A shared factor has two readings

What “whorled” was hiding opened a bucket that the earlier census had filled and walked past. The census had asked the geometry how often a random divergence gives each kind of pair, and put 35 per cent of the circle into a bucket marked whorled, meaning that the two counts share a factor. Opened, every pair in that bucket was $k$ and $2k$, the coarsest pattern repeated $k$ times, and reading the bucket up to its jugacy took the Fibonacci share from 14.7 per cent to 50.1 without describing a single extra plant. A shared factor looked there like a jugacy. It was not quite that, as a later reading showed: those pairs were a hop and its own double, one family read twice, and by the lattice’s own two families no pair in that census shares a factor and the share stays at 3.2 per cent. The arithmetic was right for what it measured, and a census of the geometry has no miscounted plants in it, so it could not have asked what a shared factor means in a report by a person.

A few essays away, the same evidence means something else. A count that can be wrong by one asked what a reported pair allows when either count might be off, and it found that some wrong reports give themselves away: if the two counts share a factor, they cannot be a true ordinary pair, so a careful reader knows something has gone wrong. There the shared factor is an alarm. It is what makes a miscount catchable at all, and the coprime check it provides is one of the few free tools a counter has.

Each of those arguments is reasonable, and each treats the shared factor as settled. They cannot both be applied to the same report without an account of how often each cause produces one, and a person holding a report of 8 and 14 spirals does not know whether to write down a bijugate plant or to count again. That account can be written down exactly.

One count from the truth #

Start with a true Fibonacci pair and ask what a single slip can produce. There are four reports one count away: either count, up or down. A census that follows the reading in the essay on the whorled bucket classifies each by dividing out any common factor and looking at what remains: a pair of consecutive Fibonacci numbers makes a whorl of Fibonacci, a pair of consecutive Lucas numbers a whorl of Lucas, and anything else no family.

The picture around 21/34 is not what anyone would expect from the phrase “a wrong report announces itself”. Of the four cells touching the centre, the one to the left is 21/33, which a census reads as three whorls of the Lucas pair 7/11. The one to the right is 21/35, seven whorls of the Fibonacci pair 3/5. Above and below it, 20/34 and 22/34 share a factor of two and name no family. All four have a factor. Two of them have a name.

The same thing happens at every pair the dial reaches. Around 8/13, a slip gives 8/12, read as four whorls of 2/3, or 8/14, two whorls of 4/7. Around 55/89 it gives 55/88, eleven whorls of 5/8, or 55/90, five of 11/18. These are not contrived. Each is a report that a person might reasonably write down after losing a spiral on the way round.

How many of the four #

Across nine pairs from 3/5 to 144/233 the count is very regular. Above the smallest pair, at least two of the four always share a factor, and at least two always carry a name. When one of the true counts is even, as with 21/34 and 34/55, every neighbour by the odd count is even too and so all four share something. The smallest pair is the only exception: 3/5 has one report with a factor, 3/6, which is named, and it also has 3/4, which shares nothing and is the Lucas pair itself.

Why names should turn up so often has a partial answer. The two families are both generated by adding, and a Fibonacci pair’s neighbours are close to pairs that are multiples of small terms of the same sequences. The multiple is not small, though, and that gives a census something to work with. Two of the four are always named, and the size of $k$ is the only thing telling them apart from a real whorl: eleven whorls of 3/5 is not a plant anyone has reported. Eight whorls of 11/18 is not either. If a census believes every factor, it is believing those.

What a census of ordinary plants reports #

That makes a straightforward calculation possible. Take a population of plants that are all Fibonacci, equally weighted over the nine pairs, with a chance $e$ that each count is off by one, up or down, independently. Nothing here is sampled: every report and its probability can be listed. At $e$ of a hundredth, 0.9 per cent of reports are called a whorl. At 5 per cent it is 4.6 per cent, and at a tenth it is 9.0 per cent. The rate is close to the chance of a slip itself.

So a census at a modest error rate reports a share of whorled plants about equal to the error rate, from a population with none. The rest of the wrong reports are divided between no family at all, 3.8 per cent at an error of 5 per cent, a factor with no family, and, rarely, Lucas, at 0.3 per cent. What is not on the figure is also a finding: the chance of a slip making an ordinary plant look Lucas is small, since a Lucas pair’s counts are mostly not one count from a Fibonacci pair’s, and the Lucas share of a census is nearly safe from this.

What a count is worth has made the point several times that a survey cannot be taken at its reported counts, and the survey that loses its second outcome has specified how to design around it. What is new is the direction of this effect, in that the error does not simply lose information. Bad counting manufactures one specific kind of finding, and it is a finding of the kind a plant could really have.

The chance a factor means a whorl #

The two causes can now be compared directly. Take a population in which some share of plants are truly whorled, with two and three organs to a whorl on the Fibonacci sequence, and ask the question a person holding a report wants answered. Given a shared factor, how likely is a true whorl?

With 5 per cent of plants truly whorled, a report with a factor is a true whorl 82 per cent of the time if a count slips one time in a hundred. At one in twenty it is 46 per cent. At one in ten, 29 per cent. With 10 per cent of plants whorled those become 90, 65 and 47 per cent, and with 1 per cent, 46, 14 and 7.

The even-odds line falls where anyone reading such a report ought to get careful. With 5 per cent of plants whorled, a shared factor is about as likely to be a slip as a plant once the error reaches about four and a half per cent. Against that should be set what is known about counts. A count of 34 or 55 spirals by eye over a head, one with spirals crowded as they are in a real capitulum, and one with a closing error of three tenths of a spiral, has earlier been estimated here as wrong about one time in eleven from the geometry of the closing alone. So at the counts where jugacy is most often claimed, the error is well within the range where a factor is ambiguous.

Refuse the absurd whorl #

A census does not have to believe every factor. Real multijugate plants have small $k$, two and three by a long way and higher only rarely, and a reader who is asked to believe in eighteen whorls of 5/8 will rightly decline. That limit does real work, and how much can be computed.

Refuse any whorl of more than three, and at an error of 5 per cent the chance that a believed factor is a true whorl rises from 46 per cent to 66. At an error of a tenth it rises from 29 to 48. Of one slip’s reports across the nine pairs, 47 per cent are named as a whorl of some size, and only 14 per cent as a whorl of three or fewer. So two thirds of the false names go away with one sensible rule, which costs nothing, since a true plant of more than three organs a whorl is rarely reported and is treated with suspicion anyway.

That leaves a stubborn core, and it has a very particular shape. Under that limit the only false whorls left are on small pairs: 3/6 as three of 1/2, 6/8 as two of 3/4, 8/14 as two of 4/7, 12/21 as three of 4/7, 21/33 as three of 7/11. Every one of these is in the range of counts where a real bijugate or trijugate plant would be reported. It is noticeable that each of those names a Lucas pair and that a false whorl with $k$ under four is commoner on Lucas than on Fibonacci. Requiring the reduced pair to be Fibonacci as well, which is what real bijugate reports mostly are, would remove them. But that is also a way of deciding the answer before counting, and it would remove a true bijugate Lucas plant, of which the essays on the Lucas seed have said such things can exist.

Large counts are the safer ones #

The shape of that list is the most practically useful result here, and it runs against what one might guess, since a large pair is much harder to count. From 34/55 upward, not one report one count away is named as a whorl of three or fewer. The nearest thing is 34/54 or 34/56, which share a factor of two and reduce to 17/27 and 17/28, a pair on no sequence at all. On 55/89 the named neighbours are 55/88, eleven whorls of 5/8, and 55/90, five of 11/18. On 89/144 and on 144/233 they are eighteen whorls and eight.

So a true bijugate plant reported at 68/110 stands well apart from anything a slip would make: reading the same count as a slip means two families off by more than ten. The same split shows up in the arithmetic. Believing factors up to three, at an error of 5 per cent and 5 per cent of plants whorled, a report on pairs below 34/55 is a true whorl 58 per cent of the time, and on larger pairs 80 per cent. At an error of a tenth those are 40 and 65 per cent. Those figures still count factors with no family attached. Requiring a name as well removes the rest of the false reports at one slip.

So the proper place for suspicion is the small pairs, where a person’s counts are at their most reliable, and the reassuring place is the large ones, where a person’s counts are at their worst. A report of 8/14 needs a rotation before anyone believes it. A report of 68/110 mostly needs the count to be nearly right.

What happens to the true whorls #

The confusion runs the other way too, and less seriously. A true bijugate plant counted with an error of 5 per cent keeps its family on 92 per cent of reports. The rest are mostly called no family: 10/17 and 11/16 share nothing and belong to nothing. So a slip takes a whorl’s factor away, and it does not change it into something else that sounds right. In this calculation not one of those reports is ever called Fibonacci at any error.

That asymmetry is what makes a factor more worrying than its absence. An ordinary plant’s slip can land it on a name, and a whorl’s slip can only land it on an unnamed report that a careful reader is likely to recount. Errors in that direction are at least noisy. Errors in the first direction are the silent kind, in the sense of the essays on silent miscounts, and a census has no internal way of noticing them.

What would settle a report #

A count alone cannot settle it. The number of spirals in each family is the same statistic for a miscounted ordinary plant and for a true whorl, so more of the same number does not help unless it comes from a different annulus. There are two independent readings that do.

The first is a rotation. The symmetry that is not there showed that a stem grown $k$ at a time and a pattern that merely counts $k$ apart differ when rotated by a $k$-th of a turn: a real whorl lands on itself, and anything else does not. A plant reported at 8/14 and truly two whorls of 4/7 maps onto itself under a half turn. One miscounted at 8/13 does not. That test needs the organs’ positions, which a photograph has.

The second is the same count taken somewhere else. A true whorl keeps its factor up the sequence of its pairs, reading 8/14, then 14/22, then 22/36. A miscount at one annulus does not repeat. So two annuli, whose reports are independent, give the same factor only if the plant has one. The rings of a jugate head carry their common factor out to the rim for the same reason. On a large head the factor can be checked from rings as well, and the three together are stronger than any one of them.

Neither is expensive. They require keeping the photograph or counting a second annulus, which is exactly the kind of thing a survey nobody has done has been specifying from the start. What cannot be done is to take one report with a shared factor as a reading of the plant, or to take it as a proof of a mistake, without at least asking how often counts of that size go wrong.

What this does not establish #

The errors here are simple: each count wrong by one, equally up or down, independently. Earlier essays have measured how a closing error depends on the family and the head, and how slips differ across families. A more careful version would use those, and two-out errors, which here are given only the small probability independence implies. The population is equally weighted over nine pairs, which no real census is, and the true whorls are limited to two and three organs on Fibonacci pairs. The share of plants that are truly whorled is not known here and is treated as a stated assumption. And the whole thing is about reports of counts. It says nothing against the earlier census, which counted no plants, and whose finding about jugacy in the geometry stands exactly as it was.

How this could be wrong #

A count’s slip that was much commoner in one direction would change the balance of factors, though not by much, since the reports are symmetrical around each pair at this size. A real census in which reports with a factor kept the factor across a second annulus at the rate of the rest would mean the reports were true whorls, whatever the arithmetic here suggests. A survey in which counters’ error was measured to be well under a hundredth would make the whole problem moot. It would be good to have one.

Still open: the factor, read across two annuli #

The second test above has a number attached that has not been computed. If each annulus is an independent report, a true whorl keeps its factor on both with probability about $(1-e)^4$, while a true ordinary plant has a factor on both only if it slips on both, with the two factors matching. The next measurement takes a population with stated shares of true whorls, slips at a stated rate and pairs taken from two annuli a transition apart, and computes how far two reports with the same factor move the posterior, whether a factor that disagrees between annuli is strong evidence of a slip, and how the answer changes once the slips at the two annuli are allowed to be correlated, as a counter’s slips on a head sometimes are.