A count that is not exact

Every reading in this thread has taken a count of growth lines as a number. On a real section it is a judgement: lines are missed where they crowd, doubled where a sub-daily increment looks like a daily one, and gone altogether where the shell is worn. So the question is what an inexact count does to the law it names — and the answer divides sharply in two, with most of the obvious worries on the harmless side and one specific worry on the other.

The arithmetic that sorts the worries #

The reading is $\hat p = \log(c_2/c_1)/\log W$. Multiply both counts by the same factor and $\hat p$ does not move at all. So a miscount that is the same everywhere costs exactly nothing, however large it is — and the size of “however large” is worth seeing rather than accepting.

What does bias the answer is a miscount that varies from whorl to whorl, and it biases it by

$$\Delta \hat p = \frac{\log (f_2/f_1)}{\log W}$$

where $f$ is the share of lines each whorl keeps. That single line is the whole of the error analysis for this instrument, and it points somewhere unexpected. It says the question to ask about a section is not how carefully it was counted. It is whether the difficulty of counting it changed along the shell.

What being counted badly, but evenly, costs #

Missing a share of the lines at random is the most natural way for a count to be wrong, and it is free. Over twenty-four seeds the mean reading is 0.9998, 1.0007, 1.0034 and 1.0049 at five, twenty, fifty and eighty per cent missing, against a law of exactly 1. Four lines in five gone moves the answer by five thousandths.

What grows instead is scatter: the standard deviation over seeds runs from 0.0067 to 0.0568 across the same range, and the worst single seed at eighty per cent is out by 0.112. That is the right shape for an error with no direction in it, and it is also the reason the reading survives a bad section at all — a scatter of a tenth still names a whole power correctly, because the laws are whole numbers a full unit apart.

Counting a share of the lines twice is free in the same way and for the same reason. At a tenth doubled the reading is 1.0014 and at four tenths it is 0.99996, because a multiplier in both counts divides out whichever direction it points.

Why the reading tolerates so much, when the other instruments on the same shell do not #

The tolerance is worth comparing with what else this subject has priced, because it is unusual. What the centre costs found that displacing an assumed centre by a quarter of the innermost whorl’s radius moves a fitted growth factor by 4.56 per cent — a small geometric error producing a real one. How far a centre must move found centres that turn a shell grown at 3.2 into a golden spiral outright. Those are instruments that read positions, and a position carries every error in the coordinate system it was measured in.

A count carries none of them. It has no centre in it, no origin, no assumed plane and no scale; it is a number of things, and the only way to get it wrong is to see the wrong number of things. That is why a missing fifth of the record costs almost nothing here and a misplaced centre costs a fitted growth factor several per cent.

The same property is what makes the one remaining failure so sharp. Everything that could go wrong with a count has been reduced to a single question — did the counting take the same share from both whorls — and a single question with one answer is either safe or badly wrong, with nothing in between.

Which end of the shell the lines crowd at #

The miscount that does vary is a resolution limit, and to see what it does the lines have to be located rather than counted. The arc between successive lines is

$$\Delta s \propto e^{k(1-p)\theta}, \qquad k = \frac{\ln W}{2\pi}$$

so the exponent changes sign at $p = 1$. Under a clock shallower than a length clock the lines spread outward: at an angular clock the spacing runs 2.42 × 10⁻⁴, 7.74 × 10⁻⁴, 2.48 × 10⁻³, 7.92 × 10⁻³ across four whorls, a factor of 32.8 — which is $W^3$, and not a coincidence. Under a steeper one they crowd outward: at a volume clock, 2.31, 0.217, 0.0209, 0.00204.

At exactly $p = 1$ the spacing is the same in every whorl — 2.853 × 10⁻³ throughout, to the last digit stored. That is what “a constant length added at the opening” means when it is said in terms of the shell rather than the animal, and it is about to matter a great deal.

What a limit leaves #

A section that cannot separate two lines closer than four thousandths of the shell’s outer radius loses 73.3 per cent of an angular clock’s record, 66.6 per cent of a length clock’s and 72.8 per cent of an area clock’s. Those are similar numbers, and they are not the point. The point is where the losses fall.

The angular clock keeps 2.9, 8.8, 25.0 and 70.0 per cent of its four whorls — a factor of twenty-four across the shell. The area clock keeps 100, 100, 61.5 and 23.0 per cent, sloping the other way. The length clock keeps 34.1, 33.2, 33.3 and 33.3 per cent, which is flat.

Those slopes are the bias, by the formula above and by nothing else. The angular clock reads 0.886 where the shell had 0. The area clock reads 1.154 where it had 2. The length clock reads 1.0011 where it had 1.

The limit’s answer is the shell’s own geometry #

The sweep at the top of this essay is the same reading at every limit, and its shape has a closed form behind it rather than a trend.

Where the limit binds completely, the surviving count in a whorl is not a property of the animal at all: it is that whorl’s arc length divided by the limit, because every pair of surviving lines is exactly one limit apart. A logarithmic spiral’s whorl arcs stand in the ratio $W$ exactly — the arc from the apex to angle $\theta$ is $\sqrt{1+k^2}\,(e^{k\theta}-1)/k$, and differencing it over successive turns leaves a geometric sequence in $W$.

So a shell whose lines are everywhere unresolvable gives counts in the ratio $W$, which reads as $\hat p = 1$: a length clock, whatever the animal was doing. An angular clock at a limit of 0.012 reads 0.916; a volume clock reads 1.16 at the same limit and 1.51 at a third of it. Every law is pulled the same way, and the pull does not stop until the answer is 1.

That is the worst shape a failure can have. It does not produce noise, it does not produce a refusal, and it does not produce an obviously absurd number. It produces a clean reading of a law that exists, sitting in the middle of the range of laws, on a shell that had a different one. It is the same class of defect as the one the residual is not the test found in a fit that accepts a circle and refuses a golden spiral — an instrument answering confidently on input it cannot handle.

How coarse a section has to be before it matters #

The threshold is not the same for every law, and the difference is useful. Sweeping the limit upward, the first point at which the reading moves by more than a hundredth arrives at 6.0 × 10⁻⁴ for a volume clock, 8.3 × 10⁻⁴ for an angular clock and an area clock, and 2.7 × 10⁻² for a length clock — a factor of thirty-two between the most fragile law and the immune one.

In terms of record lost rather than limit, the steep laws break earliest: a volume clock is already biased when 17 per cent of its lines are gone, an area clock at 16 per cent, an angular clock at 41 per cent, and a length clock not until 95 per cent. So the shells most at risk are exactly the ones whose reading was most confident, because a steep clock puts the laws furthest apart and was the easiest case for what the growth lines carry.

What a person can actually check #

The bias has a signature, and it is in data the counting already produces. If the share of lines lost varies along the shell, then the spacing of the surviving lines is the same in every whorl by construction — every surviving pair is one resolution limit apart. So a section whose measured line spacing is constant across whorls is either a genuine length clock or a section reading its own limit, and the two are told apart by whether the spacing equals the limit.

That is a check anybody counting a section can make, it needs no extra material, and it converts the worst failure mode here into a refusal. A shell whose line spacing sits at the limit in every whorl should be reported as unreadable rather than as a length clock.

What it does to a shell that changed its law #

The bias falls on a sequence of ratios exactly as it falls on one, and the consequences are worse in a specific way. A shell that changed its law reads two plateaux and a crossing, and takes each plateau as a law. A resolution limit pulls both plateaux towards 1, and it pulls them by different amounts, because the two laws are different distances from a length clock. So a shell going from an angular clock to an area clock, read through a limit, reports a change from something above 0 to something below 2 — a smaller change than happened, in the right direction, at roughly the right place.

The position survives better than the laws do, which is the one piece of good news. The position comes from the crossed ratio read against the two plateaux, and pulling all three towards 1 together leaves the inversion largely intact.

The case that does not survive is a shell whose law was near a length clock to begin with. A law that never stopped changing reads a drift as a rising sequence, and a limit flattens a rising sequence towards a level one — which is the signature of a shell with a single law. So a worn drifting shell can report one law and no change at all, and that is the one failure in this thread that produces a simpler answer than the truth rather than a wrong one.

A worn whorl is a different problem, and a soluble one #

Damage confined to one whorl behaves nothing like a resolution limit, and the difference is worth drawing because a person reading a real section will meet both.

Every ratio the worn whorl does not touch is exact. The two that do touch it are wrong — 1.399 and 2.597 against a true 2 — and they are wrong by equal and opposite amounts, because the share the whorl kept enters one ratio as its logarithm and the next as minus it. Their mean is 1.998.

So a worn whorl is correctable and not merely detectable, provided it is one whorl and it is known which one. The sequence says which: it is the whorl shared by the only two ratios that disagree with the rest, and a sequence of five ratios with two adjacent outliers of opposite sign is a very specific pattern to find by accident.

Why this is the reverse of the usual advice about measurement #

The instinct with a difficult count is to count more carefully, and for this reading that instinct is almost entirely misdirected. Counting carelessly but evenly costs scatter and no bias. Counting meticulously in the clear whorls and giving up in the crowded ones is the single worst thing that can be done to it, because it manufactures exactly the sloping share that the formula turns into a wrong law.

The same is true of the choice of material. Three points on a diameter found the calipers beating the fit on a clean section at slow expansions, and the advice there was about which instrument suits which error. Here there is one instrument, and the advice is about which part of a section to trust: a whorl counted at half efficiency is worth more than a whorl counted at full efficiency beside one counted at a tenth.

Where that leaves the reading as something to do with a fossil #

Four statements, in the order a person would need them. A count that is even can be bad: losing four lines in five costs a tenth in the worst seed and nothing on average. A count that is uneven is biased by the logarithm of the ratio of the two shares, and nothing else about it matters. The uneven count that always arrives is a resolution limit, and it always pulls towards a length clock — so a reading of $\hat p = 1$ is the one answer that should be treated as provisional until the line spacing is checked against the limit.

And a whorl that is damaged on its own is the easy case: it announces itself as two adjacent ratios of opposite sign, and averaging them recovers the law.

The uncomfortable part is that the dangerous failure and the safe one look identical in the summary statistic a person would naturally report. Both lose most of the lines. A count that has lost 73 per cent of its record evenly is a good reading; a count that has lost 73 per cent of it to a limit is a reading of the shell’s arc lengths. The share lost does not distinguish them and the share lost per whorl does, which is one more column in the same table and is not something any published count includes.

That is worth stating plainly because it is the practical recommendation this whole essay reduces to, and it is cheap: report the count and the share resolved for every whorl, not the count alone. What the growth lines carry asked for a count per whorl and the growth factor beside it. This asks for one more number per whorl, and it converts a silent failure into a visible one.

What is claimed, in one line #

The law a growth-line count names is unmoved by any miscount that is the same in both whorls of the pair and biased by $\log(f_2/f_1)/\log W$ by one that is not; a section’s resolution limit is such a miscount for every law but $p = 1$, because line spacing goes as $e^{k(1-p)\theta}$; and where the limit binds it returns the ratio of whorl arc lengths, which is $W$, so a shell too worn to read reports a length clock.

What none of this settles #

That real growth lines merge at a fixed distance. A resolution limit is the simplest model of a limit and a real section’s depends on contrast, orientation, wear and who is looking. What survives that objection is the formula, which needs no model at all: whatever the limit is, it costs nothing where it takes the same share from both whorls and biases by the logarithm of the ratio of the shares where it does not.

Nor that a real count’s misses are independent of each other, which the scatter above assumes and a person counting a crowded stretch certainly violates. Correlated misses would leave the bias where it is and widen the scatter, which is the direction that does not change any conclusion here.

And nothing here measures a shell. Every count is from a curve built to a stated law, marked at equal intervals of time, and then read as a real section would be.

What would withdraw it #

A uniform miss of any size producing a bias larger than its own scatter. A length clock moved by a resolution limit at any limit that leaves its whorls countable. Any other law moved away from a length clock by a limit. A limit binding completely and leaving counts in a ratio other than the growth factor. A worn whorl whose two touching ratios are not equal and opposite, or whose untouched ratios are not exact. Each is checked every time the measurement runs.

Still open: whether a section can report its own limit #

The check proposed above needs one number a count does not currently produce — the resolution limit itself, in the same units as the shell’s radius. A person counting lines could measure it directly, but a better question is whether the counts alone contain it. A section reading its own limit has line spacings that are constant across whorls and equal to the limit; a genuine length clock has spacings that are constant and equal to whatever the animal’s rate produced. The distribution of spacings within a whorl differs between the two — a limit truncates from below and an animal does not — and the next measurement is whether that difference survives at the sample sizes a single whorl provides.