Fifty-two errors and seven sizes

Two values born by day two that cannot be compared still have a join and a meet, because day two is a lattice: the simplest value above both exists and is unique, and so does the simplest below. What the join and the meet do not do is account for the pair. The identity a reader would expect — that the two of them together come to the same as the two values they were taken of — holds on all 201 comparable pairs and on none of the fifty-two others.

Where the order and the sum disagree established that and closed by naming what it had not done:

The fifty-two discrepancies are fifty-two distinct values and they are not random: the smallest is a star and the largest is a whole switch, and what decides the size is presumably how far apart the pair is in the order it does not have.

A function from the pair to the size would turn that list into a statement. This page looks for one, and the first thing the search runs into is that the sentence assumes something a value does not have.

A value has no size #

The word size is doing quiet work in the sentence above, and it is worth stopping on before anything is counted.

Two numbers have a size and their difference is a third number. Two games in general have neither. ∗ is not larger than nought, not smaller, and not equal to it — it is confused with it — so asking how big ∗ is has no answer in the same sense that asking how big one half is does.

What a value does have is measurements, and the collection uses four of them constantly. The two stops are the numbers the position settles at when it is played out under the two orders of play. The mean is what it is worth on average over many copies. The temperature is how much is at stake in moving. And the birthday is how many days of the construction it took to appear.

Each of those is a number, each throws information away, and the interesting fact about the fifty-two is what happens when they are applied.

Fifty-two distinct values give seven distinct pairs of stops. The error term takes fifty-two different forms and, measured by where it settles, lands in one of seven places — and every stop that occurs is one of −1, −½, 0, ½ and 1.

Not one of them is a number #

The first thing the census asserts, and the thing that decides how the rest of the page has to be written, is that no discrepancy is a number.

That is not a small remark. If the error term had come out a number the story would be arithmetic: the identity would fail by a measurable amount, the amounts would be comparable with one another, and there would be a largest and a smallest. Instead each of the fifty-two is a game confused with, above or below the others in the same tangled way the pairs that produced them are, and the words largest and smallest have no referent.

That is why the rung below could say only that the smallest is a star and the largest a whole switch. Those two are extremes in an informal sense — the star is an infinitesimal and the switch has a fight in it — and there is no ordering in which they are the ends of a list.

Three means, and the identity half-holds #

The mean is the one measurement that is additive. The mean of a sum is the sum of the means, always, so it passes straight through the identity — and asking whether the identity holds in means is asking a question the value version has already answered in the negative and which has its own answer.

It half-holds. Twenty-six of the fifty-two errors have mean nought, so on exactly half the failures the join and the meet account for the pair as far as any average can see, and the failure is invisible to it. The other twenty-six split evenly: thirteen with mean ½ and thirteen with mean −½.

Nothing in between occurs. There is no error of mean ¼, none of mean 1, none of mean −⅜. The mean of the error is one of three numbers over the whole of day two, and two of the three are the same number with a sign.

That is a much stronger regularity than the fifty-two distinct values suggested, and it comes with a caution that runs through the whole page: it is a regularity about a day, not about games. Day two contains twenty-two values, seven of them numbers and none of them born later than the second day, and half-integers are as fine as its arithmetic gets. A day further out would have quarters in it.

The direction and the mean are two measurements #

The rung below sorted the fifty-two by direction: whether the join plus the meet comes out above the pair’s sum, below it, or confused with it. The split was 13 above, 13 below and 26 confused.

The means split 13, 13 and 26 as well. Those are the same three numbers and it is tempting to read them as the same classification counted twice.

They are not.

Twenty-two of the twenty-six confused errors have mean nought, and four do not. Two of them have mean ½ and two have mean −½ — errors confused with zero that are nevertheless worth something on average. And in the other direction, two of the thirteen above and two of the thirteen below have mean nought: the identity misses in a definite direction and misses by an amount worth nothing.

Eight rows disagree out of fifty-two, and the two marginal totals conceal every one of them. That is the ordinary hazard of a summary statistic and it is worth naming, because a table with the right totals looks like a table that has been checked.

The rule, which is about temperature #

Temperature is the measurement that gives the function the ladder asked for.

The temperature of the error is the temperature of the hotter of the two values, floored at nought — on forty-four of the fifty-two pairs.

The floor is a convention rather than a repair. A number is given temperature −1 throughout this collection, which is the convention below zero sets out, so a pair of numbers would be predicted to leave an error colder than cold; the prediction is that it leaves one at nought, which is what a pair of numbers with an infinitesimal between them does.

So the sentence the rung below wanted — a function from the pair to the size — exists for one of the four measurements. It is not the mean and it is not the stops; it is the temperature, and it is read off the two values without evaluating anything.

And the eight exceptions are a class rather than a scatter. Every one of them has a bent wall in the pair: one of the two values is {1 | 0, ∗} or {0, ∗ | −1}, the two day-two values with more than one option on a side. In each of the eight the rule predicts ½ and the error comes out at nought — the rule is wrong in the same direction every time, and wrong by the same amount.

The census asserts that, which is the part worth keeping. A run in which the rule failed on a pair of plain switches would stop the build, so the exception class is a claim under test rather than an observation about eight rows.

What being bounded by the interval means #

The sentence the rung below actually wrote was about distance: how far apart the pair sits in the order it does not have. There is a candidate for that distance and it is the interval the pair straddles — everything from the meet up to the join — whose width is the game join − meet.

The error is at or below that width on all fifty-two, without exception.

That is a bound and not a formula, and the difference matters. It says the error cannot exceed the room available, which is the sort of statement one would want to be true and which nothing guaranteed. It does not say how much of the room is used, and the answer to that is very little: pairs straddling wide intervals routinely leave a single star.

So the distance the rung below reached for turns out to bound the error and not to determine it, and the thing that determines the one measurement that is determined — the temperature — is not a distance at all.

What a lattice-ordered group would have bought #

It is worth saying plainly what is lost, because the identity looks like bookkeeping and is not.

A partially ordered abelian group in which the order is a lattice and the identity holds is a lattice-ordered group, and in one of those the order and the addition are two views of a single object. Every element splits into a positive and a negative part; every element has an absolute value; the join distributes over addition. Practically, the join becomes a shortcut: knowing where two values sit relative to one another says something about what they come to when added.

The values of day two have the order and the addition. Every game has a negative, the addition is associative and commutative, and the order is a lattice. What they do not have is the bridge, and the fifty-two errors are exactly the width of the gap it would have spanned.

So the practical reading of this page is a prohibition. The join of two positions says nothing about their sum, not even approximately: half the failures are invisible to the mean, and the other half are wrong by exactly half a move. A player who has decided which of two components is better has not thereby learned anything about the board they add up to, which is the same lesson outcomes do not add teaches at the coarsest level and this page teaches at the finest.

There is one consolation and it is the temperature rule. If the order says nothing about the sum, the temperatures say something about the error: the mistake made by trusting the identity is as hot as the hotter of the two values and no hotter, on forty-four pairs of fifty-two. That is a bound on how badly the shortcut can mislead, and it is available before any addition is done.

Where the error is born #

One more measurement, and it is the one that says why the fifty-two are hard to think about.

Three of the fifty-two errors are values born by day two. The other forty-nine are not.

Every ingredient is a day-two value: the pair, the join, the meet. Addition and negation are applied twice and the result is, in forty-nine cases, something day two does not contain — { {1↓ | ↓∗}, ∗ | {↓ | −1↓∗}} is a fair sample, and reading it is the point.

That is the concrete form of a fact this collection keeps meeting. A day is closed under nothing. It is not closed under addition — the sum of two day-two values is usually not one — and the error term of an identity taken inside the day therefore lives outside it. The lattice is a structure on day two; the failure of the lattice to be a lattice-ordered group is a fact recorded in values day two does not have.

Fifty-two failures taking seven shapes #

The move from a list to a distribution is what makes this page a result rather than an appendix, and it is worth stating as a method because it applies to every counterexample set on this site.

A list of counterexamples is not a finding. Fifty-two pairs where an identity fails is a fact about fifty-two pairs, and the natural response — look at them — scales badly and usually produces nothing, because fifty-two objects are too many to hold and too few to average.

Measuring the failures turns them into a population. Not which pairs fail but what the error is: its stops, its mean, its temperature. And the answer here is that fifty-two errors take seven pairs of stops, three means and three temperatures between them, which is a very different object from fifty-two accidents.

That has two consequences. It says the failure is structural — a small number of configurations recurring, rather than a scattering — and it puts a ceiling on what a repair could buy: a corrected identity would need three cases and would still miss eight pairs, which is a worse object than the two clean theorems it replaces.

So measuring the errors answered the question of whether to attempt a repair, which listing them could not have. That is the general instruction: when an identity fails on a set, describe the errors rather than the set — because the errors are numbers, numbers have distributions, and a distribution says whether there is a phenomenon to explain.

What the census does not say #

Four limits, and the first is the one that bounds the rest.

Everything here is about twenty-two values. Day two is small enough to enumerate every pair, which is why the counts are exact and why they are counts about a day. The temperature rule is checked on fifty-two pairs; day three has 1,474 values and over a million pairs, and whether the rule survives there is a computation of the same shape and four orders of magnitude larger.

The bent-wall exception class is a description of eight rows. Two values of day two have more than one option on a side and both are in the exceptions; that is enough to state the class and not enough to say the class is the mechanism. It would be established by a day-three sweep in which every exception had a bend and no bend-free pair failed, and this page has not run one.

The three means and three temperatures are facts about a day’s arithmetic. Day two’s numbers go in halves, so its errors do too. Nothing here suggests that a day whose numbers go in quarters would leave errors in halves.

And the bound is not tight anywhere. At or below the width of the interval is satisfied on all fifty-two and would also be satisfied by a much smaller bound on most of them. What the smallest true bound is, this page has not looked for.

The convention, named #

Normal play, and every value is a canonical form. The join of two values is the unique simplest value above both inside day two, and the meet the unique simplest below; both are taken inside the day rather than inside the class of all games, which is why the join moves when a later day is available.

The temperature is the standard one — the height at which a tax on moving makes neither player want to move — computed from the thermograph rather than from a formula, and a number is given temperature −1 rather than nought. The mean is the value the two walls of the thermograph meet at. The stops are computed by playing the position out under the two orders of play.

The discrepancy is (a ∨ b) + (a ∧ b) − (a + b), reduced to canonical form, and every one of the fifty-two is computed rather than quoted.

Where the ladder goes next #

The lattice anchor has three rungs: that the order is a lattice, that the lattice and the addition do not fit together, and now how large the misfit is.

The rung above is the day-three sweep, and it is the one measurement that would turn this page’s rule into a claim about games rather than about twenty-two values. Day three has over a million pairs and the identity’s failures among them are a much larger population; whether the temperature rule survives, whether the exception class is still the bent walls, and whether the means stay at three values are three questions with the same answer-shape and one enumeration between them.

Two neighbours are worth the trip. What is at stake is where the mean and the temperature are defined and shown to be a measurement rather than a label, and it is the page that makes this one’s central move — measuring a value rather than comparing it — legitimate. And which end of the interval is open is the other page here about an interval between two stops, where the question is which of the two endpoints belongs to it and the answer is again a fact about infinitesimals.