The recount aims where the counter expects

A census of Fibonacci plants scores each specimen by its counted parastichy pair and asks whether the share that comes out Fibonacci is the geometry’s 14.7 per cent or the ninety a grown history gives. The census wants a low count found that the likeliest counting error — a counter who closes the circle a few degrees early or late, multiplying both counts by the same factor — falls hardest on the high pairs, and that most of its wrong readings at 34/55 announce themselves by sharing a factor: 34/54, 33/54, 35/56. Counting it again then counted an announced specimen afresh rather than setting it aside, and found that a fresh count changes which specimens are kept and never what a kept reading says.

That recount was blind in two ways, and the essay said so. Its second closing error was independent of the first, although a counter who closes late at one organ may well close late there again. And its second closing was not aimed, although the counter knew the first had gone wrong and might try to close a little earlier or later to correct it. This essay gives the recount both: a habit shared between the two counts of one head, and an aim. It then asks what the aim needs in order to work, and finds that what it needs is the answer the census is trying to reach.

A habit and an aim #

Each closing error on one head is written as a part the counter carries from count to count on that head — the same conspicuous organ misjudged the same way — plus a part of its own. The spread of every closing error stays $\sigma$, and the correlation between the two counts of one head is $\rho$, the share of the spread the habit carries. At $\rho = 0$ the two counts are the independent counts of the earlier recount, and this essay’s arithmetic reproduces that essay’s census exactly: twelve kept specimens at 34/55 and 7.2°, 24.4 counts on grown plants.

The counter reads the first count, and if it announces itself, counts again with an aim $\delta$: a deliberate shift of the closing mark. Given the first error $\varepsilon_1$, the second closes at a normal error with mean $\rho\varepsilon_1 + \delta$ and spread $\sigma\sqrt{1 - \rho^2}$. The counter cannot see $\varepsilon_1$. What it can do is estimate it from the reading it got, and aim against the part of it the habit will carry: $\delta = -\rho \cdot E[\varepsilon_1 \mid \text{reading}]$. Everything below turns on that expectation, and it is computed exactly — every stretch of closing error on which each candidate true pair produces the reading, weighted by the first count’s normal spread.

What the reading says about its own error #

Take the commonest announced reading of a grown plant’s 34/55 that changes only the larger count: 34/54. A counter wants to know whether the first closing ran short or long.

Ten true pairs produce 34/54 with any real chance. The likeliest is 34/54 itself, a whorled head read right, which takes 35.6 per cent. Then come 35/55, another whorled head read short, at 15.1; 33/53, a spiral head read long, at 14.6; 34/55 read short at 9.4; 34/53 read long at 8.7; and a tail of others. Their stretches lie on both sides of zero and are nearly symmetric about it, because a counter who closes short by a few degrees and one who closes long by a few degrees produce readings that are each other’s mirror across the true pair.

So the reading alone points nowhere. A counter who admits every true pair as possible — whorled heads included — puts the first error at −0.29°, which is no information at all, and this is not special to 34/54: the same counter gets −0.29° from each of the other announced readings of 34/55 read here, since the pattern of stretches is the same around each. The expectation that makes an aim possible has to come from a belief about which heads exist.

Three beliefs #

Three clean ones are read here. A counter who expects Fibonacci admits only consecutive Fibonacci pairs as true, and for 34/54 that leaves one candidate, 34/55, read short: the first error was −4.26°, and the aim at $\rho = 0.9$ is +3.83°. A counter who expects only a spiral head — any pair with no common factor — admits 34/55 but also 34/53, 33/53, 33/52 and others read long, which together outweigh it by 30.3 per cent to 9.4. That counter concludes the first closing ran long by 5.04° and aims −4.54°, the opposite way. A counter who expects nothing admits the whorled heads as well, and aims by a quarter of a degree.

None of these is a caricature. The first is how the literature talks: a count that comes out 34/54 is “obviously” 34/55 miscounted. The second is what a careful counter might say who has read that most plants are spiral but distrusts the Fibonacci claim. The third is what a census is supposed to assume.

What one aimed recount reads #

The picture below takes a grown plant whose true pair is 34/55, first read as 34/54, and recounts it once under each belief. The dial sets the habit’s correlation.

With no habit the aim does nothing, since there is nothing of the first error to aim against, and every counter reads 34/55 35.1 per cent of the time. At a correlation of 0.9 the counters separate completely. Unaimed, the recount reads 34/55 41.8 per cent of the time and announces itself again 55.2. Aimed by the counter who expects Fibonacci, it reads 34/55 69.6 per cent of the time and almost never reads another pair silently — 0.2 per cent. Aimed by the counter who expects a spiral, it reads 34/55 5.5 per cent of the time and lands on the silent 33/53, a coprime pair that passes every check a single count offers, 31.5 per cent of the time. The agnostic counter’s recount is the unaimed one, within three points.

The same thing happens at every announced reading of 34/55. The four commonest are 35/56 and 33/54, each 28.9 per cent of first announcements, and 34/54 and 34/56, each 18.7. The Fibonacci-expecting counter aims −6.58°, +6.58°, +3.83° and −3.83° at them, and its recount reads 34/55 between 67.2 and 69.6 per cent of the time whichever it started from.

The habit alone #

Before the aim, the habit has an effect of its own, and it is worth separating because it answers what the earlier recount left open: whether its independence was an optimistic assumption.

It was optimistic. An unaimed recount repeats 34/54 exactly 9.4 per cent of the time when the two counts are independent and 24.7 per cent at a correlation of 0.9, and announces itself in some way 50.1 and 55.2 per cent of the time. The recount’s right share rises a little, from 35.1 to 41.8, because the habit narrows the recount’s fresh spread to $\sigma\sqrt{1 - \rho^2}$ — 3.1° at 0.9 — and the first error that read 34/54, between 3.3° and 5.3° short, lay only just outside the stretch that reads right. But the extra announcements cost more than that gains: the census at 34/55 needs twelve kept specimens with independent counts, fourteen at 0.75 and fifteen at 0.9, and its cost on grown plants rises from 24.4 counts to 33.0.

So a counter who recounts without aiming is worse off the stronger the habit, which is what one would guess and what the earlier recount could not say. A habit is not harmless once a specimen is counted twice: it spends the second count on the error the first one made.

What the belief does to the census #

The census cares about two things per specimen: the chance it is kept, and the chance a kept reading is Fibonacci. Aimed from the Fibonacci belief at a correlation of 0.9, a grown plant at 34/55 is kept 84.1 per cent of the time rather than 68.9, and its kept reading is right 82.1 per cent of the time rather than 69.0. Both move the census’s way.

At a correlation of 0.9 and a spread of 7.2°, the census at 34/55 needs fifteen kept specimens and 33.0 counts unaimed, eight and 15.3 aimed by the counter who expects Fibonacci, and seven and 14.5 by the same counter capped, which is described below. The counter who expects a spiral needs seventeen and 37.6: its aim sends grown plants to silent readings, which lowers the grown side’s Fibonacci share from 63.0 per cent to 56.1 and narrows the gap the census has to measure. The agnostic counter spends what the unaimed one does.

The advantage appears only where the habit is strong. At a correlation of a half the four counters are within two counts of one another; the Fibonacci aim buys most of its value above a half, where the first error carries enough of the second to be worth aiming against. A counter whose two closings on a head are barely related has nothing to aim at, whatever it believes.

Against counting low once #

The question the earlier recount ended on was whether a recount at 34/55 could ever undercut the census counted at 13/21 once, with every reading scored. At 7.2° that census spends six counts, and the answer does not change. The cheapest aimed recount at 34/55 spends 14.5 counts on its worse side; at 21/34, where the high count is least exposed, 9.6. At 3.6° the aimed and unaimed recounts at 34/55 all spend 8.4 counts against 13/21’s four, and at 10.8° the cheapest spends 32.6 against fourteen.

Across three spreads, five correlations, all four beliefs and the three pairs from 21/34 to 55/89, no single recount spends fewer counts on the worse side than 13/21 counted once. The low count’s advantage was never that its readings are easier to correct. It is that they rarely need correcting, since a closing error has to be half a spacing of the larger count before a reading moves, and at 21 that is 8.6°. An aim improves the correction; it cannot remove the need for one.

The belief is the hypothesis #

The Fibonacci-expecting counter’s advantage has a source, and it is the belief. For a grown plant at 34/55 the belief is true, so aiming from it works. But the census is not run on heads known to be grown. It is run to find out whether they are, and the counter does not switch beliefs between the two sides of the question.

The geometry’s side of the census is the share of divergence angles at a rise of 0.008 that produce each pair, and about a third of its heads are whorled: 3/6, 2/4, 4/8, 5/10. Counted with closing errors spread over 7.2°, a whorled 3/6 head reads 3/6 on every count, announces itself every time, and is set aside; that is what the whorled bucket in the frequency census was. A counter who expects Fibonacci reads 3/6 and asks which Fibonacci head could have produced it. The only one is 3/5, counted with a closing error of about 37°: nothing else is within reach. So the counter concludes the first closing ran 37° long, and at a correlation of 0.9 recounts 33.6° early. The recount then reads 3/5 — a consecutive Fibonacci pair — 68.9 per cent of the time.

The census’s null then moves. With the Fibonacci aim at a correlation of 0.9, the geometry’s heads read Fibonacci 28.5 per cent of the time among those kept, against 22.5 unaimed. The census as sized still separates the sides, but only because it sizes both with the same policy. An analyst who sizes it for unaimed counting and then collects readings from a counter who aims is testing a null that is no longer true of the readings.

What an analyst who does not know is told #

With a dozen specimens the damage is small — 9.5 per cent of true nulls rejected instead of five, at a correlation of 0.9 — because a dozen specimens cannot see a six-point shift. With thirty it is 11.8 per cent, with a hundred 40.5 and with two hundred 64.7. At a correlation of 0.75, 20.4 per cent at a hundred. The larger the survey, the more reliably it finds the Fibonacci plants the counter expected. That is the ordinary shape of a bias against a growing sample, and the census is exposed to it only because its counting has become a function of its hypothesis.

A counter who will not believe a large error #

The 3/6 case is extreme, and a real counter would not recount 34° early on the strength of an expectation. A fourth counter is read for that reason: it holds the Fibonacci belief but will not act on it past twice its own spread. When the belief says the first closing must have erred by more than 14.4°, this counter concludes that the head is not what it expected and recounts without aiming.

The cap costs nothing where the belief is right and nearly removes the damage where it is wrong. Its recount of 34/54 is the uncapped one exactly, since 3.83° is well inside the cap; its census at 34/55 needs seven kept specimens and 14.5 counts at a correlation of 0.9, the best of any policy read. It never aims at a whorled 3/6 head, which it keeps about five times in a hundred thousand. And the geometry’s Fibonacci share stays at 22.5 per cent, so a census sized for unaimed counting keeps its five per cent: 5.1 at a hundred specimens.

What the cap does not stop #

The cap stops the belief from reaching far, and the geometry’s heads, whose largest count is sixteen, sit far from any Fibonacci pair with a high count. Heads with high counts that are not Fibonacci are another matter, and the census as specified has none of them on either side. They are exactly where a real survey would find something: bijugate heads such as 34/56 — twice 17/28 — or a Lucas head’s double, 36/58, or a coprime head like 33/53 that belongs to neither sequence.

For each of those, the capped counter’s aim is a few degrees, well inside its cap, and it moves the reading toward the Fibonacci pair next door. A silent 33/53 head, read Fibonacci 12.6 per cent of the time by an unaimed recount at a correlation of 0.9, is read Fibonacci 24.1 per cent of the time by the capped counter. The bijugate 36/58 goes from 3.3 per cent to 9.5, 33/54 from 37.4 to 55.4, 35/56 from 52.8 to 70.0. Uncapped, the same heads go further: 33/53 to 32.1 per cent and 36/58 to 38.6.

The census does not see this because its null has no heads with high counts. A census that set grown plants against a population of Lucas or bijugate heads, as the whorled bucket turned out to hide, would see it at once, and would find its non-Fibonacci side read Fibonacci at twice the rate an unaimed counter reports.

Why the spiral belief fails #

The coprime-expecting counter is worth a paragraph of its own, because it is the belief a sceptic would hold, and it does worse than no belief at all. Its fault is not that it is wrong about plants. It is that “any head with no common factor” is a much larger set on one side of a reading than on the other. Around 34/54 the true pairs that read it long — 34/53, 33/53, 33/52, 32/51 — all have no common factor, while of those that read it short only 34/55 does: 35/55, 35/56, 36/56 and 36/57 share five, seven, four and three. The belief is symmetric in principle and lopsided in arithmetic, and the arithmetic decides the aim.

So the same belief would aim correctly at another reading and wrongly at this one, and which way it goes is a fact about the factors of 35, 36, 55, 56 and 57 — the same kind of fact that decided which report keeps its margin with two marks. A counter who wants to aim without the Fibonacci belief has nothing better to aim with, and should not aim.

What a counter should do #

Three rules follow, each a consequence of a number above rather than a preference.

Recount without aiming unless the belief being aimed from is already established for the population counted. The agnostic counter’s recount is the unaimed one, and it is the only recount that leaves both sides of a census where the census put them.

If aiming, cap the aim. A counter who will not believe an error larger than twice its own spread gets nearly all the benefit on the heads the belief describes and none of the absurd conversions — but that cap protects only the heads the belief is far from, and a survey whose alternative population has high counts is not protected by it.

And record the policy with every count. A reading from an aimed recount is a different measurement from a reading counted once, and a census cannot be sized, or its null stated, without knowing which it has. The control a survey would need listed what a specimen record must carry; the count’s own history belongs on that list.

What these readings assume #

The habit is one number per head, shared between two counts, and the counter knows its spread. A real counter’s habit may differ between the two families, as two marks chosen by one eye already had to allow, and a counter who does not know its own spread would size the cap wrongly. The beliefs are three clean ones, each weighting every admitted pair equally; a real counter’s is some mixture, and a belief that weights 34/55 heavily among the spiral pairs would behave between the first two.

Only one recount is read, and only the one-mark closing error. A count that drifts by two read errors that move one count alone, and an aim against those would need a model of which count moved, which the reading does not give either.

Readings that would undo it #

A reading of 34/54 whose candidate true pairs, weighted equally, put the first closing error on one side of zero by more than a degree. A correlation at which the unaimed recount repeats an announced reading less often than an independent one. Any spread, correlation and belief at which one aimed recount spends fewer counts than 13/21 counted once. A whorled 3/6 head kept by the capped counter more than once in a thousand. Each would mean the arithmetic here is wrong rather than incomplete; nothing in it is sampled.

Still open: a census whose other side has high counts #

Every non-Fibonacci head in the census as specified is small, which is why its null survives a capped Fibonacci aim. The heads that do not survive it — 33/53, 34/56, 36/58 — are the ones a survey of large capitula would meet. The next measurement is the census with its null drawn from heads the geometry produces at a finer rise, where counts run into the thirties and fifties and whorled heads are bijugate rather than 3/6, asking how large a share of Fibonacci readings a capped aim adds to that side, and whether any recount policy then keeps both the census’s size and its five per cent.