The names are not built out of the old ones

Four thousand nine hundred and thirty-one regions of three positions have a side no name reproduces, and an earlier essay here guessed what they would turn out to be:

Whether they introduce names outside the ten — sums of two loopy names such as on + over, or stoppers with no short definition at all — is the next count this notation invites.

The guess is a construction and a construction can be built and tried.

The first attempt, which is the guess #

The vocabulary the census runs on is the 22 values born by day two, and each of the six stoppers with each of those values added — seventy entries once the combinations whose sums can be drawn are dropped.

The guess extends it by the pairs: on + over, on + upon, over + under, and every other way of adding two of the six, each with and without a small finite game on top.

That produces ten new entries. Most pairs do not survive — a sum of two stoppers is very often drawable against some test game, and a name that can be drawn against is not a side.

And the ten cover none of the 4,931.

The second attempt, which is larger #

If pairs of the named stoppers are not enough, the obvious next move is to stop being selective. The two-position census identifies 79 of its 256 regions as stoppers — regions with no alternating run that could go on for ever — and any one of them could be a name.

So the second extension is every one of those 79, with the same small addends: thirty-six surviving entries, several times the first attempt.

It covers none of the 4,931 either.

Why so few candidates survive #

The attrition is the most informative number in either extension and it is easy to read past.

The first extension offers 126 candidates and keeps ten. The second offers 474 and keeps 36. In both cases more than nine in ten are thrown away, and always for the same reason: the candidate can be drawn against some test game.

A side is never drawn. The convention has already awarded every infinite play, so a region’s onside and offside are games in which nothing goes on for ever without a winner — which is what a draw being outside the value theory forces a notation to arrange for before it can name anything. A candidate name that can be drawn is therefore not a side of anything, whatever else it is.

Sums of two stoppers are exactly the games that most often can. Two loops running at once give each player somewhere to keep moving, and a sum in which both can keep moving against a third game is a sum that never settles. So the guess names a family whose members are mostly disqualified before they are compared with anything — which is a second and independent reason it could not have worked.

What is actually needed #

The count that settles it does not depend on either attempt. Take the 4,931 leftovers, and ask how many distinct sides they have between them — how many names would have to be invented if somebody simply named each one.

Thirteen.

Four thousand nine hundred and thirty-one regions, 3,990 of them missing a name on one side only, and thirteen names would cover all of it. Together with the 35 the old vocabulary supplies, that is 48 names for every side of every region of three positions, against ten at two.

So the vocabulary quadruples and stays small. The regions grow by a factor of 433 and the names by a factor of 4.8, and the leftovers — the part that looked like an open-ended problem — is thirteen objects.

Why the guess was wrong #

The guess is not wrong by a little and the reason is structural.

A sum of two loopy names is a game with two independent loops in it, and a region of three positions has one token moving on one graph. Two independent loops need two places for the token to be at once, which a region does not have. So the family the guess names is a family of sums rather than of regions, and the regions were never going to be in it.

The second half of the guess — stoppers with no short definition at all — is nearer and is still not what happened. The thirteen are not exotic; they are the sides of small three-position regions, and the smallest of them is {0 | off} read with the infinite plays given to Left. What makes them new is only that a two-position region cannot be that shape.

The missing names are three-position games, and the reason the old vocabulary misses them is that it was built out of two-position ones. That is a much duller answer than the guess and it is the one the census gives.

The thirty-five that were already there #

Before the thirteen, the 35 are worth a paragraph, because they are a finding the census makes almost silently.

The two-position regions use ten names. The three-position regions that the same vocabulary can read use thirty-five — and twenty-five of those are entries the vocabulary already held and two positions never reached.

They are the ordinary things — the values the special symbols name and a few that need braces: a half, two, minus two, up, down, star-two, one-star, and the stoppers carrying those values — over plus a half, over plus minus two, upon plus down, minus upon plus up. Nothing was invented for them. They were sitting in the vocabulary because the vocabulary was built as every stopper with every day-two value added, and a region of two positions is too small to produce most of them.

So the growth from ten to forty-eight has two quite different halves. Twenty-five of it is the old vocabulary being used properly for the first time; thirteen of it is the old vocabulary running out. Only the second half is news, and a count that reported the names went from ten to forty-eight without the split would be reporting the wrong thing.

What a vocabulary of forty-eight means #

The number is worth reading against the alternative it was compared with.

A notation whose vocabulary grew with the objects would be a list. If naming every side of every three-position region took thousands of names, the two-name form would be a way of saying this region has two readings and nothing more — the readings would have to be written out as graphs, and the notation would carry no information beyond the fact of there being two.

Forty-eight is not that. It is a table somebody could print, and it names 110,934 regions — against the 1,474 values born by day three, every one of which the brace form writes exactly and at length. So the two-name notation is doing real work at three positions, and it is doing it with a vocabulary that has to be extended rather than replaced.

What is not established is whether it keeps behaving. Four positions is 2³² graphs, which this census cannot take, and the growth from ten to forty-eight is two points on a curve.

What the notation is for, restated #

It is worth putting the two-name form beside the brace form once more, because this essay changes the comparison.

The brace notation names every game exactly and gets long: twenty-two characters for the middle value born on day three, with the abbreviations everybody writes covering one game in twenty-three. Its failure mode is length, and a long expression is still an expression.

The two-name form does not get long. A side is one name however large the region, and a region of three positions writes as two names exactly as a region of two does. Its failure mode is vocabulary: the names run out, and when they do there is nothing to write at all.

This essay prices that failure and the price is thirteen. So the two-name form’s weakness is real and it is bounded — at this size — which makes it a better notation than the brace form in exactly the respect this sequence has been measuring, and worse in the respect nobody had measured until the census above.

A notation that is short and incomplete and a notation that is complete and long are two different bargains, and the two-name form is the first of the two to turn up here.

The one thing both attempts did establish #

A failed extension is not a wasted one, and these two say something the count alone does not.

Both extensions are closed under the old vocabulary’s operations: the first adds sums of stoppers, the second adds every two-position stopper, and both then add every small finite game on top. Between them they exhaust what can be built from what was there.

Covering nothing, from a family that large, is a stronger statement than the guess was wrong. It says the leftover sides are not reachable from the two-position vocabulary by any of the constructions that vocabulary was built with — not a sum, not a stopper of the old size, not either with a value added.

That is the shape of a genuine extension rather than an oversight. The two-position census did not miss thirteen names it could have found; the objects were not there to be found.

What a guess is worth when it fails #

An earlier essay here made a prediction and this essay refutes it, which is worth one paragraph on its own terms because the prediction was a good one.

It was specific: sums of two loopy names such as on + over. That is a family somebody can build, and building it is the whole of the test. A vaguer guess — the vocabulary will need extending — would have been unfalsifiable and would have earned nothing.

It was also the right guess to make from where it was made, since the notation exists so that positions can be added and a guess about names is naturally a guess about sums. At two positions, every name in use is either a finite value or a stopper with a finite value added, so the pattern a name is a loopy thing plus a finite thing is the pattern the data showed. Extending it to a loopy thing plus a loopy thing is the next term in the obvious sequence.

What it missed is that a region is not a sum. A sum of two games is two objects being played at once and a region is one token on one graph, so the family the guess names cannot arise as a region at all — and the census is what says so rather than any argument available beforehand.

The value of the guess is that it was cheap to test and that its failure names the mistake. A region is not a sum is a sentence the two-position census could not have produced and this one does.

The half-named regions #

One column of the census deserves a closer look, because it says the gap is on one side rather than in the middle.

Of the 4,931 leftovers, 3,990 have a name on one side and nothing on the other — four in five. Only 941 are blank on both.

That asymmetry is exactly what the mechanism predicts. A convention that awards infinite plays to one player can delete an option for the other; it deletes nothing for the player it favours, whose looping move becomes a win. So one reading of the region is an ordinary game with all its options intact, and the other is the game with a branch removed — and it is the second that has no name.

Which is a small thing and a useful one. A region with no name is usually a region with one name, so a reader meeting the notation is not faced with two blanks; they are faced with a familiar name and a gap beside it. That is a much more workable failure than the count alone suggests, and it is the sort of detail a total count hides.

Why a candidate name has to be undrawable #

The loopy convention throughout, with a side being the game a region becomes when every infinite play is awarded to one player, and a name being a game whose column of winners against a test set matches the side’s.

Three conventions of the extensions.

A candidate name must never be drawn against a test game. A side is never drawn, by construction — the convention has already decided every infinite play — so a candidate whose sum with some test game can go on for ever is not a side and is dropped. That is why the first extension yields ten entries out of a hundred and twenty-six candidates — twenty-one pairs of stoppers, six addends apiece — and the second thirty-six out of four hundred and seventy-four.

The addends are six small games: nothing, star, one, minus one, up and down. A wider set of addends would add entries and cannot change the outcome, since a leftover side unmatched by a stopper is unmatched by that stopper plus anything the test set can see.

And the test set is the 22 values born by day two, the same as the census it extends. A name here is a name against those 22.

Thirteen is a count and not a proof #

Thirteen is a count of distinct sides, not a proof that thirteen names suffice. Each of the thirteen is a side and each would need writing down; whether any shorter description of them exists is a question about notation that a census cannot answer.

Both extensions are constructions, and neither is exhaustive over all games. What is established is that these two families miss everything, not that no family built from the old vocabulary could hit it.

The regions are single positions, not boards. A board written as a sum of its parts is a different measurement and the one the earlier essay makes; nothing here adds a region to anything except a test game.

And the day-two test set is doing the identifying. A side and a name matching against 22 games might part company against 1,474, in which case the thirteen would be an undercount and some of the 35 would split.

Where the thirteen come from #

The thirteen leftovers have a description once the region above is followed, and it is a shape rather than a list.

Every one of them is the onside or offside of a region in which the two conventions do not merely change who wins but change what the game is. In the smallest case, Right’s move into a loop is a win under one convention and a loss under the other, so the option is present in one reading and effectively absent in the other, and what is left after removing it is a game nothing two positions can build.

That is the mechanism and it needs three positions: one to be the root, one to be the loop, and one to be the option that survives when the loop is thrown away. Two positions can hold a root and a loop and nothing else.

So the thirteen are not a ragged collection. They are the sides of the smallest regions that can have an option deleted by a convention, which is a family somebody could describe properly — and describing it properly is a piece of work this census does not do.

Still open: the fourth position #

The obvious next census is four positions and it is out of reach, so the question is what could be measured instead.

The curve has two points: ten names at two positions, forty-eight at three. A third point would say whether the vocabulary is growing like the square of the positions, like some polynomial, or like the regions themselves — and only the last of those would make the two-name notation useless at scale.

What is affordable is a sample: draw a few thousand four-position regions at random, identify their sides against the day-two values and the forty-eight-name vocabulary, and count how many need something new. That would not be a census and it would place the third point, which is the whole of what the curve needs. It would also say whether the thirteen new names of three positions get used at four, or whether each size brings its own — and those are very different notations wearing the same description.