The high heads the geometry rarely makes

A Fibonacci census scores each specimen by its counted parastichy pair and asks whether the share that reads as consecutive Fibonacci numbers is the geometry’s or the ninety per cent a grown history gives. The geometry’s share is not a number anyone measured on plants. It is the share of divergence angles that give a Fibonacci pair on a cylinder of stated rise, and at the rise the census was specified at, 0.008, it is 14.7 per cent, from heads whose largest count is sixteen.

The recount aims where the counter expects found that small number carrying more weight than it looked. A counter who recounts an announced reading and aims the second count at the Fibonacci pair they expect will read the geometry’s whorled 3/6 heads as an overrun 3/5 and raise the null’s Fibonacci share; capped at twice their own spread, the same counter leaves the null alone. But the cap left the null alone only because every non-Fibonacci head in it is small. The heads the capped counter does move — a silent 33/53 read as Fibonacci 24 per cent of the time rather than 12.6, and 34/56 and 36/58 with it — are the heads a survey of large capitula would meet, and the null as specified contains none of them. The measurement here draws the null from the geometry’s heads at finer rises, where the counts run into the thirties and forties, and asks what a capped aim then adds, and whether any recount policy keeps both the census’s size and its five per cent.

What the null is made of at a finer rise #

Halving the rise raises the counts by about the square root of two. At 0.008 the geometry’s heads count to sixteen; at 0.004, to 22; at 0.002, to 32; at 0.001, to 45, and there 23.5 per cent of the null is spiral heads with a count of thirty or more; at 0.0005 and 0.0003, to 64 and 82, with half the null’s spiral heads past thirty. The null’s other shares move less than one might expect. Whorled heads — whose two counts share a factor, 5/10, 7/14, 11/22 — grow from 35.4 to 43.7 per cent, and the rest of the null is spread over more and more distinct pairs — 48 at the coarsest rise, 1,007 at the finest read here, 0.0003, where the counts reach 82. And the consecutive Fibonacci heads fall, from 14.7 per cent to 4.7 at 0.001 and 2.5 at 0.0003, almost all of what is left being 1/2 at angles near a half turn: at a rise of 0.001, 21/34 and 13/21 together are two tenths of a per cent of all angles.

That last change is the one that decides the census. The null’s Fibonacci share is the thing the grown plants have to beat, and at a fine rise there is far less of it to beat — which is the observation Fibonacci is a branch made from the angles’ side: the Fibonacci readings are one branch of the angles among many, and a finer rise divides the angles among more branches, each taking its share of the half turn. The null’s Fibonacci share is not a property of plants; it is a property of how finely the surveyor’s band divides the angles, and it should be quoted with the rise it was computed at.

The census gets easier #

The counts are the same as before: closing errors spread over 7.2°, the two counts of one head correlated at 0.9, one recount of an announced reading. Recounted unaimed, the null’s Fibonacci share among kept specimens falls from 22.5 per cent at a rise of 0.008 to 16.0, 11.0 and 7.5 at the finer rises, while the grown plants’ stays at 63 per cent; at 0.0005 and 0.0003 the null reads 5.1 and 3.7. The census needs fifteen kept specimens at 0.008, nine at 0.004, seven at 0.002 and five at each finer rise.

The capped aim does exactly what it did at 0.008, at every rise. It leaves the null where the unaimed recount put it — 22.5, 16.0, 11.0 and 7.5 per cent, to the tenth, and 5.1 and 3.8 at the finest two — and lifts the grown plants from 63 to 75 per cent, because a grown plant’s announced reading of 34/54 or 33/54 is exactly the reading a capped aim is built to correct. The census then needs seven, seven, six and four kept specimens, and four at the finest two rises. The uncapped counter lifts both sides, the null to 28.5, 21.1, 15.0 and 10.6 per cent and on to 5.4 at the finest rise: the same overrun of whorled heads read as the Fibonacci pair beside them that the earlier essay found in 3/6, spread now over more whorled heads.

So the question the earlier essay left open has an answer, and it is not the one feared. A null drawn from high-count heads does not make the census harder. It makes it easier, and the capped aim is still pure gain on the grown side.

Why the capped aim finds nothing to move #

The earlier essay’s high heads were one count off a Fibonacci pair: 33/53 is 34/55 closed three per cent short, 34/56 is 34/55 with its larger count one over. Those are the heads a capped aim moves, because the closing error that would have made them out of a Fibonacci pair is within twice the counter’s spread. The geometry’s high heads are not like that.

At a rise of 0.001 the null holds thirteen heads whose ratio is within five per cent of the golden ratio but which are not consecutive Fibonacci numbers, 2.7 per cent of the null between them. Seven of the thirteen — 16/25, 16/27, 17/27, 13/22 and others — are not what any closing error within a fifth of a turn makes out of a Fibonacci pair at all. The other six are, but from far away: 17/28 is 21/34 closed 17.6 per cent short, 18/29 is 14.7 short, 19/30 is 11.8 short, 19/31 is 8.8 short, 14/23 is 13/21 closed 9.5 per cent long. One, 20/33, is 21/34 closed 2.9 per cent short, inside the cap, and it is 0.17 per cent of the null.

The heads that sit within the cap’s reach are rare at every rise, and the rise decides which they are. None does at 0.008, 0.004 or 0.002. At 0.001 it is 20/33 alone, 0.17 per cent of the null; at 0.0005, 22/35 and 20/33, a quarter of a per cent. Only at 0.0003, where counts reach 82, does the geometry make a head one count off 34/55: 33/53, which a cylinder reads as its dominant pair at divergence angles near 54.4° — nowhere near the golden angle — and 35/57 beside it, 0.11 per cent of the null together. The capped aim does move them there, exactly as it moved them before, and the null’s Fibonacci share rises by four hundredths of a point for it: 3.73 per cent unaimed, 3.77 capped.

So the silent near-miss heads the earlier essay worried about are real heads of the geometry, but they are a sliver of it at every rise. A near-golden ratio in the null is common — two to three per cent at every rise from 0.002 down — and a near-golden ratio a single closing error away from a Fibonacci pair is not, because the non-Fibonacci heads with golden-looking ratios sit on other branches of the continued fractions, 17/28 or 19/30, whose nearest Fibonacci pair is ten or fifteen per cent away.

The five per cent is lost to the habit #

The census’s size came through. Its false-alarm rate did not, and the aim had nothing to do with it.

An analyst sizes the test for the null’s Fibonacci share under independent counting: 6.8 per cent at a rise of 0.001. If the counter’s two counts of one head are correlated at 0.9, the null’s kept share is 7.5 per cent instead, whether or not the recount is aimed, and a true null is rejected 7.1 per cent of the time in a census of thirty, 7.3 in a hundred and 7.8 in two hundred — recounted unaimed or aimed and capped, to within a few tenths. At the specified rise of 0.008 the capped counter’s census held five per cent; at 0.001 it does not, and nor does the unaimed recount beside it. At 0.0005 a census of two hundred rejects a true null 8.7 per cent of the time unaimed and 8.9 capped, and at 0.0003, 7.3 and 7.7; smaller censuses at the finest rises happen to sit near five per cent only because so few Fibonacci readings are expected that the test’s cut moves in whole specimens.

The mechanism is the habit’s, and it is quiet. A head whose first reading announced itself was, more often than not, read through a habit that would announce it again, so the recount sets aside a different mix of heads than independent counting would. Among the geometry’s heads, those that announce are mostly whorled, and removing more of them raises the Fibonacci share of what is left. At a coarse rise the whorled heads’ second readings were mostly announced anyway, and the effect vanished in rounding; at a fine rise there are more whorled heads and more high counts, and it shows. The uncapped counter, for comparison, rejects a true null 20.6, 36.7 and 54.7 per cent of the time — the overrun effect, as large at a fine rise as at a coarse one.

Setting aside keeps it, at a price #

There is one policy whose five per cent does not depend on the counter’s habit: count once and set announced readings aside. One count carries nothing from count to count, so its null share is the same at every correlation.

Setting aside needs fifteen kept specimens at a rise of 0.008 and 29.5 counts to get them; nine and 17.8 at 0.004; seven and 13.9 at 0.002 and 0.001 alike; five and 9.9 at the two finest rises. The capped recount needs seven, seven, six and four kept specimens, for 14.5, 15.7, 13.8 and 9.2 counts, and four for about nine counts at the finest two. At a coarse rise the capped recount is far cheaper and keeps its five per cent; at 0.002 the two cost the same; at 0.001 the recount is a third cheaper and rejects a true null 7.4 per cent of the time; at the finest rises the difference is one kept specimen and one count.

So the answer to whether any recount policy keeps both the census’s size and its five per cent is that at a fine rise none does, unless the analyst knows the counter’s habit and sizes the test against the null the habit produces — 7.5 per cent rather than 6.8. A counter’s correlation between two counts of one head is not a number anyone has published, and two marks chosen by one eye found no way to read it from the counts themselves. Setting aside needs one to three more kept specimens and costs nothing in trust.

The spread decides more than the policy #

Everything above is at one closing spread, 7.2°, the value the earlier essays carried. The spread turns out to matter more than any choice of policy.

At half that spread, 3.6°, a census against the null at a rise of 0.001 needs three kept specimens whatever the counter does: set aside, recounted or aimed. Grown plants read Fibonacci 91 per cent of the time unaimed and 93 capped, because a closing error of 3.6° rarely moves 34/55 at all, and the null reads 8.1. At half again the spread, 10.8°, the same census needs fourteen specimens set aside, eleven recounted unaimed and eight recounted with a capped aim, since a grown plant now reads Fibonacci only 43 per cent of the time unaimed and 58 aimed. The null barely moves across the whole range, 7.1 to 8.1 per cent: its heads are mostly whorled or far from Fibonacci, and a closing error cannot turn them into anything the census counts.

So the size of a census against a high-count null is set almost entirely by how often grown plants survive their own counting, which is the argument the census wants a low count made about the alternative and a count that can be wrong by one made about a single report. What the policies change is the price of a poor count: a capped aim buys back most of what a wide spread costs the grown side, and it does so without touching the null. A counter who does not know their own spread should plan the census at the wide end. A counter who knows it is under four degrees can forget about recounting altogether, since at that spread counting it again buys nothing a single count does not already give.

What this changes about the census #

The census wants a low count argued from the alternative’s side: grown plants counted high lose their Fibonacci reading first, so a census should count low pairs. The null’s side says the opposite, weakly: the geometry’s heads read at a fine rise have less Fibonacci to offer, so the census is cheaper against them. The two arguments are about different rises. The null’s rise is the size of the band a surveyor counts in, and a survey of large capitula counted at 34/55 is implicitly comparing against a null at a fine rise — where it needs fewer specimens than the specified census, not more, and where the capped recount helps. What it must not do is recount with a habit it has not measured and then use a test sized for independent counts.

What this does not establish #

That a cylinder’s dominant pair at a fine rise is what a large non-Fibonacci capitulum shows. A seed head is a disc read in a band, not a cylinder, and the rise here stands in for the band’s size. That a real non-Fibonacci plant’s divergence is spread evenly over the half turn, which is what drawing the null from every angle assumes — the control a survey would need asked for exactly that population and it has not been measured. And that the rises read here are the ones that matter: a cylinder at 0.0003 counts to 82, past any capitulum a survey is likely to count at 34/55, and nothing finer was read.

What would overturn it #

A null drawn from the geometry at a fine rise in which the capped aim moves the Fibonacci share by more than a tenth of a point. A cylinder’s null, at any rise, whose heads within a capped aim’s reach of a Fibonacci pair are more than a fraction of a per cent of it. A census at a fine rise, set aside rather than recounted, whose false-alarm rate depends on the counter’s correlation. Each would mean the account here of which heads the geometry makes is wrong.

Still open: the null read on a disc #

Every null in this census is a cylinder’s: one rise, one dominant pair per angle. A capitulum is a disc whose rise falls toward the centre, and a surveyor counts it in a band where the pair changes across the band’s width — which is how the counts change up a head in the first place. The next measurement draws the null from discs grown at every divergence angle and counted in a band the way a surveyor would, at several band radii: whether a disc’s null holds more of the heads one count off a Fibonacci pair than a cylinder’s sliver, since a band straddling a transition can mix two pairs into a reading that neither pair is, and if it does, whether the capped aim finally finds something on the null’s side to move.