Two marks chosen by one eye

A reported parastichy pair pins a plant’s divergence angle to a band about $221°/mn$ wide, and what a count is worth made that the reason to count high: 34 and 55 fix the angle to about a tenth of a degree. Three essays since have asked what the report is worth when a count is wrong, and they differ in one assumption. A count that can be wrong by one and a count that drifts by two let each count err on its own. Two counts that slip together tied them to one closing error: a counter who runs past the starting mark by a fraction $\varepsilon$ of a turn counts both families round $1 + \varepsilon$ of the head, and both counts move together.

The two assumptions gave opposite advice. With independent errors the second annulus — a second ring of the head, whose pair must continue the first — caught every single error of any size, and a rough estimate of the angle was a help at the high pairs only. With one shared closing error the second annulus passed a sixth of the wrong readings of 34/55 and three quarters of 13/21’s, and a protractor good to twelve degrees did the work instead.

Neither assumption describes a person counting. To count a family, a counter picks an organ, follows the spirals round the head and stops on returning to it. The 34-family and the 55-family are traced separately, usually from different organs, so there are two closing marks and two closing errors. One eye chose both, and the habits that make it close one family late — a crowded rim, a mark hard to find again — make it close the other late too. The errors are correlated, but not equal. This essay reads the report across the whole range between, and asks where the advice changes.

One correlation, four marks #

The model needs one new number. Each closing mark errs by a normal amount of spread $\sigma$, and every pair of marks is correlated by $\rho$: the error is a part shared by every mark the counter makes and a part of the mark’s own, weighted so that the two parts’ variances are $\rho$ and $1 - \rho$ of the whole. At $\rho = 1$ every mark is the same mark and the model is the one-mark reading exactly. At $\rho = 0$ the four marks — two families on the first annulus, two on the second — are independent.

Given the shared part, the marks are independent, so every share below is one integral over the shared part of products of normal interval masses. It is taken numerically on sixteen hundred points and needs no sampling. At $\rho = 1$ the integral degenerates, and that end is read from the one-mark essay’s exact interval sum; at $\rho = 0.9999$ the integral already agrees with it to three decimals, which is the check that the two computations are of the same thing.

The picture is the whole of the model. Each axis is one family’s closing error, and each rectangle is the set of pairs of errors that round to one reported pair. The one-mark counter lives on the diagonal. A counter with two marks lives in the ellipse, which is a circle at $\rho = 0$ and closes onto the diagonal as $\rho$ rises. Everything this essay finds is a statement about which rectangles the ellipse covers.

The rectangles off the diagonal #

On the diagonal, the rectangles next to the true one are the readings the one-mark essay found: 34/54 and 34/56 when the 55-family moves first, then 33/54 and 35/56 when both move. All four share a factor — two, two, three and seven — so a reading of 34/55 closed at one wrong mark announces itself until the mark is 9.82° out.

Off the diagonal sit the readings one mark cannot reach: the 34-family moved one way and the 55-family the other. They are 33/56 and 35/54, and neither shares a factor. 33 is three times eleven and 56 is eight times seven; 35 is five times seven and 54 is twice twenty-seven. Both pass as ordinary reports, and they report 109.2° and 113.3° — the same two sharp bands the single-miscount reading found beside 34/55, twenty-four degrees from the truth, reached now by a different route.

To land there, the 34-family’s mark has to be out by half a spacing of that family, a sixty-eighth of a turn, 5.29°, and the 55-family’s by at least 3.27° in the other direction. So two marks cost 34/55 its margin: the smallest closing error that makes it silently wrong falls from 9.82° to 5.29°, on the larger of the two marks. The price is the direction. The two errors must fall on opposite sides of the start, one family closed late and the other early, and that is exactly what a correlated eye makes rare.

Whose margin survives #

The same question can be asked of every Fibonacci report, and the answer is arithmetic rather than anything about counting.

21/34 keeps its margin of 8.57° with two marks. Its off-diagonal neighbours are 20/35 and 22/33, which share five and eleven, so moving its families in opposite directions announces itself just as moving them together did. 13/21 keeps 8.57° as well, because its first silent readings, 13/20 and 13/22, are on the diagonal already. 55/89 loses more than 34/55 does: its margin falls from 9.82° to 3.27°, because 54/89 and 56/89 share nothing, and a counter with two marks can move the 55-family by one while the 89-family stays put — something one mark cannot do, since 89 spirals cross a sliver of the head before 55 do.

The run of reports up the sequence also corrects something the one-mark essay said. It found that 34/55 and 55/89 each held a margin of about ten degrees, and read that as a property of high counts. It is not. From 89/144 upward every margin is under a degree and a half, with one mark or two: 89/144 goes silent at 1.25°, as 89/143, because 143 is eleven times thirteen and shares nothing with 89; 233/377 at 0.48°. A margin is half a spacing of whichever count moves first into a coprime pair, and 34/55 and 55/89 were protected only because the larger count’s neighbours, 54 and 56, 88 and 90, happen to share factors with the smaller. Two marks shorten the margin at four of the twelve reports — 34/55, 55/89, 610/987 and 987/1597 — and lengthen it at none.

What the correlation does to the readings #

A margin says where silence begins. How often a counter lands there depends on the spread, and the one-mark essay’s middle spread, 7.2° a mark, is where both effects show clearly.

With independent marks 34/55 is read right 18.9 per cent of the time, announces itself 55.6 per cent and is silently wrong 25.6 per cent. With one mark it is right 35.1 per cent, announced 50.1 and silent 14.9. Correlation helps twice over: it keeps the two families’ errors on the same side, where 34/55’s neighbours share factors, and it makes a right reading likelier outright, because on the diagonal both families have to be within half a spacing of the start together rather than separately. Of the wrong readings, the share that is silent falls from 31.5 per cent to 22.9.

The same bars for 21/34 run the other way, and the reason is the arithmetic of the last section, seen as a frequency.

With independent marks 21/34 is silently wrong 6.2 per cent of the time; with one mark, 20.7 per cent. Its same-sided neighbours, 20/33 and 22/35, share nothing, and a correlated eye walks straight into them. Of its wrong readings, the silent share rises from 10.5 per cent to 44.8. So whether it is better for a counter’s two marks to agree depends on the report: for 34/55 agreement is protection, for 21/34 it is exposure. No counting habit is safe for every pair, and which way a habit cuts can be read off the counts before any head is counted.

Where the second annulus goes blind #

The check that the independent readings relied on is a second annulus. Reading $(a, b)$ in one ring and $(c, d)$ in the next, a golden head’s pairs continue each other when $c = b$ and $d = a + b$. The one-mark essay found that a shared closing error passes that relation whenever rounding lets it, since a linear relation survives multiplying every count by $1 + \varepsilon$.

The fair way to judge a check is not how many wrong readings it rejects but how differently it treats right and wrong ones. A reading that shares a factor is set aside whatever else is done, so what the relation has to separate is a right reading from a silent one. Its worth is the ratio of the two acceptance rates: the share of right readings it passes, over the share of silent readings it passes. Above one, passing the relation is evidence for a reading. At one it is a coin.

With independent marks, at 34/55, the relation passes right readings 2.84 times as readily as silent ones. At a correlation of a half, 2.15 times. At 0.9, 1.25. It crosses one at a correlation of 0.978, and with one mark it is 0.82 — below one, so a reading that passes the relation is slightly more likely to be silently wrong than one that fails it. 13/21 starts much higher, at 16.3, and falls further, going blind at 0.945. 21/34 goes blind at 0.982 and 55/89, which starts lowest at 1.72, only past 0.998.

Almost the whole loss happens in the last tenth of the correlation, and that is worth stating plainly because it cuts against intuition. A counter whose marks are half-correlated — a strong habit — has lost a quarter of what the relation was worth at 34/55. What destroys the check is marks that are nearly the same mark, which is what a counter produces by closing both annuli, and both families, at one conspicuous organ.

At a narrower spread the picture keeps its shape. With marks spread over 3.6°, 34/55 goes blind at 0.991, and 21/34 does not go blind at all: even with one mark its relation passes right readings twice as readily as silent ones.

Two ways through the relation #

One detail of the 3.6° curve does not fit the story so far. At 34/55 the relation’s ratio is 4.44 with independent marks, rises to 12.1 at a correlation of a half, and only then falls, to 0.87 with one mark. If agreement between the marks were simply bad for the relation, the curve would fall from the start.

The explanation is that a silent reading has two ways through the relation, and each end of the correlation opens one. The relation asks two things of the second annulus: that its smaller count repeat the first annulus’s larger one, and that its larger count be the sum. The readings independent marks reach first, 33/56 and 35/54, are opposite-sided errors, and an opposite-sided error keeps the sum: $33 + 56 = 35 + 54 = 89$. The second annulus’s 89-family has nothing to disagree with, and all that has to happen for the reading to pass is that its 55-family repeat the first annulus’s error. With independent marks at 3.6°, 99.9 per cent of the silent readings the relation passes are those two; at 7.2°, 78.6 per cent.

The one-mark route is the one the earlier essay found — multiply every count by the same factor and hope the rounding agrees — and it passes 33/53 and 35/57, which do not keep the sum. By a correlation of 0.75 the sum-keeping route carries under one per cent of what passes, because the marks now rarely err in opposite directions, while the shared-mark route has not yet opened, because they still rarely err by the same amount. The relation discriminates best in that gap, where both routes are narrow.

21/34 has no such gap to exploit, and does not need one. Its sum-keeping readings one step out are 20/35 and 22/33, which share factors, so the first route is closed; the first silent readings that keep its sum, 19/36 and 23/32, are two steps out in each family and carry under one passing reading in ten thousand.

What the relation keeps #

A ratio above one is not the same as a useful check, because the relation can be harsh on right readings too. At 34/55 the second annulus’s larger count is 89, and 89 spirals cross a sliver of the head so often that the 89-family gains or loses one at a closing error of two degrees.

With independent marks spread over 7.2°, nine right readings in ten fail the relation, because their second annulus’s 89 has moved. It keeps 2.2 per cent of all readings, and of those 67.6 per cent are right, against 42.4 per cent of the non-announcing readings before it was applied. So it does improve the count — but a counter would need about forty-six counts of a head to keep one reading, and a third of the readings kept would still be wrong. At a correlation of 0.9 it keeps 14.5 per cent, about one reading in seven, and moves the right share from 66.2 to 71.1 per cent. With one mark it keeps a third of all readings and lowers the right share, from 70.2 to 65.9 per cent.

So the relation is caught between two failures. Where the marks are independent enough for it to discriminate, it discards nearly everything, including nearly every right reading. Where the marks are correlated enough for it to keep readings, it cannot tell the right ones from the silent ones. There is no correlation at which it both keeps a fair share of readings and doubles the odds that a kept reading is right.

The protractor the relation is worth #

The alternative the drift essay set against the second annulus is a rough estimate of the angle, from a photograph or from the counts at a lower ring. A protractor good to $T$ degrees cannot separate a silent reading whose band lies within $2T$ of the true band and separates every other one. It rejects no right reading at all, which already puts it ahead on what it keeps. The comparison left is how rough it can be and still reject as many wrong readings as the relation.

At 34/55 with independent marks the relation is worth a protractor good to 12.1°: the silent readings that marks spread over 7.2° reach most often are 33/56 and 35/54 at 109.2° and 113.3°, about twenty-four degrees from the truth, and a protractor coarser than twelve degrees lets them through. As the marks come to agree those readings become rare, and the relation’s equivalent coarsens — 21.1° at a correlation of 0.75, 27.6° at 0.9, 41.5° with one mark. At 55/89 with independent marks the relation is worth one good to 5.4°, a demanding instrument, because 54/89 and 56/89 sit at the same two bands and are reached by moving the 55-family alone.

Read the other way, the line says when a counter should stop trusting the second annulus. A protractor good to twelve degrees — a line laid across a photograph with a school protractor — beats the relation at 34/55 for every correlation of the marks, and beats it at 13/21 and 21/34 by a wide margin. Only at 55/89 with marks that are nearly independent does the relation earn its place, and there it keeps one reading in a hundred.

What this reading assumes #

It gives every pair of marks the same correlation. A counter may close the two families of one annulus at nearly the same organ and the two annuli at quite different ones, which would be two correlations rather than one, and the relation’s worth depends on the second. The model has one number where a real habit might need two, and no measurement of real counters to fix either.

It keeps the lattice ideal, so that a family’s spirals are spread evenly round the head and a sliver holds a fair share of each. Near a transition the counts change with radius, and an annulus holds three families unevenly — a coupling neither this model nor the one-mark model contains.

And it measures what a check does to one reading of one head. A survey reads many heads, and the census that wants a low count found that the count which pins an angle best is the one a census should avoid. Whether two marks change that ranking is a separate calculation, since the census cares about the silent share and not about the angle a silent reading reports.

Readings that would overturn it #

A Fibonacci report whose opposite-sided neighbours at a step of one both share a factor and whose margin nonetheless falls with two marks. A correlation below 0.9 at which the relation passes silent readings of 34/55 as readily as right ones, at a spread of 7.2°. A silent reading of 34/55 within twenty-four degrees of the true band reached by marks that each err less than eight degrees. Any of them would mean the arithmetic here is wrong rather than incomplete; nothing here depends on a sample.

Still open: marks that are chosen, not drawn #

Every mark here is a random draw. A careful counter does not choose a starting organ at random: they choose one that is easy to find again, usually on the rim or on a conspicuous spiral, and that choice is where both families’ closing errors come from. The next measurement would model the mark as a choice — the organ nearest a fixed place on the photograph, for each family — and ask whether the correlation it produces is a property of the head’s geometry, fixed by where the two families’ spirals cross the rim, rather than a habit of the counter. If it is, the correlation can be computed for each head rather than assumed, and the protractor the relation is worth becomes a number a survey could print beside every count.