A question the record is too small to answer

The interval is wider than the number ends by pointing at a column of the record nothing has ever read as a distribution: how many independent sources survive for each claim. Every entry has carried that number since the record was written, and it has been used only one entry at a time — to say that a claim rests on one source and is therefore fragile.

Read together, the fifteen numbers are the raw material for a question this field cannot otherwise approach. How many claims left no surviving source at all?

Why the question has a method #

A claim with no surviving source is not in the record, and it is not in the record in a way that leaves no trace — nobody is making it, nothing cites it, and there is no entry with a zero in the source column.

The shape of the observed distribution is the trace. If claims left many sources each, a claim leaving none would be very unlikely, and few could be missing. If most claims left one source and barely survived, then plenty of others must have left none. The information is in how heavy the low end is, and there is a standard estimator that reads exactly that: the number missing is about $f_1^2 / 2f_2$, where $f_1$ is the count seen once and $f_2$ the count seen twice.

Here $f_1 = 5$ and $f_2 = 5$, so the estimate is $25/10 = 2.5$ claims. Fifteen observed and about two and a half missing: a record holding six sevenths of its own subject. That would be a striking thing to know.

What the interval says about it #

The ninety-five per cent interval runs from 15.4 to 30.6 claims against fifteen observed. The lower end says nothing is missing; the upper end says as much again is missing as survives. An interval covering “none” and “half of everything” is an interval that excludes nothing anyone would have proposed.

Re-reading one entry moves it by about as much as the estimate itself. Move a single claim from two surviving sources to one — a judgement a historian might make either way about whether two accounts are independent — and the estimate goes from 17.5 to 19.5. Move one the other way and it goes to 16.33. The whole quantity is a ratio of two counts of five, and neither five is robust to a reasonable disagreement about what counts as an independent source.

The assumption that is actually false #

Interval width is a symptom of a short record. The deeper trouble is an assumption the estimator makes and documents do not satisfy.

The estimator treats each source as an independent draw: whether a source for one claim survives is unrelated to whether a source for another does. Documents do not fail that way. A library burns and takes everything in it; a language stops being read and a whole genre becomes inaccessible; a period leaves almost nothing and the periods either side leave a great deal. Survival is correlated by place, by date, by material and by language, and every one of those correlations violates the assumption directly.

The direction of the error is knowable even when its size is not. Correlated loss makes the observed distribution look less heavy at the low end than independent loss would — claims that survive at all tend to survive in clusters — so the estimator understates how many claims are missing. Two and a half is a lower bound on a quantity whose upper bound is unknown, which is a different sort of statement from an estimate and not a better one.

A record is not a proof makes the general version of this point as a table of what the record cannot establish. This is one line of that table computed rather than declared, and the computation adds something a declaration does not: it says why the question fails, and what a record that could answer it would look like.

What answering it would have been worth #

It is worth saying what was at stake, because a failed computation is only interesting if the question mattered.

A number for the claims that left nothing would put a figure on this field’s ignorance rather than on its disagreements. The record holds the claims people argue about; a count of the ones nobody can argue about, because nothing survives, would say whether the arguments are over most of the subject or over a fragment of it. Six sevenths and one third are very different pictures of a discipline, and the essays that discuss the record’s shape have no way to choose between them.

It would also change how a single discovery should be read. If the record holds nearly everything, a newly found document is a refinement; if it holds a third, a newly found document is a sample from a large unseen population and may well overturn things. One lost source and the story changes shows how much a removal moves the field’s statistics, and the symmetric question about additions cannot be answered without knowing how many additions are available to be made.

So the estimate is not an ornament. It is the missing denominator in most of what this field says about itself, and the honest position is that the denominator is unknown and the record as constituted cannot supply it.

What it would take #

The estimator’s instability is a function of how many entries sit at the low end, and that gives a usable target.

The spread scales roughly as the square root of $f_2$, so halving the interval’s relative width takes about four times as many entries at two sources — twenty rather than five, on a record of about sixty. A record of sixty claims with the same shape would give an interval about half as wide, which is still not tight and would at least exclude something.

That is a concrete thing to want and it is not obviously out of reach. The fifteen entries here are the claims this subject argues about; a record built to answer this question would include claims nobody argues about, which are precisely the ones whose attestation is uncomplicated. The entries that would make the estimate work are the boring ones, which is the opposite of how a record of disputed claims gets assembled.

Reading the estimator’s own shape #

The formula deserves a sentence of its own, because its shape explains both the answer and its instability.

$f_1^2 / 2f_2$ is large when many classes are seen once and few are seen twice — a distribution with a long thin tail, which is what a heavily depleted record looks like. It is small when the singletons are few relative to the doubletons, which is what a well-preserved record looks like. The estimator is reading how quickly the counts fall off, and it needs the fall-off to be measurable.

With five and five there is no fall-off to measure. The distribution reads 5, 5, 3, 2 — nearly flat at the low end and then declining — and a flat low end is the case the estimator handles worst, because the ratio $f_1/f_2$ is then near one and every one of its powers in the variance formula contributes equally. That is exactly why the interval comes out several times the estimate rather than a fraction of it.

A record with, say, 12 singletons and 3 doubletons would give $144/6 = 24$ missing against fifteen observed, and a much tighter relative interval, because the shape would be unambiguous. This record’s problem is not that it is small; it is that it is small and flat, and the second is what leaves the estimator nothing to read.

Which claims are at the low end #

The five single-source entries are worth naming, because the estimator is about them and because the list is not the one a reader would guess.

They are paper’s arrival in Japan, recreational folding, the senbazuru, Beloch’s paper and the one-cut star. Three of those are practices dated by tradition; one is a twentieth-century mathematical result; one is an eighteenth-century curiosity. They have nothing in common except that one document carries each of them, and they are spread across eighteen centuries and three continents.

That heterogeneity is itself evidence against the estimator’s assumption. Five claims with a single source each, drawn from completely different documentary situations, is not a sample from one survival process. It is five separate accidents, and an estimator that reads them as five draws from one distribution is reading them as something they are not.

One lost source and the story changes removes those five together and finds the field’s headline statistic nearly halving. That is a statement about the record as it stands and it needs no distributional assumption at all — which is why it holds and this does not.

The other end of the distribution #

The estimator reads the low end, and the high end is worth looking at because it says what a well-attested claim in this field looks like.

Two claims rest on four sources: ceremonial wrapping and the kindergarten syllabus. Three rest on three: paper’s arrival in Europe, the vertex conditions, and the Miura fold. Four of those five are dated later than their earliest evidence — the popular date lags the document rather than running ahead of it — which is the ordinary situation for something whose first appearance is on the record and whose adoption took time.

That is a pattern worth stating even though the estimator cannot use it. Well-attested claims in this field have small or negative gaps and badly attested ones have large positive gaps, which is not a coincidence and not a discovery: a claim with four surviving sources is a claim somebody documented at the time, and a claim documented at the time cannot be dated nine centuries before its documentation.

The ceremonial-wrapping entry is the exception and it is the interesting one: four sources and a gap of four hundred years. Two traditions and a merge says why — it is a well-documented practice whose popular date is inherited from a story about continuity rather than from any of the four documents.

A source count is a judgement #

The whole estimate turns on two counts of five, and it is worth being explicit about what kind of number a source count is.

Two accounts of the same thing are one source if one derives from the other, two if they do not, and often nobody knows which. An account that cites a lost work and another that plainly paraphrases it are one source wearing two coats. A translation is not a second source. A later compiler drawing on an earlier compiler is not either, unless the earlier one is lost and the later preserves material the first did not.

So the column this essay is reading is a column of judgements, each defensible and several of them arguable. The figure shows the consequence: moving one entry either way moves the estimate by more than a tenth in each direction, and a historian who disagreed with two of the fifteen classifications could move it by a quarter.

The paper had to arrive first turns on exactly such a judgement: how many independent lines of evidence date the arrival of a material, and whether accounts that agree are agreeing because they are right or because one copied the other. Fifty years in the wrong language is the standing example of what independence does in this field — a result that existed, in print, and was unavailable — and it is a reminder that “independent” is about transmission rather than about publication. Two papers that never reached each other’s readers are independent in a way two papers in one journal are not, and no count of documents sees that.

What the estimate cannot show #

Almost everything, and the point of the figures is to make the almost precise.

It cannot show what a missing claim would be. The estimator counts classes, not contents: two and a half missing claims is a number with no description attached, and there is no way to ask what they were about, when they were, or whether they would change anything.

It cannot show whether the record’s inclusion rule matters. These fifteen entries are the claims the subject disputes, which is a selection made on grounds having nothing to do with attestation. A different rule would give a different distribution and therefore a different estimate, and nothing here says how sensitive the answer is to that.

It cannot see what a second source year would add. Some discoveries would make it worse asks the neighbouring question about the record’s dates rather than its counts, and finds the record able to answer a hypothetical it cannot answer as a measurement.

And it cannot distinguish a claim with no source from a claim nobody thought to make. A practice that left no document and that nobody has since proposed is not a missing entry in this record; it is outside the record’s universe. The estimator counts the first and cannot see the second.

What the model assumes #

Sources survive independently. This is the assumption named above as false, and it is the reason the estimate is a lower bound rather than an estimate.

The source counts are right. A count of independent surviving sources is a judgement — two accounts derived from a lost third are one source or two depending on what is known — and the figure shows what one such judgement is worth.

The fifteen entries are a sample from a population of claims. They are a curated list of disputes, which is not a sample of anything.

And a claim is a class. Treating a claim about a practice and a claim about a theorem as two instances of one kind of object is what allows them into one distribution, and they are not obviously alike.

How the numbers were checked #

The estimate’s interval is required to be several times the estimate itself, which is the essay’s claim and is computed from the estimator’s own variance rather than described.

The two counts the estimator reads are required to be equal — five and five — because that is the fact that makes the ratio so unstable, and a record where they differed would make a different argument.

The disturbance is computed rather than described: the estimate is recomputed with one entry moved in each direction, and the swing is required to exceed half the estimate.

And the source counts are required to account for every entry. Five plus five plus three plus two is fifteen, checked, so no claim is missing from the distribution the whole argument is about.

Still open: whether a discovery would help #

The question this account cannot answer suggests one it can, and it is more useful.

A new document does not only add a source — it moves a claim from one bin to the next, which changes $f_1$ and $f_2$ and therefore the estimate. A source found for one of the five single-witness claims takes $f_1$ to 4 and $f_2$ to 6, and the estimate falls from 17.5 to 16.33. So a discovery lowers the estimated number of missing claims, which is the right direction and a very small step.

More interesting is what a discovery would do to the field’s other statistics, and that has nothing to do with estimators. A source found earlier than the current one for a claim changes its gap directly, and the field’s headline number is a mean of gaps. Whether a given discovery would raise or lower that number is computable from the record as it stands, claim by claim, and the answer is not obviously the one everybody would expect: a document that closes a small gap removes a small number from a mean of large ones.

Sideways from here, the source-count column has a second use nobody has made. When two of them are first attested together builds a joint date from pairs and finds it firmer than either half; a joint date built only from multi-source claims would be firmer again, and the record has the column needed to build it.

The habit worth carrying is about estimators applied to short data. Compute the interval before deciding whether the estimate was worth computing. An estimate whose interval covers everything is not a weak answer to the question; it is a demonstration that the data cannot address it, and that demonstration is worth more than a refusal, because it says what would be enough.