A count that drifts by two

What a count is worth measured what a reported parastichy pair pins the divergence angle to: a band about $221°/mn$ wide, so that a report of 34 and 55 fixes the angle to a little over a tenth of a degree. A count that can be wrong by one asked what that report is worth when either count may be off by one, and found two things. A high count that is wrong is not blurred but moved: 34/55 read with a tolerance of one allows three narrow bands, at 137.5°, 113° and 109°, costing $\log_2 3$ bits. And two Fibonacci pairs in every six, 21/34 and 34/55 among them, cannot be miscounted silently at all by a single error of one, because every such error lands on a pair sharing a factor.

Every tolerance there was one. A person tracing a family of eighty-nine spirals by eye — following each one round without an index — is as likely to lose two as one, and the arithmetic of a neighbour two away is different from the arithmetic of a neighbour one away. This essay repeats the reading at tolerances of two and three.

The reading being extended #

At a tolerance of one the picture was sparse and legible. 34/55 had two coprime neighbours, 33/56 and 35/54, whose bands sit at 109.23° and 113.27°, each about an eighth of a degree wide, twenty-four and twenty-eight degrees from the true band at 137.52°. The six other neighbours shared a factor with each other’s counts and announced themselves as whorled. 21/34 had two coprime neighbours, at 54.37° and 82.13°, and the same six announcing ones.

That picture is what a wider tolerance adds to. The question at each step is whether the new bands fall nearer the truth, whether they stay as narrow, and whether the arithmetic that made six of eight neighbours announce themselves carries over.

Eight neighbours, then twenty-four, then forty-eight #

A report of $(m, n)$ read with a tolerance $T$ allows every pair $(m + a, n + b)$ with $a$ and $b$ each between $-T$ and $T$: eight pairs besides itself at a tolerance of one, twenty-four at two, forty-eight at three.

Each is one of four things. Some are impossible, with the smaller count not smaller. Some share a factor, which makes them a whorled plant’s pair, and a counter who knows the plant is not whorled knows the count is wrong. The rest share no factor, and each of those is the two shortest families of some lattice at some divergence — and the new thing at a tolerance of three is that for a few of them that divergence is below twenty degrees.

A fourth kind of neighbour #

All of the survey’s scanning is done between 20° and 180°, which covers every divergence a plant has been recorded at. At a tolerance of one every coprime neighbour of a Fibonacci pair has its band in that range. At three, some do not. Around 21/34 the pairs 18/37 and 19/37 are the two shortest families nowhere between 20° and 180°; scanned lower, 18/37’s band is at 19.45° to 19.72°. Around 34/55 the pairs 37/55 and 37/56 are the same.

The reason is visible in the numbers. When the larger count is a little more than twice the smaller, the angle that makes both families short is about one turn divided by the larger count, and for counts in the high thirties that is under twenty degrees. A miscount that lands there reports a divergence nobody would accept, so it announces itself as surely as a shared factor does, and it is counted with the pairs that announce themselves.

What 34/55 allows at each drift #

Read with a tolerance of one, 34/55 allows the true band and two others: three bands, 0.354° in all. With a tolerance of two it allows thirteen bands, 1.550° in all, spread across 150° of the range. With three it allows twenty-nine, 3.445° in all, plus the two pairs below the range.

The total width grows tenfold between a tolerance of one and three, and it is still a small number — three and a half degrees scattered over a hundred and fifty. The report does not blur. It fragments, into many sharp candidates rather than a few, and each candidate is still about as narrow as the truth.

The nearest wrong answer did not move #

A fragmented report is still easy to use if every wrong fragment is far from the truth, because any rough second estimate of the angle then picks the right one. So the number that decides the practical cost is how near the nearest wrong band comes.

At 34/55 the answer is surprising. The nearest band a tolerance of one allows is 35/54’s, 24.13° from the edge of the true band. A tolerance of two adds ten more bands, and not one of them comes nearer: the nearest is still 35/54. Only a tolerance of three brings a nearer one, 37/54, whose band sits at 126.60°, 10.80° from the true band’s edge.

The other reports do not share that luck. At 21/34 the nearest wrong band is 55° away at a tolerance of one, 24° at two, and 1.73° at three, where 18/31’s band sits at 139.55°. At 13/21 the gaps are 54.6°, 4.6° and 0.9°. At 8/13 a tolerance of three produces a band overlapping the true one, and at 5/8 a tolerance of two already does.

Why the cost is only the count #

Counted as information, a report at 34/55 carries about ten bits about the angle when it is right. A tolerance of one costs 1.58 bits of that, a tolerance of two 3.71 and a tolerance of three 4.87. Those are the base-two logarithms of 3, 13 and 29 — the number of bands each tolerance allows — to within a fiftieth of a bit.

That equality is what “fragments rather than blurs” means, stated as a number. When a report allows $k$ bands that are all as narrow as the true one, the only thing lost is which of the $k$ it is, and a choice among $k$ things is $\log_2 k$ bits. Every Fibonacci report from 21/34 upward, at every tolerance, lies within 0.02 bits of that. The low reports lie above it, because their wrong bands are wider than their true one and overlap it.

A low count drifts into everything #

The same reading at the bottom of the sequence shows what fragmenting turns into when the bands are wide. A report of 5/8 read with a tolerance of one allows 35.2° of angle in five pieces. With a tolerance of two it allows 67.5° in eight pieces spanning 114°, and some of those wrong bands overlap the true one, so no second estimate can separate them. With a tolerance of three it allows 90.8°, spanning 126°, and one of the wrong bands — 3/5’s — contains the golden angle itself.

That last fact sounds reassuring and is the opposite. A 5/8 report that may have drifted by three is consistent with the golden angle through a pair it did not report, and consistent with most of the range besides, so it cannot be evidence for the golden angle or against it. What a count is worth already found low counts nearly worthless when right. Under drift they are worthless outright, and the loss is not a matter of degree.

What a second estimate now has to do #

The recommendation the one-step reading made was to count high and record one more thing: a rough angle, or the count in the next annulus. A rough angle settles a tolerant report when it is nearer the true band than any wrong one, so it has to be good to half the gap.

At 34/55 that was twelve degrees at a tolerance of one, and it is still twelve at two, which is the good news: a protractor laid on a photograph gives that. At a tolerance of three it is 5.4°. At 21/34 it goes from 27.5° at one, to 11.9° at two, to 0.86° at three — a precision no photograph offers and a round trip through the counts at a lower count would be needed for. So a rough angle rescues a drift of two at the high pairs and does not rescue a drift of three below 34/55.

The two sequences stop differing #

The one-step reading found a contrast between the two sequences of counts a head can show. From 13/21 up, six of a Fibonacci report’s eight neighbours share a factor and announce themselves; on the Lucas branch only two of 11/18’s or 18/29’s do, so most of a Lucas report’s miscounts pass silently. That was a strong statement about which counts a survey can trust.

It weakens with the drift. At a tolerance of two, 34/55 has twelve of its twenty-four neighbours sharing a factor and 18/29 has seven; at three, 34/55 has eighteen of forty-eight, 18/29 eighteen, 11/18 nineteen, 55/89 twenty-four. Every report from 11/18 up lies between 38 and 50 per cent at a tolerance of three. The share of all pairs of whole numbers that share a factor is $1 - 6/\pi^2$, 39 per cent. A wide enough drift samples the integers near the report broadly enough that the report’s own arithmetic stops mattering.

The Lucas pairs’ candidate bands behave like the Fibonacci pairs’ too. 18/29 allows 7, 18 and 31 bands at the three tolerances and loses 2.80, 4.15 and 4.94 bits, each within two hundredths of the logarithm of its band count; its nearest wrong band is 14.9°, then 6.4°, then 2.4° from the truth. So the distinction between the two branches that the one-step reading drew is a distinction about the integers adjacent to a Fibonacci or Lucas number, and a counter who drifts by two is no longer sampling only those.

The protection belonged to a drift of one #

The sharpest result of the one-step reading was about single miscounts: one count wrong, the other right. For most Fibonacci pairs two of the four single miscounts by one land on a pair that shares no factor and so pass as an ordinary report. For seven of the first twenty-three — 21/34, 34/55, 377/610, 610/987 and three more, two in every six — none does.

At a drift of exactly two, no pair among the twenty-three is protected. Every one has between two and four silent single miscounts, the seven included. At 34/55 they are 32/55, 36/55, 34/53 and 34/57; at 21/34 they are 19/34, 23/34 and 21/32, with only 21/36 sharing a factor.

Why two breaks what one kept #

The protection was arithmetic, and it can be read off the numbers. Moving 34 by one gives 33 and 35, and $33 = 3 \times 11$ and $35 = 5 \times 7$, while $21 = 3 \times 7$ and $55 = 5 \times 11$. Each neighbour of 34 shares a prime with 55, and each neighbour of 55 — 54 and 56 — shares one with 34. That is a coincidence of small primes that repeats every six Fibonacci pairs because of how Fibonacci numbers factor.

Moving 34 by two gives 32 and 36. $32 = 2^5$ shares nothing with 55, and $36 = 2^2 \times 3^2$ shares nothing with 55 either. Nothing in the structure that made 33 and 35 share primes with the neighbouring Fibonacci numbers reaches 32 or 36. The protection was a property of the nearest integers to a Fibonacci number, not of the pair, and a drift of two steps past it.

What drifting does to a survey #

The consequence for a census of heads is the one that matters. Take each count as wrong a twentieth of the time, and let a wrong count be off by two with some probability relative to being off by one. When it is never off by two, 34/55 is silently wrong 0.125 per cent of the time and 13/21 is 4.75 per cent — the thirty-eightfold advantage of the protected pairs. When drifting by two is a quarter as likely as drifting by one, 34/55 is silently wrong 2.5 per cent of the time. When it is as likely, 9.5 per cent.

At that point 34/55 is silently wrong exactly as often as it announces itself — 9.5 per cent each — and not much less often than 13/21, at 11.9. The arithmetic that made high counts safe against silent error is gone. A survey that trusts its high counts because a census taken high almost never manufactures a non-Fibonacci ordinary pair is trusting a property that holds only if its counters never drift by two.

The other half of the same arithmetic runs the opposite way. A miscount that shares a factor is recorded as a whorled pattern, and at a drift ratio of one 34/55 produces such a report 9.5 per cent of the time — close to the one head in ten the one-step reading found. Opening the census’s whorled bucket found it full of genuinely jugate pairs, k and 2k; a drifting counter adds pairs that share a factor without being k and 2k at all, and the pattern that arrives two at a time is the check that tells the two apart. How often a census is Fibonacci therefore depends on counting errors in both directions at once: silent non-Fibonacci pairs added, Fibonacci pairs moved into the whorled bucket.

The second annulus still works #

The other check survives intact. Counting two adjacent annuli, $(a, b)$ and then $(c, d)$, a golden stem’s pairs continue each other when $c = b$ and $d = a + b$. Every one of the four counts appears in that relation, so a single error of any size breaks it: a drift of two in one count is caught as surely as a drift of one.

Only combinations pass. Reading 21/34 with 34/55, a wrong reading passes 0.106 per cent of the time when drifting by two is as likely as by one, against none at all when counts only drift by one. Reading 34/55 with 55/89, it passes 0.110 per cent of the time. Those are nearly a hundred times smaller than the silent rate of a single annulus at the same drift.

What counting further out now costs #

The recommendation changes shape again. Counting high is still worth it, and a rough angle still resolves a report that may have drifted by two, at 34/55 and above. What no longer holds is the idea that some high pairs are safe without a check. A count carries no error argued that a published count should say how it could be wrong; drifting by two is one of the ways, and the only check that survives it is a relation that ties every count to another.

So the record that makes a high count trustworthy is two annuli rather than one, whichever pair is reported. That was optional under a drift of one at 21/34 and 34/55. Under a drift of two it is the whole of the protection there is.

What this does not say #

It does not measure how often counters drift by two, or by three. The drift ratio here is a parameter swept from none to one, and where a real counter sits on it is a question about people counting spirals, which is not measured here.

It does not treat drifts in the two counts as related. A counter who loses track of the 55-family has probably lost track of the 34-family in the same way at the same moment, and correlated drifts land on different pairs than independent ones.

And the lattice is ideal. The counts change with radius, so a real annulus near a transition shows two pairs at once, and that ambiguity is a different error from a drift.

What would withdraw it #

A Fibonacci pair among the first twenty-three with no silent single miscount by two. A report from 21/34 upward whose tolerance costs more than a fiftieth of a bit beyond the logarithm of the bands it allows. A single miscount of any size that passes the two-annulus relation. A tolerance of two at 34/55 bringing a wrong band nearer than 35/54’s.

Still open: drifts that move together #

Every drift here is independent between the two counts. The failure that seems likeliest in practice is the one that is not: a counter who skips a spiral in one family while tracing it round the head skips the crossing spiral too, so the two counts move together, by the same amount or in a fixed ratio. Pairs such as 33/54 and 35/56 — both counts down, or both up — are then more likely than 33/55 or 34/56. The measurement is the same tolerant reading with the two drifts coupled: which of the coupled neighbours share a factor, whether any Fibonacci pair is protected against a coupled drift, and whether the two-annulus relation still catches it.