The thirty that cancel themselves

The values that are their own negatives found thirty of them among the 1,474 born by day three — the elements of order two in a group whose other elements have infinite order — and closed by naming the instrument it had not pointed at them:

what the 30 look like under temperature — several are switches, so they are hot, and a hot game that is its own negative has a mean value of exactly zero.

Both halves are true. The first is a measurement and the second is a theorem, and the interesting thing is how little the theorem buys.

Why the mean has to be nought #

The theorem is one line and it is worth having the line rather than the assertion.

The mean value of −G is minus the mean value of G. That is not an extra fact about means; it follows from the construction, since negating a position swaps the two players and therefore reflects the thermograph in the vertical axis, which sends the meeting point of the two walls to its negative.

So if G = −G then the mean of G is its own negative, and a number that is its own negative is nought.

The same argument gives the stops. The left stop of −G is minus the right stop of G, so a self-negative value has stops that are exact opposites — and all thirty do. Both are asserted in the census rather than reported, because a value that broke either would mean the evaluator and the thermograph had stopped describing the same object.

And how little that picks out #

Four hundred and ninety-six values of day three have a mean of exactly nought. Thirty of them are their own negatives.

So mean zero is necessary and is satisfied by seventeen values for every one it identifies. A condition with that ratio is not a weak test; it is not a test at all, and the distinction matters because a reader who checks it and finds it holding has learned nothing they did not already know about a value picked at random from the middle of the day. Nought is by a wide margin the commonest mean on the day, for the same reason it is the commonest value: the construction is symmetric, so every value with a positive mean has a partner with the negative of it, and the ones in the middle pile up. The stops are no better: symmetric stops is a weaker condition still, and every value with a mean of nought and a symmetric interval around it passes it.

The temperature scale can see the subgroup and cannot pick it out, which is the useful form of the finding. An instrument that returns a number for a position cannot detect an equation between a position and its reflection, because the number is invariant under things the equation is not.

That is worth stating because the reverse mistake is easy. A reader told that self-negative values have mean nought will reach for the mean as a test, and the test passes on 496 values, of which 466 are not self-negative at all.

Half of them are fights #

The count by temperature is the part the rung below guessed at and did not compute: one number, fifteen at temperature nought, five at a half, eight at one, and one at two.

Fourteen of the thirty are hot. That is the finding, and it undoes an association a reader is very likely to have made.

The best-known self-negative values are the nimbers — star, star two, star three — and every nimber is an infinitesimal, sitting at temperature nought and worth nothing on any count. It is natural to slide from the two-torsion contains the nimbers to the two-torsion is a cold place, and it is wrong by half.

{1 | −1} is its own negative and has a temperature of one: a fight over two points, with nothing underneath it, in which whoever moves takes the lot. {2 | −2} is its own negative and has a temperature of two, and it is the hottest value day three produces at all — the record-holder of its day, and it is in the subgroup.

Worth nothing and worth moving in #

That a position can have a mean of nought and a temperature of two is not a paradox, and it is the clearest small illustration of what the two numbers are for.

The mean is what the position contributes once the fighting is over: how the count stands when neither player wants to move there any more. The temperature is how much is at stake in getting the next move there. A switch between a number and its negative has both — the fight is worth four points and whoever wins it wins two and loses two relative to the middle, so on average nothing changes hands.

A mean of nought is a statement about the settled position and says nothing about urgency. On a board with {2 | −2} on it and nothing else hot, the first player takes two points and the second loses two, and a reader who priced the region at its mean has mispriced it by four.

This is what is at stake restated in the sharpest available case. The pair of numbers is a measurement rather than a label, and here the two halves of the measurement are as far apart as they go: nought and the day’s maximum, on the same position.

The thirty split into three recognisable kinds and there is nothing left over.

• The symmetric switches. {a | −a} for a a value of the previous day: {1 | −1}, {2 | −2}, {1/2 | −1/2} and their relatives, together with forms carrying extra options that are themselves in negative pairs. These are the hot fourteen.
• The nimbers and their kin. Nought, star, star two, star three — and four more all-small values that are not nimbers but whose option sets are closed under negation. Seven of the thirty are all-small.
• The number nought, alone, since no other number is its own negative.

The structural test the rung below drew — the Left options are, as a set, the negatives of the Right options — is what all three kinds have in common, and it is a property of the written form rather than of the value. It can be checked by looking, which is what makes it worth having beside an equality test that needs a search.

And it is not equivalent to the equality. A form whose options are conjugate is its own negative; a form whose options are not can still be worth a value that is, because a different form of the same value may have conjugate options. That gap is the reason both tests are on the census rather than one.

The five at a half, which are the odd ones #

The temperature counts have a shape and one bar of it is strange.

Fifteen at nought and eight at one are what the two obvious families give: the all-small values and the nimbers sit at nought, and the symmetric switches between the integers of the previous day sit at one. Five values sit at a half, and a half is not a gap between two day-two integers.

They are the switches between 1/2 and −1/2, with and without extra option pairs — {1/2 | −1/2} itself and four forms that add conjugate options to it without changing the walls. So the bar at a half is one value’s worth of position with four decorations, which is what happens whenever the two furthest-apart options are a pair of fractions rather than a pair of integers.

That is a small observation with a use: it says the distribution of temperatures across the subgroup is a distribution over the gaps between day-two numbers, halved, which is exactly the rule the whole day’s temperatures follow. The subgroup does not have a temperature distribution of its own; it inherits the day’s, restricted to the symmetric pairs.

What being in the subgroup is worth #

There is a use for the classification and it is not the one the group theory suggests.

G + G = 0 means two copies of the position cancel. So a board with two copies of a self-negative region on it can have both struck out, whatever else is on the board, and cancellation is exactly what licenses that. On a board with four copies of {2 | −2}, a region worth two points a move each, the whole eight points come off the analysis at once.

That is a real saving, and it is the one place the arithmetic is worth more than an inspection: recognising that two regions are the same is easy, and recognising that a region is worth a value equal to its own negative needs the evaluator. That is a real saving and it applies to the hot members as much as to the cold ones — more usefully, in fact, since a hot region is the expensive kind to evaluate and the cancellation removes two of them.

What it does not give is a strategy. Knowing that two copies cancel says the pair is a second-player win and does not say which move to make in the position that remains, which is the standing complaint against every theorem of this shape.

What a player would do with a symmetric region #

There is a piece of play advice hiding in the fourteen and it is worth extracting, because it is the only thing on this page a reader could use over a board.

A region worth {a | −a} is worth nothing on average and everything to whoever moves in it. So it is a region to move in first, and the amount by which it matters is the amount by which its temperature exceeds everything else on the board — the ordinary hottest-first reasoning, with no correction.

What is unusual is that a reader who has priced the region by counting will price it at nought and skip it. Counting territory is a mean-value estimate, and the mean is exactly the number that hides this region: a symmetric switch contributes nothing to the count and is the most urgent thing on the board.

So the class this page is about is precisely the class a counting player under-values, and the thirty are the small examples of it. A Go player would recognise the shape immediately — a symmetrical exchange worth the same to both sides — and would also recognise that the counting method they were taught says nothing about it.

That is the same failure the orthodox account is built to avoid: an account that adds means and stops has thrown away every temperature on the board, and on a board of symmetric switches it has thrown away all of the game.

What the census does not say #

Three limits.

Thirty is a count about day three. The subgroup grows with the construction and this is its intersection with one day. What the proportion does — thirty of 1,474 is one value in forty-nine — is a question about day four and is not answerable here.

Two countings of the same set are in circulation on this site and they differ by four, so it is worth saying which is which. Thirty are born by day three and twenty-six are born on it: nought arrives on day nought, star on day one, and $\ast 2$ and $\{1 \mid -1\}$ on day two. Every count on this page is the cumulative one, because the temperature scale is a question about a set of values rather than about an increment; the rungs above count the increment, because growth is what they are measuring.

Temperature is one reading. The thirty have been sorted by the height at which their walls meet, which is what the rung below asked for. Sorting them by birthday, by width, or by how many positions realise them would give three other tables, and the first of those would probably be the most informative: a self-negative value has a conjugate form, and a conjugate form has an even number of options at every level, which bounds things.

Nothing here is a characterisation. Three families and no leftovers is a description of thirty values, not a theorem about which values are their own negatives. The theorem is the structural one — a form whose options are conjugate is self-negative — and it runs one way; the families above are what that produces on a day small enough to list.

And the mean-zero count is a count of values. Four hundred and ninety-six of 1,474 have a mean of nought, which is a third of the day. Weighted by position rather than by value the proportion would be different and probably much larger, since most positions are worth numbers and the commonest number is nought.

The convention, named #

Normal play. A value is its own negative when its canonical form equals the canonical form of its negative, which on this site is one key comparison; the negative of a position is the position with the two players’ roles exchanged, drawn as the mirror.

The mean and the temperature are read off the thermograph — the height at which the two walls meet and the value they meet at — and a number is given temperature −1 rather than nought, which is why nought appears in the count as its own class and not among the fifteen at temperature nought.

Where the ladder goes next #

The negation anchor has four rungs to here: that every game has a negative, which values are their own, what may be struck out because of it, and now where the subgroup sits on the temperature scale.

The rung above answers the birthday question this page could only pose, and answers it exactly. A self-negative value costs a day finds that the earliest self-negative value of temperature $t$ is born the day after $t$ itself — which accounts for the shape of the bar chart above, for the four temperatures that carry exactly one member and the four that carry none. The guess offered here, that the first value of each temperature to appear is a self-negative one, holds at four temperatures of five and is not the form the answer takes.

Above that the counting is turned round. At least five hundred and seventy-one drops the idea of sifting a day for its self-negative values and builds them instead: a value is its own negative exactly when it has a mirror form, so the subgroup of one day is constructed from subsets of the day below rather than found in the day above. That gives a floor of 571 on day four against day three’s twenty-six, and — the part that could not have been guessed from thirty values — the share of a day that is self-negative keeps falling, one in nine on day two, one in fifty-six on day three.

The two rungs after that are about the mirror map itself, and they take the construction somewhere this page gives no sign of. What identifies two subsets finds the collapse from 1,793 subsets of day two to thirty values happening in two stages of quite different character, with almost all of the second stage landing on two values. And a mex with no impartial game in it describes those two fibres with one rule: the mirror of a set is the least nimber no element of the set reaches — a mex, arrived at inside a construction with no impartial game anywhere in it.

Two neighbours are worth the trip. What is at stake is where the mean and the temperature are defined and where the pair is shown to be a measurement rather than a label; this page is the case where the two halves are furthest apart. And what can be struck out is the cancellation these thirty license, which is the one thing being in the subgroup is actually worth.