and how far the top of it reaches. The five above it all take the weighting this page asks for and could not do — counting positions rather than values — and each one lowers the answer again.

How hot a real position is applies the weights directly. Counted one value at a time, a tenth of the subject is hot; counted over every board this site has enumerated, all 11,397 of them, it is a twentieth. Two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all. The proportions this page calls backwards are more backwards than it says.

What a game actually produces then narrows it once more, from a catalogue to a game. Fifty-three per cent of Domineering regions of at most eight squares are hot; of the components eleven hundred random games actually produce, sixteen per cent are. The figure holds across three board sizes, so it is a property of play rather than of the board — and it means every temperature census taken over a catalogue, this page’s included, overstates the heat by about a factor of three.

The three rungs after that repair the picture rather than deflating it further. One fight makes a board a fight tallies at the board instead of at the piece and finds 32 per cent hot — twice the piece figure and not ten times it, because a Domineering board carries only 1.68 pieces and the hot ones cluster together. The obstacle was the catalogue removes the limitation that made the early game unmeasurable, and reverses the reading again: a board is hot four times in five three moves in, and cools as it breaks up. And eight squares and no hotter finds the ceiling — no Domineering position hotter than three halves, attained by a region of eight squares, holding at nine and ten where the obvious extrapolation predicted seven quarters.

Read together they say something this page could not. The construction’s temperatures climb by one a day without limit, and a real ruleset has a ceiling of its own, reached at a small region and never exceeded. The day-by-day record is about how far the integers get; a game’s record is about how much board a fight can occupy, and the two have nothing to do with each other.

Two neighbours are worth the trip. What is at stake is where the temperature is defined and where the pair of numbers is shown to be a measurement rather than a label. And how old a value is is the same day-by-day reading applied to the birthday, where the quantity being counted bounds much less than a reader expects.