Each size is built from the last

A loopy region — a small game whose move graph has a cycle, so that play can go on for ever, the kind loopy games introduces — is written with two names. Give every infinite play to Left and the region behaves like one game, its onside; give every infinite play to Right and it behaves like another, its offside. A loop is written with two names introduced the notation, and the essays since have been counting how many names it needs: ten for every region of two positions, forty-eight for every region of three.

Four positions, sampled placed a third point on that curve. Four positions are over four billion move graphs, too many to count, so it drew three samples of three thousand regions at three densities and identified both sides of each. The forty-eight names cover between 95.9 and 99.5 per cent of them, the thirteen names invented for three positions come back at four almost all of them, and the samples between them meet sides that nothing earlier reproduces. It ended on the test that had failed one size down: are the new four-position sides sums of two earlier names, or of an earlier name and a small game?

At three positions the answer was no, without exception. At four it is yes for nearly half of them.

The test, one size up #

The names are not built out of the old ones asked whether the sides three-position regions needed could be written with what two positions supplied. It built two extensions of the vocabulary and tried both. Every pair of the six stoppers — on, off, over, under, upon and its negative — with a small game added, and every two-position region that is a stopper, with the same small games added: dozens of new names each. Neither extension wrote a single one of the regions that needed a name. Thirteen names had to be invented, and the conclusion was that at three positions the notation grows by invention, not by composition.

The four-position version asks the same question with more parts. The parts are the twenty-two day-two values, the six stoppers, and one region for each of the thirteen three-position names — a region whose side, read with the right convention, is that name. Every sum of two parts is formed, 861 sums in all, and each sum’s side is read exactly as a region’s is: add it to each of the twenty-two day-two values, solve each sum by retrograde analysis with every infinite play given to Left and then to Right, and take the column of winners as its key. A new four-position side is written as a sum when some pair’s key matches it.

The sides being tested are every side the three samples met that neither the two-position vocabulary nor the thirteen three-position names reproduce: fifty distinct sides, turning up 209 times across the nine thousand regions.

Twenty-three of fifty #

Twenty-three of the fifty new sides are sums of two parts, and they are not the rare ones: between them they account for 152 of the 209 times a new side turned up. The other twenty-seven are written by no sum of two parts and need names of their own.

The split by occurrences says more than the split by sides. Seventy-three per cent of the times a region in these samples needed a new side, the side it needed was a sum of two earlier games; the sides that are not sums are, as a group, the rarer ones — fifty-seven occurrences spread over twenty-seven sides, against 152 over twenty-three. A reader drawing four-position regions at random would mostly meet compositions and only occasionally an invention. That is the opposite of what a list of new keys suggests when every key counts once.

That is a different answer from the three-position one in kind, not just in degree, and the difference is where the parts come from.

At three positions the sums were built from two-position material, and two-position material turned out to be the wrong raw stuff: whatever a three-position region does that two positions cannot, a sum of two-position things cannot do either. At four positions the sums can use the thirteen names that three positions forced, and those do reach. The inventions of one size become the parts of the next. That is a statement about how the vocabulary grows that the earlier curve could not make — it had only counts of names, and a count cannot say whether the names are related.

What the sums are made of #

The twenty-three are not one kind of sum, and sorting them by their simplest decomposition separates something important from something trivial.

Eleven are a three-position name with a day-two value beside it — the commonest shape, and exactly the shape the three-position test looked for with stoppers in the name’s place and did not find. Two are a three-position name and a stopper, and two are sums of two three-position names. So fifteen of the new sides are built from the three-position vocabulary, and a notation for four positions that already has the thirteen needs no new name for them: three-position name 4 plus −1 is a name.

Eight are sums of two day-two values, and they are the trivial part.

A sum of two day-two values is a finite game, and every one of these eight is a finite game born on day three or later — $\{−1 \mid −2\}$, $\{2 \mid 1\}$, $1/2∗$, $↑∗3$ and the rest. The vocabulary the samples are identified against names the finite games born by day two and no others, because that was the range the two- and three-position censuses needed. A four-position region can reach further, and when its side is simply an older finite game it reads as new. These eight need no invention at all, only a longer list of finite games, and they are a gap in the reference list rather than in the notation.

What a composed name is for #

A composed name is not merely a shorter label; it carries the reason for the name. The two sides of a region are ordinary games, and ordinary games add. If a region’s onside is identified with three-position name 4 plus −1, then in any sum where the infinite plays go to Left, the region can be replaced by the three-position region that carries name 4 together with the integer −1 — and everything already known about name 4, its behaviour beside each small game, transfers with a shift of one point. That is the use the notation was built for, as the notation was the argument puts it: a loopy region is to be handled by handling two finite things, and a composed name hands over one of them already decomposed.

An invented name does not. The twenty-seven sides with no decomposition are known only by their columns of winners against twenty-two test games; nothing about them transfers from any smaller region, and reasoning about a sum that contains one means solving that sum. So the split between twenty-three and twenty-seven is also a split between sides the theory can reason about from smaller cases and sides it can only look up.

At every density #

A sample’s answer might depend on how it was drawn, and the earlier essay found that the share of regions needing new names depends strongly on density. The sums were therefore checked sample by sample.

The densest sample, with each possible move present at a chance of one half, meets twelve new sides and four of them are sums. The sparser samples meet thirty-one and thirty-five new sides, and seventeen and eighteen of those are sums. The share written as sums does not fall as the regions get sparser and stranger; if anything it rises from the densest sample to the others. The three samples share many of their sides, which is why their counts add to more than fifty.

The sides that need names #

What is left after the sums are taken out is the real measure of what four positions add.

Twenty-seven sides, and twelve of them turned up exactly once in nine thousand regions. Six turned up twice; nine three times or more, the commonest seven times. That is the shape the earlier essay’s growth curve had for all the new sides together — a population the samples have not exhausted, in which each further sample would meet some sides it had not met before. So twenty-seven is a floor on the names four positions need beyond composition, not a count of them.

It also changes the third point of the vocabulary curve. Four positions, sampled read its floor from the sparsest sample: forty-eight names plus the thirty-five new sides that sample met, a floor of eighty-three. With sums of two allowed as names, eighteen of those thirty-five are compositions or finite games, and the floor becomes forty-eight plus seventeen — sixty-five. Across all three samples it is forty-eight plus twenty-seven, seventy-five. The curve is still rising from ten through forty-eight, and it rises by less than the list of new keys suggested.

It also puts a number on the notation’s cost at four positions that the earlier curve overstated. The curve counted every side not in the forty-eight-name list as a new name. With sums allowed, eight of those are finite games the list simply did not include and fifteen are compositions of names already in hand; only twenty-seven — about half — are inventions.

Why composition should begin at four #

There is a reason to expect exactly this pattern, and it makes the three-position result look less like a law and more like a boundary.

A three-position region’s side is new because the region does something two positions cannot: a cycle through three points, or a position that can reach two different loops. A sum of two two-position things has no such structure in either part, so it cannot produce it. A four-position region has more room, and one way to use it is to be, in effect, a three-position region with a fourth position attached — a position that offers a move into the three-position part and something small besides. The side of such a region is naturally the three-position part’s side plus a small game, and that is exactly the commonest decomposition found: a three-position name with a day-two value beside it.

The pattern also has a precedent one size further down, which the counts alone concealed. The ten names that write every two-position region are not ten inventions. Each is either a day-two value — 0, 1, −1 and ∗ — or a stopper, alone or with a day-two value added: on, off, over, over plus −1/2, upon plus ∗ and its negative. So the two-position vocabulary is itself built by composition, from a few loops and a list of small games. A stopper and how to find one is where those loops come from. Read that way the three sizes run: composition at two positions, from the stoppers; invention at three, where thirteen sides are sums of nothing older; composition again at four, now with the thirteen as parts, together with a new residue of invention. Three positions were not the start of a notation that only accumulates. They were the size at which the parts ran out, and four positions is where the new parts start to be used.

That is an explanation after the fact, and the decompositions found here are matches of a key, not proofs that a region is a sum. A four-position region whose side equals three-position name 4 plus −1 on every test game is a region whose side is identified with that sum against twenty-two test games, which is the same standard every name in this vocabulary has been held to — and, as four thousand nine hundred regions with no name found at three positions, a standard that can call two different sides the same when the test set is small.

How the sums were formed and read #

Every part is a move graph. A day-two value is its game tree as a graph; a stopper is its defining loop; a three-position name is represented by one three-position region that carries it on the right side. The sum of two graphs is their product graph with moves in either component. Each sum is added to each of the twenty-two day-two values, the result solved by retrograde analysis to find who wins moving first on each side, and the column read twice, once with every draw given to Left and once to Right, as the census reads a region. A sum explains a new side when either reading matches it. Each explained side is credited to its simplest decomposition, with finite values ranked before stoppers and stoppers before three-position names, so that the eight finite games are not counted as uses of the three-position vocabulary. The three samples are the ones four positions, sampled drew, with the same seeds. The test itself is cheap: 861 sums, each read against twenty-two games, take a few seconds, far less than drawing and identifying the nine thousand regions the new sides came from, so a test of triples — fourteen times as many sums — is well within reach.

What the sample cannot show #

Only sums of two parts are tried. A side that is a sum of three — a three-position name, a stopper and a value — is counted among the twenty-seven, and some of them may be exactly that. The twenty-seven are an upper bound on what composition from these parts cannot reach, as well as a floor on what four positions need.

A match is a match against twenty-two games. A sum explains a side when their columns of winners agree against the twenty-two day-two values, which is the standard every name here is held to and not a proof that the two sides are equal games. The censuses measured what shrinking the test set from day three to day two gets wrong at two positions, and a larger test set could split some of the twenty-three matches into a sum and a side that merely agrees with it on these games.

The parts are a choice. One region represents each three-position name, and two regions with the same side on one convention can differ on the other; the parts could be chosen differently and write a slightly different set.

And the sample is a sample. Fifty new sides from nine thousand regions, drawn uniformly over move graphs rather than over regions up to relabelling, is a view of four positions, not a census of it. The names are not built out of the old ones could say none because it counted every three-position region; this essay can say twenty-three of the fifty it met.

Still open: whether three parts close the gap #

The twenty-seven sides no pair writes are the next object, and the measurement is the same test with one more part: every sum of three games from the same pool, 12,341 triples, read the same way. If most of the twenty-seven fall to triples, the four-position vocabulary is generated by the three-position one with a bounded amount of composition, and a notation for regions of any size might be a finite list of inventions per size plus the rule for adding them. If they do not, then four positions invent as three did, and composition is a partial economy — it explains the commonest new sides and leaves a residue that grows with the regions. The notation was the argument is the reason the answer matters: a notation that composes is a theory, and one that only accumulates names is a table.