Composition at every size

A loopy region — a small game whose move graph has a cycle in it, 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 that notation, and the essays since have been measuring what it costs: how many names the regions of each size need, and whether the names are related.

Each size is built from the last took the fifty sides that four-position regions needed and that nothing earlier named, and asked whether each was a sum of two earlier games: two of the twenty-two values born by day two, the six stoppers, and the thirteen names three positions had forced. Twenty-three were, and twenty-seven were not. It closed on the obvious next measurement — one more part — and stated what hung on it. If most of the twenty-seven fell to sums of three, a notation for regions of any size might be a finite list of inventions per size plus the rule for adding them; if they did not, four positions invented as three had, and composition was only a partial economy.

Most of them fall. But the more consequential result is what happens when the same test is run one size down, because it changes the story the whole sequence of counts has been telling.

Sixteen of twenty-seven #

The parts are the forty-one of the earlier essay: twenty-two day-two values, six stoppers — on, off, over, under, upon and its negative — and one region carrying each of the thirteen three-position names. Every sum of three of them, repetition allowed, is 12,341 sums. Each is read exactly as a region’s side is read: it is added to each of the twenty-two day-two values, each sum is solved by retrograde analysis with every infinite play given to Left and then to Right, and the column of winners is the sum’s key. A side no pair wrote is written by a sum of three when some triple’s key matches it.

Sixteen of the twenty-seven are written by sums of three, and eleven are written by none. Adding the twenty-three the pairs already wrote, thirty-nine of the fifty new four-position sides are sums of at most three earlier games.

The occurrences say the same thing more strongly. Across the nine thousand sampled regions a new side turned up 209 times. The sides a pair writes account for 152 of those; the sides only a triple writes account for 40; and the eleven that nothing writes account for seventeen — about one time in twelve that a region needed a side outside the forty-eight names. A reader drawing four-position regions at random would meet an invention rarely.

What the triples are made of #

Sorting the sixteen by the simplest triple that writes each splits them the way the pairs split.

Nine use a three-position name, and most of them use two: six are a day-two value beside a pair of three-position names, such as $\{1 \mid -1\}$ with names one and nine, or $\{1 \mid 0, \ast\}$ with name seven twice. Two more put a star and a stopper beside a three-position name, and one is two day-two values and a three-position name. So the names three positions forced keep being the parts four positions are built from, as the earlier essay found for pairs, and a four-position notation that has the thirteen needs no new name for any of these nine.

Seven are sums of three day-two values, and those are finite games.

A sum of three finite games is a finite game, and each of these seven is born after day two — $\{\{2{\ast} \mid 1{\ast}\} \mid 1, \{1 \mid 0\}\}$, $\{\{1{\downarrow}{\ast} \mid {\downarrow}{\ast}\} \mid \{ {\downarrow} \mid -1{\downarrow}\}\}$ and five more. The reference list the samples are identified against names the finite values born by day two and stops there, so a region whose side is an older finite game reads as new. With the eight sums of two day-two values the pairs found, fifteen of the fifty new sides are finite games the list was too short to name. They are regions with a cycle in their move graph that nevertheless behave, against every test game and under that convention, exactly like a finite game from a later day. The loop is there and does no work.

The same test, one size down #

The earlier essay’s argument rested on a contrast. At three positions, it said, sums of older things wrote none of the sides that needed names, so the notation grew there by invention; at four positions sums wrote nearly half, because the three-position names were now available as parts. The contrast was drawn from two different tests. The three-position test — in the names are not built out of the old ones — tried pairs of stoppers with a small game added, and every two-position stopper with a small game added. It never tried two finite values together, which is the first thing the four-position test tried and which wrote eight of the four-position sides.

So the fair comparison is to run the four-position test’s shape at three positions: the thirteen names three positions forced, against every sum of two and of three parts drawn from what existed before three positions — the twenty-two day-two values and the six stoppers.

Two of the thirteen are sums of two day-two values: $-1 + \{1 \mid -1\}$, which is the finite game $\{0 \mid -2\}$, and $1 + \{1 \mid -1\}$, which is $\{2 \mid 0\}$. Both are born on day three. They are the same kind of side as the fifteen at four positions: a finite game the list stopped too early to name, counted among the inventions because nothing looked for it there.

Six more are sums of three. Two are finite — $\{0 \mid -1\} + \{1 \mid 0, \ast\} + \ast 2$ and its mirror image, finite games born later still. The other four are loopy, and they are sums of two small infinitesimals and a stopper: ↑ + ↑ + (−upon), ↑ + ↑∗ + (−upon), and their negatives with upon. Upon is the stopper $\{upon \mid \ast\}$, which Left can keep re-entering. A three-position region that behaves like upon shifted by two small infinitesimals is precisely the kind of thing the earlier test could have found, had it put two values beside a stopper rather than one.

On the four-position test’s own terms, then, eight of the thirteen three-position names are compositions and five are inventions.

Their birthdays say how near the old list they were. $\{0 \mid -2\}$ and $\{2 \mid 0\}$ are born on day three. The four loopy ones are upon or its negative beside ⇑, ⇑∗, ⇓ or ⇓∗ — and each of those four infinitesimals is itself a single finite game born on day three, so each of the four names is a stopper with one day-three value beside it. That is exactly the shape the old vocabulary already used, a stopper with a finite value added, with the list of values extended by one day. Only the two finite games that need three parts are further out: both are born on day six. Six of the eight compositions are one day beyond the list the census was identified against, and a reference list running to day three would have named them without any sums at all.

What a longer list would have done #

The finite games deserve to be separated from the rest, because they are not really about loops.

A side that is a finite game is a region whose cycle never pays: under the convention that settles infinite play, the region behaves against every test game as a plain position would, and the plain position happens to be born later than the reference list reaches. Across the two sizes there are nineteen of them — the two day-three games and two day-six games among the thirteen three-position names, and the fifteen at four positions, the eight sums of two day-two values all born on day three and the seven sums of three born later. None of them says anything about how loopy regions compose. Each says only that the census was holding a list of finite games too short for the regions it was reading, and the length of that list was set by what two-position regions needed.

That changes how the invention counts should be read. The thirteen names three positions forced were counted as the price of a third position. Two of them were the price of a list that stopped at day two, and four more were a stopper with a day-three value beside it. The measurement that looked like a boundary in the regions was partly a boundary in the reference list, and the two are easy to confuse, because both make a region’s side look new.

What survives the separation is the loopy composition, and it is the part that matters for the notation. At three positions it is four sides — upon shifted by a small infinitesimal. At four positions it is twenty-four of the thirty-nine sums — fifteen pairs and nine triples — and every one of them uses a three-position name. Those are loops genuinely being built out of loops, and the stoppers they are built on are the ones a stopper and how to find one found at two positions.

The residue of invention #

That changes what the sequence of counts says.

Each size is built from the last read the three sizes as composition at two positions, invention at three, and composition again at four, now using three’s inventions as parts. The first and last of those stand. The middle one does not. Three positions compose too: eight of its thirteen names are sums of two or three older games, in the same proportions as four positions, where thirty-nine of fifty are. What each size genuinely invents is a residue: five sides at three positions, which is a census, and eleven at four, which is what three samples of three thousand regions met.

The floor the samples put under the four-position vocabulary falls with it. Four positions, sampled read its floor from the sparsest sample: the forty-eight names, plus the thirty-five new sides that sample met, eighty-three. Allowing sums of two brought that to sixty-five. Allowing sums of three brings it to fifty-three — forty-eight plus the five sides of the sparsest sample that no sum writes. The curve of names rises from ten through forty-eight to at least fifty-three, and almost all of the rise past forty-eight was composition that nobody had checked for.

The picture the earlier essays built — that three positions were a boundary, the size at which the parts ran out — was a product of the test rather than of the regions. A test that looks only for pairs of stoppers finds no stoppers to pair, and the finite values that were sitting in its list of inventions were never offered to it as parts.

At every density #

A sample’s answer can depend on how it was drawn, and the samples were drawn at three densities: every possible move present with chance one half, about a third, and a quarter.

In every sample the triples take most of what the pairs leave: five of eight in the densest, ten of fourteen and twelve of seventeen in the sparser two. The remainder is three, four and five sides. The samples share many of their sides, so the three remainders overlap and together make the eleven. No density produces a large residue, and the sparser samples, which meet the most new sides, compose at least as well as the densest.

The inventions are the rare sides #

The last reading is by frequency, because a notation’s cost is paid per region, not per name.

The sides pairs write include every common one — met up to thirty-three times each. The sides only triples write are rarer, the most frequent seven times. And the eleven that no sum writes are rarest of all: eight were met exactly once, one twice, one three times and one four times. So the residue is not only small in number but small in use. That is also a warning, taken up below: the rarest sides are the ones a sample is least sure to have seen, and the eleven are exactly the population a larger sample would add to.

How the sums were formed #

Every part is a move graph: a day-two value as its game tree, a stopper as its defining loop, a three-position name as one three-position region carrying it. A sum of graphs is their product, with moves in any one component. A sum of three is formed as a sum of two and then a third, 12,341 of them at four positions and 4,060 at three, and each is added to each day-two value and solved by retrograde analysis twice, once with every draw given to Left and once to Right. A sum writes a side when either reading’s column of winners matches the side’s. Each side is credited to its simplest writer, values before stoppers before three-position names, so a finite game is never reported as a use of the loopy vocabulary. The three-position run uses no three-position names as parts, because it is asking what existed before them; the four-position run uses them, as the earlier essay did.

What a match against twenty-two games is #

Every “written by” above is an identification: a sum and a side that agree on who wins against each of the twenty-two day-two values, under a given convention. That is the standard every name in this vocabulary has been held to, and four thousand nine hundred regions with no name showed how it can mislead: at three positions a test set of day-two values called some different sides the same. A larger test set could split one of the matches here into a sum and a side that merely agrees with it. The counts are of identifications against day two, and so, symmetrically, are the counts of inventions they replace.

The convention being named matters in the same way. A region’s two sides are read with draws given to Left and to Right, and a sum writes a side if it matches under either reading. A side matched only under one convention is written as a sum for that convention’s purposes, and the other side of the same region may still need an invented name.

What the samples and the sums cannot show #

Four positions are sampled. The eleven are what nine thousand regions met, and eight of them were met once. A census of four positions would meet more sides, and the new ones would come disproportionately from the rare tail where the residue lives. Eleven is a floor on four-position inventions, not a count.

Three parts is a bound. A side written by no sum of three might be a sum of four. At three positions the parts are twenty-eight and sums of four would be cheap; at four positions sums of four of the forty-one parts are 135,751, eleven times the triples, and they were not run.

And the parts are a choice. One region stands for each three-position name, and the three-position names include eight compositions; a four-position side written with a three-position name is therefore, in some cases, a longer sum of older parts in disguise. The accounting here credits each side to its simplest writer among the parts given and does not unfold them.

Still open: whether four parts leave anything #

The residue is five at three positions and eleven at four, and the question that decides whether the notation is a theory is whether each residue survives one more part. At three positions the test is small — sums of four from twenty-eight parts — and it would say whether the five are genuine inventions or longer compositions. At four positions it is 135,751 sums, affordable once, and it would say the same of the eleven. The notation was the argument is why the answer matters: if every residue falls to longer sums, the notation for loopy regions is a handful of loops and the rule for adding them; if a hard core remains at every size, each size invents, and the vocabulary grows with the regions however long the sums are allowed to be.