A self-negative value costs a day

A value equal to its own negative is an element of order two in a group whose elements otherwise have infinite order. The values that are their own negatives counted them — thirty among the 1,474 values born by day three, of which four are nimbers — and the thirty that cancel themselves put them on the temperature scale and closed with a question about time rather than about heat:

A self-negative value has a form whose options come in negative pairs, so its width is even at the root and its structure is constrained all the way down; whether that forces anything about when it can be born — whether, for instance, the first value of each temperature to appear is a self-negative one — is a question the day-three data can start and cannot finish.

It forces something exactly, and the specific guess offered alongside is not it.

The rule #

The earliest self-negative value of temperature $t$ is born on the day after $t$ itself.

Nine temperatures occur among the day-three values, and the rule accounts for every one of them.

A temperature of one: the number 1 is born on day one, so the rule predicts day two, and the census finds the first self-negative value of that temperature on day two. A temperature of a half: $\tfrac12$ is a day-two number, so day three, and day three is where the first one appears. A temperature of two: $2$ is a day-two number, so day three, and there is exactly one day-three value of temperature two and it is self-negative.

And the four empty temperatures are predicted too. A quarter, three quarters, one and a quarter, one and a half — each is a number born on day three or later, so the earliest self-negative value carrying it is a day-four value at best, and no day-three sweep can contain one. Four temperatures, 464 values between them, and not a single self-negative among them.

That the rule gets the absences right is what makes it a rule. A description fitted to the four temperatures that have self-negatives would have said nothing about the four that do not, and those four are where a wrong description would have been invisible.

Why #

The mechanism is one sentence and it is visible once stated.

A value of temperature $t$ that is its own negative has to contain the switch $\pm t = \{t \mid -t\}$ in it — that is the shape a self-negative fight has, since negation swaps the two options and negates them, and only a form symmetric under that operation comes back unchanged.

A switch $\{t \mid -t\}$ is born the day after its options. So $\pm 1$ is a day-two value because $1$ and $-1$ are day-one numbers, and $\pm\tfrac14$ is a day-four value because $\tfrac14$ is a day-three number.

Self-negativity therefore costs a day, and the day it costs is charged against the temperature rather than against anything else. That is why the constraint bites earlier at hotter temperatures than at colder ones: 2 is a simpler number than $\tfrac14$, so $\pm 2$ arrives before $\pm\tfrac14$ does, and the hottest self-negative value on day three is hotter than most of what surrounds it.

The guess, and where it fails #

The rung below’s specific conjecture was that the first value of each temperature to appear is a self-negative one. It is a reasonable guess — a constrained form ought to be a simple form — and it is not right.

At a temperature of a half the earliest values arrive on day two and the earliest self-negative one on day three. So a temperature can be reached a full day before it can be reached self-negatively, and being self-negative is not what gets a value in first.

The rule above says exactly why. A temperature of a half is first reached by $\{ \tfrac12 \mid 0 \}$-shaped values, whose options are a day-one number and a day-two number — cheaper than the symmetric pair $\pm\tfrac12$ the self-negative version needs. An asymmetric form is cheaper than a symmetric one, which is the opposite of what the guess assumed.

And at four temperatures the guess is not false so much as silent: there is no self-negative value at all, so nothing is first.

The one value at temperature two #

One row of the table is a single value, and it is worth looking at because it is the rule at its tightest.

Exactly one value born by day three has a temperature of two, and it is self-negative. That is not a coincidence and it is not evidence for the rung below’s guess either — it is the rule with no room in it. A temperature of two needs the number 2, which is a day-two value, so $\pm 2$ is a day-three value; and a day-three form has no room left to be anything other than $\pm 2$ decorated, since every option is already spent reaching the height.

The census reads that as a hundred per cent and it is a population of one. Reported here rather than in the finding, because a row with one member is where a rule and a coincidence look identical, and this one is a rule for reasons that do not depend on the row.

How fast the subgroup thins #

The other reading of the same data is the size of the subgroup as the days go on, and it falls faster than the guess would suggest.

Day nought: the one value there, $0$, is self-negative. Day one: one of three. Day two: two of eighteen. Day three: twenty-six of 1,452 — under two per cent.

The share falls by roughly a factor of three a day while the population grows by a factor of eighty, which is the arithmetic of a constraint that has to hold all the way down a form. Every option has to be matched by the negative of an option on the other side, and each new day multiplies the ways of failing that faster than it multiplies the ways of satisfying it.

So the two-torsion is a young phenomenon. The subgroup is not small because there is something rare about cancellation; it is small because symmetry is expensive to maintain, and the expense compounds.

What that says about the group #

The thirty that cancel themselves established that these thirty are the values a comparison may strike out unconditionally, which is what being in the two-torsion is worth. This page adds a timetable to that.

The practical form: a position whose value is self-negative is a position whose value is simple in a specific sense — it carries a temperature whose number is old. A fight worth $\pm 2$ can cancel itself and a fight worth $\pm\tfrac14$ cannot, not because a quarter is a smaller stake but because a quarter is a younger number.

That is a statement about the interaction of two orderings this site keeps finding to be independent. Big is not the same as hot establishes that birthday and temperature vary freely; the two-torsion is one place where they do not, and the rule above is the exact form of the dependence.

It is worth being precise about how narrow that dependence is, because the independence result is the general one and this is the exception. Among all 1,474 values a temperature says nothing about a birthday and a birthday says nothing about a temperature — the sweep that establishes it finds values of every temperature at every day that can carry them. Restrict to the self-negative ones and the two lock together completely: the birthday is the temperature’s birthday plus one, on every row of the table, with no spread at all.

A constraint that couples two free quantities is more informative than one that bounds either. The two-torsion is not a small subgroup of the values; it is a thin diagonal slice of them, and the slice is describable in a sentence.

And it explains the one thing the rung below found genuinely surprising: that only four of the thirty are nimbers. A nimber is self-negative and cold, so it sits at the $t = 0$ end of the diagonal, and the diagonal has twenty-six further places on it by day three. Where the impartial theory stops is the general account of why the nimbers are a small part of the picture, and this is that account at one specific temperature.

Reading it as a construction #

There is a constructive form of the rule, and it is the more useful statement for anybody building examples.

To make a self-negative value of temperature $t$: take the number $t$, which is born on some day $d$; form the switch $\pm t$, which is born on day $d+1$; and decorate it with any self-negative infinitesimal, which costs nothing extra as long as the decoration is no deeper. The result is self-negative, has temperature $t$, and is born on day $d+1$.

Running that in the other direction is what produces the timetable. Every self-negative value the census finds is of that shape, and the 26 that arrive on day three are $\pm\tfrac12$, $\pm 1$ and $\pm 2$ with the available decorations hung on them, together with the fifteen cold ones — which are the same construction with $t$ nought, where the switch degenerates and the decoration is the whole value.

That also says which self-negative values do not exist, and the answer is the practically interesting half: there is no self-negative value at any temperature the numbers have not reached yet. A fight over an amount that takes a long time to name cannot be a fight that cancels itself.

What the timetable is worth #

A date is a stronger thing to know than a count, and it is worth saying what this one buys.

It bounds a search. Looking for a self-negative value of a given temperature, the timetable says which day to start on and — more usefully — says when to stop looking on the days below it. Four temperatures in this sweep carry none, and a search that did not know the rule would have spent 464 comparisons finding that out.

It explains a shape rather than reporting one. The rung below observed the subgroup clustered at temperatures 0, ½, 1 and 2 and could say only that those were the temperatures it happened to occupy. They are the temperatures whose numbers are old enough, and the next one to fill will be a quarter, on day four, and then three quarters and one and a half, also on day four.

And it is a prediction. That is the part worth carrying: the rule was fitted to nine rows and it says what day four will contain. A day-four sweep is beyond this evaluator, so the prediction stands unfalsified rather than confirmed — which is the right state for a rule that costs one sentence to state.

A timetable is a lower bound on arrival, not a schedule #

The result reads like a schedule — the earliest value of temperature $t$ arrives on day $t+1$ — and it is worth being exact about which half of that is established, because the other half is where the count lives.

What is proved is an earliest. No self-negative value of temperature $t$ exists before that day, and one exists on it. That is a two-sided statement about the first arrival and it says nothing whatever about how many arrive then, or later, or whether any arrive at all on the days after.

A timetable of first arrivals is compatible with almost any population. The four temperatures carrying exactly one member and the four carrying none are the visible consequence: knowing when a class may first appear tells a reader nothing about whether it does appear more than once, and on this day most of them do not.

That is why the counting question the rung above takes is genuinely a separate piece of work rather than a refinement of this one. It needs a construction — build the values rather than dating them — and the construction is what turns a statement about first arrivals into a floor on a population.

And it is why the guess this page offers is the wrong shape. The first value of each temperature to appear is a self-negative one is a claim about which value arrives first among all values of that temperature, which is a competition rather than a bound, and a competition is decided by the whole population rather than by the earliest date either side could manage.

What the census does not say #

Four limits.

Three days. The rule is checked on the nine temperatures a day-three sweep reaches, and its content at the four empty ones is a prediction of absence rather than of a date. Day four would test the dates — $\pm\tfrac14$ should arrive there and nowhere earlier — and day four is beyond what this evaluator can enumerate.

The mechanism is a sketch. A self-negative value of temperature $t$ contains $\pm t$ is stated here in one sentence and is not proved. Its weak point is nameable: it assumes the symmetric form is the cheapest way to reach the temperature self-negatively, and a value whose symmetry came from somewhere else entirely would break the timetable without breaking the group theory.

A temperature is not a birthday. The rule predicts a day from the birthday of the number equal to the temperature, and a temperature is a number by construction here — it is a height on a thermograph. Nothing in the argument survives a game whose temperature is not a dyadic rational, and this site has none of those.

The rule is about the earliest, not about the rest. It says when the first self-negative value of a temperature can appear and nothing about how the others are spread after that. Twenty-six of the thirty arrive on day three together, which the rule permits and does not predict.

And the thinning is four points. One hundred per cent, thirty-three, eleven, two: a clean curve on four data points, three of which come from populations of one, three and eighteen. The last point is the only one with any statistical weight.

The convention, named #

Normal play, and all values are canonical forms computed by the recursion. Self-negative means $G = -G$, tested by comparison rather than by inspecting the form — a form symmetric under swap-and-negate is certainly self-negative and the converse needs the test.

A value’s birthday is the depth of its canonical form: nought for the zero game, one more than the deepest option otherwise. Born on day $n$ means birthday exactly $n$.

The temperature is the height at which the two walls of the thermograph meet, and a number is given $-1$ by convention so that a stack of temperatures holds only genuinely hot components. The row labelled a number in the table is the fifteen number-valued day-three values, whose only self-negative member is nought.

Where the ladder goes next #

The negation anchor has five rungs to here, and this one has just dated the two-torsion subgroup. The three above count it, describe the map that builds it, and end somewhere nobody would have predicted.

At least five hundred and seventy-one answers the counting question this page hands upward, and answers it by changing what is enumerated. A value is its own negative exactly when its form is a mirror, so the subgroup of a day can be built from subsets of the day below rather than sifted out of the day above — which turns an impossible enumeration into a large one, and gives a floor of 571 on day four against day three’s twenty-six. It also settles the two hypotheses: the share of a day that is self-negative falls, one in nine on day two and one in fifty-six on day three, so the population is growing faster than the constraint loosens.

What identifies two subsets then takes the mirror map’s fibres — 1,793 subsets of day two giving thirty values — and finds the collapse happening in two stages of different character. Domination takes 1,793 to 96 and is a theorem; the rest takes 96 to 30 and is concentrated almost entirely 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 built entirely out of partizan values, with no impartial game anywhere in it.

So the ladder that begins with a timetable ends with the impartial theory’s own operation turning up where it has no business being.