The band nobody can be placed in

The line was already exact located the boundary between shells whose whorls run into one another and shells whose whorls run free, and found that the textbook D = 1/W is not an approximation of it: the residual over 481 expansions is 2.2 × 10⁻¹⁶. Six essays since have sharpened the line further — what a spire buys, whether it is a kink or a tangency, what share of a box lies either side, which picture of a shell can decide it, what an aperture’s shape moves.

All of that is about where the line is. None of it is about where a specimen is, and a specimen does not arrive as a point. It arrives as a measurement with an error on it, and the size of that error has just been counted.

The band, in one line #

Whorls are in contact when W·D < 1. A specimen’s expansion is measured as W with a relative error of at most b, so the smallest expansion consistent with the reading is W(1 − b) and the largest is W(1 + b). The specimen is therefore decidably in contact only when even the largest keeps the product under one, and decidably free only when even the smallest keeps it over:

$$W(1+b)\,D < 1 \quad\text{or}\quad W(1-b)\,D > 1$$

Between those the measurement says nothing about which kind of shell it is, and the band it says nothing about runs from $1/(W(1+b))$ to $1/(W(1-b))$. That is a closed form with nothing fitted and no search in it, and it straddles the exact boundary at every expansion and every error, which is what a band has to do to be a band rather than a bias.

At an expansion of 3.2 and an error of 54.4 per cent it runs from 0.2024 to 0.6853 around an exact 0.3125 — a band nearly half the width of the whole axis. At ten per cent it is 0.2841 to 0.3472; at one per cent, 0.3094 to 0.3157.

Where the number 54.4 comes from, and the worse one behind it #

The error budget for a nautilus assembled every measured way a growth factor read off a section can be wrong and added them: 332.6 per cent in the worst case, falling to 54.4 once the dividers are set aside on the grounds that a walk with enough steps is exact.

Both numbers matter here and the first matters more than it looks. An error past a hundred per cent allows an expansion of one or less, and at an expansion of one every distance from the axis under one puts the whorls in contact. So at the whole of the measured budget the band is open above: no shell whatever can be shown to have free whorls. That is not a degenerate case to be excluded. It is the case the arithmetic above is in until the dividers are excluded from it.

What share of a morphospace that is #

On the box the site’s own morphospace figure draws — expansion from 1.1 to 6, distance from 0.02 to 0.9 — the band covers 93.2 per cent at the whole budget, 48.2 per cent with the dividers set aside, 7.8 per cent at a tenth, 3.9 at a twentieth and 0.78 at a hundredth.

Forty-eight per cent is the number to carry. It says that a shell picked at random out of the drawn morphospace, measured by every technique this collection has priced except an under-stepped pair of dividers, has a better than even chance of not being classifiable at all — not misclassified, but undecidable, with the measurement consistent with both answers.

The share falls roughly in proportion to the error below that, which is what a band of width proportional to $b$ does over a region whose area is roughly linear in the band’s width. There is no threshold in it and no cliff: halving the error halves the undecidable share, all the way down.

And that share is a statement about the box #

A fraction of nothing established that a share of a morphospace is a statement about the box rather than about the geometry: six boxes differing only in where their edges were drawn gave between 4.64 and 52.81 per cent for one and the same contact region, and sampling one axis geometrically rather than uniformly multiplied the answer by 4.60.

The undecidable band inherits that exactly, which is worth checking rather than assuming. Across the same six boxes its share runs from 6.7 per cent on two decades of expansion sampled uniformly to 59.5 on the slow-expansion box — a factor of 8.9. So “half the morphospace is undecidable” is a sentence about the drawn box and not about shells, and it should be said with the box attached every time, as its predecessor should have been.

The reason is the same in both cases. The boundary is a hyperbola, the region under a hyperbola is a logarithm, and a box is a rectangle; the ratio of a logarithm to a rectangle has no limit as the rectangle grows.

What a spire does to it #

What a spire buys found that translation along the coiling axis enters the contact boundary as its square, so a shell that has only just begun to walk along its axis has not moved the boundary at all, and there is a threshold above which no distance from the axis whatever puts the whorls in touch.

That has a consequence here which is easy to state and easy to get backwards. A quantity entering as a square is less sensitive to an error in itself near zero, so a small error in a small translation moves the boundary hardly at all — the band does not widen appreciably for a nearly planispiral shell whose spire was measured badly. But the same squaring means the boundary moves quickly once the translation is real, so a shell with a genuine spire has a boundary whose position depends sharply on a third number, and how well that third number can be read has not been priced either.

So the band drawn above is the planispiral case, which is the case where the arithmetic is cleanest and the error smallest. A spired shell is worse off in a way nobody has quantified, and the honest reading of the shares above is that they are the best case in a second respect as well as the first.

Which criterion carries the error better #

One angle decides contact found a second criterion for the same question: seen from the apex of the coiling axis every whorl subtends the same half-angle, and two whorls touch exactly when the sine of that angle exceeds $(W-1)/(W+1)$. The two criteria agree — that essay checked them against the drawn discs at 400,000 random shells — but they do not carry an error in the expansion the same way.

Under the distance boundary the expansion enters the answer directly: the boundary is $1/W$ and its relative sensitivity to $W$ is exactly one. Under the angle criterion the expansion enters the threshold, and the relative sensitivity of $(W-1)/(W+1)$ is

$$\frac{2W}{W^2 - 1}$$

Those are equal when $W^2 - 2W - 1 = 0$, which is at $W = 1 + \sqrt{2} = 2.41421$ — the silver ratio, arriving here for no reason anybody would have guessed. Below that expansion the angle criterion carries the error worse and above it better, and the advantage grows without limit: at an expansion of nine it is exposed to about a fifth of what the distance boundary is.

So the two criteria are not interchangeable once a specimen has an error on it, and which to use depends on the shell. A slowly coiling shell should be classified by its distance from the axis; a fast one by the angle its whorls subtend. That is a recommendation nothing in the earlier essays could have made, because they were comparing the criteria on exact shells, where the two agree by construction.

Whether the boundary’s own sharpness matters #

A boundary with no edge found that two continuous measures cross the contact line and disagree about whether it is sharp: one falls to zero as a straight line and makes the line a kink, the other leaves it as a three-halves power and makes it a tangency. That disagreement is about the geometry near the line, and the obvious question is whether it changes the band.

It does not, and the reason is worth a sentence because it is the one place in this essay where an earlier complication turns out not to propagate. The band is defined by which side of the line a measurement can put a shell, and which side is a matter of the sign of $1 - WD$ and nothing else. How steeply some continuous measure of contact approaches zero as the line is crossed does not enter, because the classification is not that measure.

What the sharpness question does bear on is a different reading — how much the whorls overlap, rather than whether they do — and that quantity carries the expansion’s error with a different exponent on each of the two measures. Nobody has priced it, and it is the natural companion to what is priced here.

What a given shell needs #

The question turns round easily. A shell is decidable when its distance from the boundary exceeds the error, so the error it can tolerate is $|1 - WD|/WD$ — one line of arithmetic with no search in it.

The site’s own hero shell, at W = 2.4 and D = 0.42, has a product of 1.008 and needs an error under 0.8 per cent. It is eight thousandths of the way off the boundary, and nothing measured so far comes close to placing it. A shell at W = 3.2 and D = 0.42 needs 25.6 per cent; at W = 4.5 and D = 0.15, 48.1; at W = 6 and D = 0.1, 66.7 per cent, which the trimmed budget clears.

A shell exactly on the boundary needs an error of nothing, which is the right answer and the reason no measurement ever classifies one. That is not a failure of the instrument. It is what a boundary is.

What it does to a published morphospace #

The reason Raup’s cube is famous is a claim about occupancy: real shells fill a small and characteristically shaped part of it, and the empty regions are supposed to say something about what animals cannot or do not do. Raup’s three numbers is the account here of the compression that makes such a claim possible at all.

A band forty-eight per cent wide does not refute that claim, and it does change what the cube’s regions can be read as. A cluster of specimens near the boundary is not evidence that animals prefer to sit near it; it is what a scatter of measurements with an error of tens of per cent looks like when the thing being measured is a ratio. And an empty region narrower than the band is not evidence of anything at all, because no specimen could have been placed in it.

The measurement that would settle which of those is happening is not a better morphospace. It is an error bar on each point, computed from the span of arc fitted and the residual left behind — both of which the fitting already produces, and neither of which is normally reported. That is the same conclusion a section seen from the wrong angle reached from the other end, arriving at a different figure.

And what the aperture does #

The fourth number divides the third measured what the shape of the opening does to the boundary, and found that with no translation it does nothing at all for any convex opening symmetric about the plane of coiling. That is the one result in this thread which narrows the problem rather than widening it: the band drawn here does not need the aperture’s shape to be known, which removes a fourth number from the error budget before it could be added to it.

It is worth noticing what made that possible. The result was a negative one — a quantity measured and found to be zero — and it has now done work twice: once in closing the hedge three earlier essays ended on, and once here, in keeping the band a function of one measured number instead of two. A negative result is the only kind that can be reused like that, because it is the only kind that removes a term.

Why this is not an argument against the boundary #

The line is still exact and the essays that located it still stand. What has changed is the reading of what those essays are about.

A boundary located to 2.2 × 10⁻¹⁶ invites the thought that the classification it supports is correspondingly sharp. It is not, and the two facts are about different things: the geometry is exact and the measurement is not, and the exactness of the first is what makes the second’s error propagate cleanly rather than what removes it. The whole of this essay is that propagation, and it needed the boundary to be exact before it could be done at all. A boundary known only to a per cent would have had its own error in the sum and the two would not have been separable.

That is the general shape and it is worth stating once. An exactly located line is not a sharp classification. It is a classification whose sharpness is entirely the measurement’s, which is a better position to be in and a different one.

What is claimed, in one line #

Carrying the whole measured error budget onto the exact contact boundary turns it into a band from $1/(W(1+b))$ to $1/(W(1-b))$ which covers 48.2 per cent of the box the morphospace is drawn on at the trimmed budget and between 6.7 and 59.5 per cent across the six boxes it uses; a shell’s tolerance is $|1-WD|/WD$ exactly; and the angle criterion carries the same error better above an expansion of $1+\sqrt2$ and worse below it, the crossing being where $2W/(W^2-1)$ meets one.

What none of this establishes #

That the distance from the axis is known exactly. Nothing has measured how wrong D can be read off a section, so the band here is the error from one of the two numbers the boundary is a relation between. The real band is wider by an amount nobody has measured, and the shares above are therefore lower bounds rather than estimates.

Nor that the budget is complete — it is the sources measured so far, and one nobody has thought of is not in it. Nor that a real specimen carries the whole budget: each entry is a worst case under a stated condition, and a worker who controls the conditions is not in this band at all.

And nothing here classifies a shell. Every point is a stated pair of numbers and the question asked of it is arithmetic.

What would withdraw it #

A band that does not straddle the exact boundary, or does not close as the error closes. An undecidable share that does not fall with the error. A box whose share falls outside the range the boxes give. A crossing of the two criteria’s sensitivities anywhere other than $1+\sqrt2$, or either sensitivity departing from $1$ and $2W/(W^2-1)$. A shell whose stated tolerance does not match $|1-WD|/WD$. Each is checked every time the page is built.

Still open: what the distance from the axis costs #

The band above is half a band. D is the second number the boundary is a relation between and nothing has priced reading it off a section — which is odd, because it is a length measured against another length and should be easier than the expansion rather than harder. The next measurement is the same exercise for D: what an assumed centre, an oblique view and a partial section do to the distance from the axis, whether the two errors are independent, and what the band becomes when both are carried rather than one. If D turns out to be the cheaper number, the recommendation at the end of this essay changes — a criterion that leans on D and not on W would be better than either of the two compared here.