A resampled maximum can only fall

Two quantities in this field’s record have been given two different kinds of treatment, and neither has been given the other’s.

The first is the headline mean — how far ahead of its evidence a popular date runs, averaged over the claims dated ahead of it. The interval is wider than the number resampled it and found ninety per cent of its weight spread over 441 years around a value of 357. The second is the joint date, the year by which every claim in a set is on the record: the latest of their earliest sources. When two of them are first attested together built it for the five lineages of paper-folding practice, found it at 1838, one year from where the same claims are popularly dated together, and called it firmer than any of its members, because a maximum inherits its best member.

The obvious thing to do is to give the joint date the treatment the mean got — resample the record and put an interval on it — and the essay that proposed this expected it to be treated kindly. It is treated kindly. The kindness is the problem.

Two in three, and never later #

Resampling the five lineages means drawing five of them with replacement and taking the latest earliest source among those drawn. Twenty thousand draws give the distribution above: 1838 in 67.1 per cent, 1797 in 25.5, 1793 in 6.4, 1680 in 0.9, and nothing after 1838 in any of them.

Read the way the mean’s interval was read, that looks like an excellent result. The observed value is the mode by a wide margin, the whole distribution sits within a century and a half of it, and there is no upper tail at all. A historian shown this beside the mean’s 441-year spread would conclude that the joint date is a far better number, and that is the conclusion the earlier essay predicted.

The first number gives the game away. Sixty-seven per cent is not a measurement of anything in the record. It is $1 - (1 - 1/5)^5 = 0.672$, the probability that five draws with replacement from five entries include one particular entry. Whenever the kindergarten syllabus is drawn, the maximum is its source year, 1838, whatever else was drawn beside it; whenever it is not, the maximum is the next latest source that was. The resampled distribution is the distribution of which entry the draw happened to contain, and the record’s dates enter it only as labels.

The missing upper tail says the same thing from the other side. A resample is built from the record, so it can never hold a source later than the latest the record has, and a maximum over it can equal the observed maximum or fall below it and do nothing else. The resampled joint date is bounded above by construction. A distribution with no mass above the observed value is not evidence that the value could not have been later. It is the only shape this procedure can produce for a maximum, on any data at all.

Why a mean can be resampled and a maximum cannot #

The difference is not a quirk of fifteen entries and it is worth stating properly, because it decides which of the field’s numbers can be given an interval this way.

A mean is a sum, and every entry contributes to it in proportion. Redraw the record and the entries that are drawn twice pull it one way and the entries that are missed let it go the other, and the spread of the result is an honest picture of how much the value depends on which claims happened to survive. That is why the mean’s interval was worth computing: it measured how much the headline figure is made of two documents, and one lost source and the story changes had already found the same two.

A maximum is decided by one entry and ignores the others entirely. Redrawing the record can only ever remove that entry or keep it. Removing it gives the runner-up, keeping it gives the observed value, and nothing in the procedure can produce a value above either. The statistician’s name for this is that the bootstrap is inconsistent for an extreme: however large the sample, the resampled distribution of a maximum does not converge to the true one, and it piles about $1 - 1/e$, or 63 per cent, of its weight onto the observed value in the limit. The standard counter-example — the maximum of a uniform sample — is Bickel and Freedman’s, from 1981, two years after Efron introduced the method.

So the joint date’s resampled “interval” says nothing about how firm the joint date is. It is the right answer to a question nobody asked: how often does a random sub-record contain the latest-attested lineage?

The one-year agreement is one entry’s gap #

The earlier essay’s headline for the joint date was not only the year but how close it sits to the popular one. The five lineages are popularly dated together from 1837 and jointly attested from 1838 — one year apart, against 980 years for the worst single claim. That closeness looked like the payoff of combining claims.

The same twenty thousand draws say where it comes from. The joint gap is one year in exactly the 67.1 per cent of draws that contain the kindergarten entry, and in every other draw it is between 293 and 980 years. There is no draw anywhere in between. The one-year agreement is not a property that emerges from combining five lineages; it is the kindergarten entry’s own gap, which is one year because Froebel’s syllabus is popularly dated to 1837 and first printed in 1838.

The reason is simple once seen. The joint popular date is the latest of the popular dates and the joint attested date is the latest of the source dates, and in this set both maxima belong to the same claim. Two maxima that share their holder subtract to that holder’s own gap, and the other four lineages drop out of the arithmetic entirely. Remove the syllabus and the joint gap becomes the thousand cranes’ 1797 against the pajarita’s popular 1500 — 297 years, which is a statement about two different claims and is about as uninformative as a number here can be.

The earlier essay saw half of this. It noted that every joint date in the set is the kindergarten syllabus’s date and called the construction “fortunate rather than robust”. The resampling makes the other half precise: the joint date’s closeness to its popular date is a one-entry fact, and the chance of losing it under resampling is the chance of missing that entry, one in three.

Whose half the joint date is #

The general claim — that a maximum inherits the better-attested member of a set — can be checked against the record directly, without resampling, because the record says how many independent witnesses survive for each claim.

Among the five lineages the answer is already mixed. The kindergarten syllabus rests on four independent witnesses and holds every pair it is in, so those four pairs inherit the best-attested claim in the set. But the thousand cranes rest on one book, the Hiden Senbazuru Orikata of 1797, and hold the pairs with ceremonial wrapping, the pajarita and recreational folding; recreational folding rests on one poem and holds its pair with ceremonial wrapping. Four of the ten pairwise joint dates belong to a single surviving witness, and in four the later source is the worse-attested claim’s.

That is not what “inherits the better-attested half” means, and the reason it was said is visible in the first table the earlier essay drew. Its argument was about contest, not survival: a joint date is never argued about by more than its later member, and the later member of these pairs is the one argued about least, since a claim attested late has less room between its source and its popular date. “Better attested” meant nearer to its own evidence, which a later source often is. It did not mean attested by more witnesses, and the record’s witness column is the one that says how much a date would survive losing a document.

A hundred and five pairs #

The five lineages are one set. The record has fifteen claims and therefore a hundred and five pairs, and the question can be put to all of them at once.

The holder has more witnesses than its partner in 46 of the 105 pairs, as many in 24, and fewer in 35. Twenty-eight of the hundred and five joint dates — more than a quarter — rest on a single surviving document, so that one lost book, poem or chronicle would leave the pair with no joint date at all.

So a maximum does not inherit the better-attested half. It inherits the later half, and the later half is the better-attested in a minority of pairs. The rule that made the joint date look firmer than its members is a rule about dates, and it was right about dates: the joint date is never contested by more than its holder. Stated about witnesses, which is the reading a reader would naturally give it, it is wrong in more than half the record.

The distinction matters because the two kinds of firmness answer different threats. A date’s distance from its popular date measures how much it is argued about. A date’s witness count measures how much it would survive a fire. The joint date is firm against the first and has no special protection against the second, and in 28 pairs it is as exposed to the second as a date can be.

Whose date the record’s joint dates are #

Counting pairs the other way — by holder rather than by verdict — shows which part of the record a joint date is a statement about.

A claim holds every pair in which its source is the later one, so the latest-attested claim holds fourteen pairs and the earliest holds none. The latest-attested claims are the twentieth-century results — the fold-and-cut theorem of 1998, the vertex conditions in 1979, the Miura pattern in 1970 — and the seven results hold 69 of the 105 joint dates, the practices 33 and paper itself three.

Set that beside what some discoveries would make it worse found about the headline mean. That statistic averages the claims dated ahead of their evidence, and every twentieth-century result was invisible to it — half the record with no influence on the field’s most-quoted number in either direction. The joint date has the opposite blind spot. Taken over the whole record it is a statement about the twentieth century almost entirely, and the pre-modern practices the field’s arguments are about hold a third of it between them.

So the two numbers are not two measurements of one thing. They are measurements of the two halves of a record that two kinds of claim had already found to behave differently: the mean reads the practices, which are dated ahead of their evidence, and the maximum reads the results, which are dated after. Neither summarises the record, and each is silent about exactly what the other reports.

What a discovery can do to a maximum #

Resampling cannot say how firm a joint date is, but a direct question can: what would a newly found document do to it? The earlier essay on discoveries asked this of the mean and found that five of fifteen discoveries would raise it. A maximum behaves differently, and more simply.

A discovery for any lineage but the holder moves the joint date by nothing. A document two centuries earlier for ceremonial wrapping, recreational folding, the pajarita or the thousand cranes leaves the five lineages jointly attested from 1838, because the maximum never read those entries. A discovery for the holder moves it year for year — until the holder’s source passes the runner-up’s. The runner-up is the thousand cranes at 1797, so the most any single document can take off the joint date is 41 years, and past that the cranes hold the date and a further discovery for the syllabus is worth nothing.

That has two consequences the mean did not have. First, no discovery can ever make the joint date look worse: it can only move earlier, which is what the earlier account of discoveries called its virtue as a target for effort. Second, and less comfortably, the joint date is bounded below by its runner-up, so moving it far needs discoveries for several lineages at once, taken in the order of their sources. To bring the five lineages’ joint date back to 1700 would need new documents for the syllabus, the cranes and the pajarita together, earlier than anything now known for them by 138, 97 and 93 years respectively; recreational folding and ceremonial wrapping are already there.

The loss direction has the same shape reversed, and it is where the witness counts finally matter. A lost document can only move a maximum later, and only if it is the holder’s earliest. The syllabus has four witnesses, so losing its 1838 source would move the joint date to its second witness — a year the record does not carry. The thousand cranes have one, so for the three pairs they hold, losing the Hiden Senbazuru Orikata would not move the joint date later. It would leave those pairs with no joint date at all.

What the procedure cannot show #

It cannot show that the joint date is firm. That is this essay’s argument turned into a limitation, and it bears repeating as one: a resampling of the record can only ever lower a maximum, so an interval computed that way is a statement about the procedure and never about the date. Nothing in these figures says the 1838 joint date is right. It says which entry it belongs to and how many documents stand behind that entry.

It cannot price the second witness. The loss direction needs the year of each claim’s second-earliest source, and the record does not carry it. One lost source and the story changes asked for that column and it has still not been built; without it, “losing the syllabus’s 1838 source would move the joint date later” is a direction with no distance attached.

It cannot say which pairs are meaningful. A joint date for the fold-and-cut theorem and paper reaching Japan is arithmetic on two rows and says nothing about a world in which both were true, since nobody would ask when those two were jointly on the record. The 105 pairs are every pair, not every pair worth asking about, and the verdicts in them are about the record’s shape rather than about any merge that happened.

It cannot see a claim that left no source. Every maximum here is taken over claims that survived to be counted, which is the survival problem a record is not a proof sets out as the field’s standing limit. A lineage with no surviving source cannot hold a joint date, and an unrecorded lineage later than the syllabus would move the five-lineage date in a direction no figure here can show.

The idealisation, named #

The resampling treats the record as a draw from a larger population of claims, the same assumption the mean’s interval rested on. It is a convenient fiction for a curated list of disputes, and a question the record is too small to answer shows how far this record is from a sample. Here the fiction does less harm than usual, because the argument is that the procedure fails for a maximum whatever population the record is drawn from.

Witnesses are counted, not weighed. Four surviving witnesses to the syllabus and two to the pajarita are compared as four against two, although a printed book of known date and an engraving of uncertain meaning are not the same kind of evidence. The record’s column for the kind of source grades them and the kindergarten was a geometry class reads the syllabus’s documentation properly; the pair verdicts use only the count, because the count is what decides whether a date survives a single loss.

Ties are excluded by the data rather than handled. No two claims in the record share a source year, so every pair has exactly one holder. A record with ties would need a rule for which claim holds a tied pair, and the verdicts would depend on it.

How the numbers were checked #

No resample may return a joint date later than the record’s latest source. That is the essay’s central claim and it is checked on all twenty thousand draws, not argued.

The share that returns the observed date must match the chance of drawing the holder, $1 - (1 - 1/n)^n$, to within two points. If it did not, something other than the holder’s presence would be deciding the resampled maximum and the argument would be wrong. At five lineages the two are 67.1 and 67.2 per cent; over the whole record’s fifteen they are 65.6 and 64.5.

The joint gap must split cleanly: its observed one year in the draws that contain the holder, and more than a century in every draw that does not. A distribution with values in between would mean the agreement was built from several entries rather than belonging to one.

The pair census must account for every pair once, with more, as many and fewer witnesses summing to 105, and the “better-attested” count must fall short of half. And the discovery figure checks that a discovery for any lineage other than the holder, at two centuries, leaves the joint date where it was.

Still open: a maximum with a second witness #

The measurement that would turn this essay’s directions into distances is the same one these measurements of the record have been asking for since the leave-one-out: the year of each claim’s second-earliest source. With it, the loss direction becomes arithmetic — the joint date after losing the holder’s earliest witness is the later of its second witness and the runner-up — and a joint date could be quoted with the honest one-sided bracket a maximum actually has: no earlier than this, unless a document is found; no later than that, unless one is lost.

There is a cheaper intermediate, and the census above is its first reading. A joint date whose holder has two or more witnesses survives the loss of any single document as a date: it may move later, to the holder’s second witness, but it cannot vanish. Seventy-seven of the 105 are of that kind and twenty-eight are not. For the five lineages the multi-witness answer is still 1838, because the syllabus is both the latest and the best attested of them — a coincidence of this set that the whole record does not share. What the intermediate cannot say is how far the seventy-seven would move, and that is the second column again.

Sideways, the same failure awaits any statistic in the field that is decided by one entry. The earliest attestation of a practice is a minimum over its sources, and resampling sources would produce the same illusion from below: two in three redraws would return the earliest known date, and none could ever return an earlier one. Nothing here is as old as it sounds rests on minima of exactly that kind, and it is the next place the procedure should be refused before it is run.

The habit worth carrying is about instruments and extremes. Before resampling a statistic, ask whether a resample could ever produce a value on both sides of it. A mean can move either way and its interval means something. A maximum or a minimum can move only one way, and an interval with one wall is a picture of the procedure rather than of the evidence.