At least five hundred and seventy-one

A self-negative value costs a day found the timetable of the two-torsion subgroup: the earliest value of temperature $t$ that is its own negative is born the day after $t$ itself is. It closed on the quantity the timetable says nothing about:

The rung above is the width. The timetable here is set by the temperature; what it does not touch is how many self-negative values a day supplies once it can supply any — twenty-six on day three against two on day two is a jump of thirteen-fold, and whether that is the population growing or the constraint loosening is a question a day-four count would settle and a day-three one cannot.

A day-four count is not available to anybody. What is available is the construction, and it answers the question in a form the enumeration could not have.

A self-negative value is a mirror #

$G = -G$ means $G + G = 0$, and the negative of a game is its form with the two option lists swapped and every option negated. So a canonical form is its own negative exactly when

$$\mathcal{R} = -\mathcal{L},$$

the Right options being the negatives of the Left ones as a set. That is a condition on the form, and because the canonical form is unique it is a condition on the value.

Two things follow immediately, and one of them is a small theorem the rung below did not have. Both are about the form rather than about play, which is what makes them cheap: the canonical form is unique, so a condition on it is a property of the value and can be read off without any comparison.

Every self-negative value has an even number of options. The two lists are in bijection, so their lengths agree, so the width is even. Six hundred and twelve of day three’s values have odd width and every one of them is disqualified by that alone — which is nearly half the day removed by a parity argument.

There is a second way to say the parity result that is worth having, because it is the version a reader can apply. Comparison is a search and testing $G + G = 0$ costs one; testing whether a form’s two option lists have the same length costs nothing, and it eliminates 612 of day three’s 1,452 new values before any search is run.

And the share does not fall as the form widens. Five of 167 at width two, 17 of 527 at width four, seven of 67 at width six. A reader expecting the mirror condition to bite harder on wider forms — more options to match up — gets the opposite: the widest forms in the day have the largest share.

Counting without enumerating #

The mirror condition turns the count into a construction. If $\mathcal{R} = -\mathcal{L}$ then the whole value is determined by $\mathcal{L}$, so the self-negative values born by day $n+1$ are exactly the distinct values of $\{S \mid -S\}$ as $S$ ranges over sets of values born by day $n$.

That is checked rather than asserted: building $\{S \mid -S\}$ for every subset of day two of size at most three produces exactly the thirty self-negative values born by day three, with none of the thirty missing and nothing spurious.

The gain is enormous and it is not enough. Day three has 1,474 values and its subsets are beyond counting, so the construction converts enumerate a day and test each value into enumerate the subsets of the day below — which is a smaller impossible problem.

What it does buy is a floor. The narrowest mirrors can be built exhaustively.

Every $\{a \mid -a\}$ with $a$ born by day three is a self-negative value, and 98 of them are born on day four. Adding a sample of pairs $\{a, b \mid -a, -b\}$ — 19,900 of them, drawn evenly through the day — takes the count to 571.

That number is a floor and a weak one. On day three the width-two forms supplied only 2 of the 26 self-negative values, the other 24 having width four or six, and this floor counts no form wider than four. The true day-four count is far above 571 and nothing here estimates how far.

Which of the two hypotheses #

The two hypotheses were the population is growing and the constraint is loosening, and the instrument that separates them is the share rather than the count.

Day two: 18 values born, 2 self-negative — one in nine. Day three: 1,452 born, 26 self-negative — one in fifty-six. So between those two days the subgroup grew thirteenfold and the day grew eighty-onefold, and the share fell by a factor of seven.

That comparison is the one the rung below wanted and it needed no day four at all. The population is growing, and the count of self-negatives is a slow shadow of it. A day supplies more of them because it supplies more of everything, and the mirror condition is a constraint that costs more the larger the day gets, not less.

The floor on day four is consistent with that and does not confirm it, which is worth saying plainly: 571 against 26 is a twenty-twofold rise, and the day’s own growth is far larger than twenty-twofold, but nobody knows how much larger. What the day-two-to-day-three comparison establishes is the direction, and it establishes it on complete counts rather than on floors.

What it costs to ask properly #

The cost is worth recording, because it is what made the rung below leave the question open and because it is the same shape as the other refusals on this site.

A day-four enumeration is not a large computation; it is an impossible one. Day three has 1,474 values, and day four is built from subsets of them, which is a number with four hundred digits in it. No sweep reaches that and no sweep ever will, so count day four’s self-negatives was never a piece of work that could be scheduled.

The mirror construction changes what has to be enumerated and not by enough. Restricted to one option a side, day four’s mirrors are 1,474 forms and take a second and a half; restricted to two, they are 1.1 million forms and about ten minutes — which is affordable and would raise the floor by an unknown amount. Restricted to three, they are 530 million and out of reach again.

So the honest position is a floor that can be pushed and never closed, and the value of the construction is not that it counts the day but that it turns the question into one about subsets, where an argument might do what a sweep cannot. That is the same trade the lattice ladder made when its own day-three sweep was refused on cost: the enumeration is priced, the price is stated, and the work moves to the structure.

What the count is a count of #

There is a subtlety here that the arithmetic hides and it is worth surfacing, because it is the reason the rung below’s question was hard.

The self-negative values form a group — a subgroup of the game values, in which every element is its own inverse, which makes it an elementary abelian 2-group. Its known members include all the nimbers, since $\ast n + \ast n = 0$ for every $n$, and the nimbers alone contribute one new element on each day: $\ast n$ is born on day $n$.

So part of the growth is simply the nimbers arriving one per day, and the rest is everything else. On day three the 26 break into $\ast 3$ and 25 others; on day two the 2 are $\ast 2$ and $\{1 \mid -1\}$. The interesting count is therefore the non-nimber part, and it is 1 on day two and 25 on day three, which is a twenty-fivefold jump rather than a thirteenfold one.

That is a sharper version of the same finding, and it makes the shape of the subgroup clearer: the nimbers are a thin regular spine and the rest of the subgroup is a much larger and less orderly body growing around it.

Thirty is not a power of two, and that is a finding #

The subgroup is an elementary abelian 2-group, so any subgroup of it has a power of two for its size. The counts on this page are 1, 2, 4 and 30 — and 30 is not a power of two.

Which is not an error and is not a curiosity. It says, in one arithmetic step, that the self-negative values born by a given day are not closed under addition. Two of them can be added and the answer can be born later than either.

The sweep says how often. Of the 465 unordered pairs from the thirty, only 68 have a sum still born by day three; 397 escape the day. $\ast$ and $\{2 \mid -2\}$ are the smallest example — both born by day two, and their sum $\{2\ast \mid -2\ast\}$ is not born until day four.

So the intersection of the subgroup with a day is a set and not a structure, and every count on this page is a count of that set. The first three days hid it: by day one the self-negatives are $0$ and $\ast$, by day two they are those plus $\ast 2$ and $\{1 \mid -1\}$, and each of those is closed by accident of being small. Day three is the first day large enough for the closure to fail, and it fails on six pairs in seven.

That also sharpens what the floor of 571 is a floor on. It is not a count of a group that could be pinned down by finding generators; there is no smaller set of day-four mirrors from which the rest follow by addition, because adding two of them usually leaves the day. The construction on this page — enumerate subsets, mirror them, canonicalise — is not one method among several. It is the only handle there is, which is why the rungs above go after its fibres rather than after its arithmetic.

What a mirror looks like #

It is worth seeing a few of the thirty, because the phrase its own negative sounds exotic and the objects are not.

$\ast$ and $\ast 2$ are mirrors in the simplest possible way: their option lists are equal and negating a nimber changes nothing. $\{1 \mid -1\}$ is a mirror because $-1$ is the negative of $1$ — the switch a player fights over, worth a move to whoever takes it and nothing to either otherwise.

Day three’s additions are less familiar and have the same shape at one more remove — $\{1\ast \mid -1\ast\}$, $\{2 \mid -2\}$, and forms like $\{\uparrow, \uparrow\!\ast \mid \downarrow, \downarrow\!\ast\}$ where the mirror is over a pair of options rather than one. Every one of them is a position in which the two players face exactly the same prospects with the colours swapped, which is what being one’s own negative means at the board rather than in the notation.

That reading also explains the parity. A position in which Left and Right have mirrored prospects must give them the same number of moves to choose between, and a form with three options on one side and three on the other is even-width by construction.

What this does not say #

Four limits.

The floor is a floor. Five hundred and seventy-one is what two families of narrow mirrors supply on day four. The wider families are not counted, and on day three they held 24 of the 26, so a true count could be an order of magnitude higher or several.

Nothing here is about play. Every count is over values of the abstract construction, not over positions of any game — which values a ruleset actually reaches is a different census, and the self-negative values a real game produces are a subset of these whose size nobody has measured.

The pairs are a sample. The 19,900 pairs are drawn at a fixed stride through day three rather than exhaustively, because the full set is 1.1 million forms and about ten minutes of arithmetic. Sampling can only lower the floor, which is the safe direction.

The share is measured on two days. One in nine to one in fifty-six is two data points and a direction. It is consistent with the population growing and it does not measure a rate, and a third point is exactly what is unavailable.

And a mirror form need not be canonical. $\{S \mid -S\}$ is self-negative whatever $S$ is, but two different sets can give the same value, which is why the counts above are of distinct values after canonicalisation rather than of forms. That is also why the construction cannot be counted by counting subsets: the map from sets to values is many-to-one and nobody has described its fibres.

The convention, named #

Normal play, canonical forms, computed by the recursion.

A value is self-negative when $G + G = 0$, equivalently $G = -G$, equivalently — and this is the form used throughout — its canonical form’s Right options are the negatives of its Left options as a set.

Born on day $n$ means the canonical form has depth $n$, so a value born on day 3 is one that day 3 supplies and day 2 does not. The tables above separate born this day from born by this day everywhere, because the second is a cumulative count and the growth question is about the first.

Width is the total number of options in the canonical form, Left’s and Right’s together.

Who cares about a subgroup of order two #

The two-torsion of the game group is not a curiosity of this site’s making. It is where misère theory keeps landing: under the misère convention a great deal of the normal-play arithmetic fails, and the fragments that survive are usually statements about positions that behave like their own negatives. And it is where the nimbers sit inside the partizan world — the impartial values are all self-negative, so the subgroup contains the whole of impartial theory and rather more.

Rather more is the part this page counts. Day three’s twenty-six self-negative values include exactly one nimber, and the other twenty-five are partizan positions with mirrored prospects. That ratio is the honest measure of how much larger the subgroup is than the impartial games inside it, and it is growing.

Where the ladder goes next #

The negation anchor has six rungs to here: that every game has a negative, which values are their own, what may be struck out because of it, where the subgroup sits on the temperature scale, when its members can first appear, and now how many of them there are.

The rung above takes the fibres of the mirror map, which is the obstruction this page names and cannot get past. What identifies two subsets finds the collapse from 1,793 subsets of day two down to thirty values happening in two stages of quite different character. The first is domination: a subset and its antichain of maximal elements give the same mirror, which is a theorem and which takes 1,793 to 96. The second takes 96 to 30 and is not a theorem at all — it is concentrated almost entirely on two values, nought and star, which between them absorb nearly every remaining identification while the other twenty-eight are barely touched.

So the many-to-one map this page could not describe is nearly one-to-one once domination is taken out, and the residue is two large fibres rather than a general blur. That is a much better position to be in than a floor, because it means a count of day four’s mirrors needs a description of two sets rather than of a map.

A mex with no impartial game in it supplies it, and the answer is the one thing nobody would have predicted from a construction with no impartial game anywhere in it. Nought’s fibre has an exact description and star’s is “some element is at least nought, and none is at least star” — and the two are one rule: the mirror of a set is the least nimber no element of the set reaches. A mex, arrived at from subsets of partizan values, which is the impartial theory’s own operation turning up where it has no business being.

Two neighbours are worth the trip. The values that are their own negatives is where the subgroup is found and its thirty day-three members are listed, and it is the page this one supplies a construction for. And how old a value is is where the birthday is established as a measure of complication, which is the axis every count on this page is taken along.