A floor no better fit can lift

The line where a coiled shell’s whorls stop running into one another is D = 1/W, and it is exact — located by bisecting the drawn circles at 481 expansions, its residual is the last bit a double holds over the whole range.

A specimen is not a point on an exact line. It is two measurements, each with an error, and carrying the first of those errors onto the boundary turned the line into a band: a shell inside it cannot be shown to have whorls in contact or free of them. At the measured budget with the dividers set aside, that band covers 48.2 per cent of the box the morphospace figure is drawn on.

It also carried only one of the two errors, and said so. The distance from the axis has now been priced, and this is what the two make together.

The band widens by a third #

Take the independent calculation first, because it is what a reader would construct and because it turns out to be nearly right.

A shell measured at W̃ and D̃ might really be anywhere in the box that the two error bars allow, and it is undecidable when that box straddles the product one. The band in D at a given expansion then runs from 1/(W(1+b)(1+c)) to 1/(W(1−b)(1−c)) instead of from 1/(W(1+b)) to 1/(W(1−b)).

At an expansion of 3.2, an expansion error of 54.4 per cent and the 16.15 per cent the averaged axis-distance reading costs at the same assumed centre, the band runs from 0.2323 to 1.0897 around an exact 0.4167. The one-error band ran 0.6439 wide; this one runs 0.8573, a factor of 1.33.

The share of the drawn box goes from 48.2 per cent to 59.7.

The widened band is not a worse measurement than the narrow one. It is the same measurement with the half that was missing put back, and the narrow band was never a statement about shells — it was a statement about one of the two numbers a shell is placed by. A reader who took 48.2 per cent as the size of the region nobody can be placed in was taking a lower bound for an answer.

That is a real widening and it is not the interesting one. The interesting numbers are further down the same table.

Where the two columns come apart #

At the whole budget the two columns barely differ: 93.2 per cent against 94.4. When the expansion is known to within a factor of three, adding a sixth of the other number changes nothing anybody would notice.

Tighten the expansion and the columns separate. At a tenth, 7.8 against 20.2. At a twentieth, 3.9 against 16.4. At a hundredth, 0.8 against 13.4 — a factor of seventeen.

The one-error column behaves the way a reader expects an error budget to behave: measure the thing better and the region nobody can place shrinks toward nothing. The two-error column stops.

Going from a tenth to a twentieth on the expansion removes 3.8 points of the box. Going from a twentieth all the way to a hundredth removes 3.0 more. Past about a tenth, effort spent on the expansion is effort spent on the smaller of the two errors, and the marginal return on a better fit to the curve is close to nothing.

The floor, and what sets it #

Set the expansion’s error to zero — not small, zero — and ask what is left.

12.6 per cent of the box cannot be placed however perfectly the expansion is measured. At an expansion controlled to a hundredth the two-error share is 13.4 per cent, so 94 per cent of what survives a perfect fit belongs to the second number.

The floor moves with the second error and with nothing else: 12.6 per cent at the 16.15 the averaged reading costs today, 7.8 at a tenth, 3.9 at a twentieth, 1.5 at a fiftieth, 0.8 at a hundredth. It is almost exactly linear in it, which is what a band whose width is a product of two small factors should do and is worth checking rather than assuming.

This is the result the whole boundary argument was missing, and it inverts a recommendation made here more than once. Every previous essay on this boundary ended by saying what a worker should control, and every one of them named the expansion — because the expansion was the only number with a price on it. The honest version is that controlling the expansion below about a tenth is wasted effort until the other number is controlled too, and the cheapest way to control the other number is the azimuthal average, which costs nothing and is worth a factor of 4.91.

A share is not a property of the geometry #

Before any of those numbers is carried further, one check is owed, and the essays on this boundary are the reason it is owed.

Six boxes differing only in where their edges were drawn gave between 4.64 and 52.81 per cent for the same contact region, and sampling the expansion axis geometrically rather than uniformly multiplied the answer by 4.60. The share of a box is not a property of the geometry; it is a property of the box, and the region under a hyperbola is a logarithm where a box is a line. Quoting 12.6 per cent as though it described the space would repeat the exact mistake that essay exists to record.

So the floor was computed on all six.

The floors run from 1.5 per cent on the widest box to 17.3 on the tightest — a factor of eleven, which is the box-dependence behaving exactly as the earlier measurement said it would. Taken alone, 12.6 per cent says almost nothing.

The ratio does not move. Floor against one-error share at an expansion controlled to a hundredth: 15.7 on the drawn box, 16.5 on the unit box, 16.1 tight, 15.6 open, 15.0 wide, 14.5 on the geometrically sampled decades. Six boxes, a range of 14.5 to 16.5.

That is the number worth carrying, and it is worth carrying because it is a ratio of two shares over one box rather than a share. The statement it supports is box-free: carrying the second error leaves about fifteen times more of the space undecidable than the one-error calculation says, at the tight end where anyone would want to work. Every earlier figure in this essay is a share and inherits its box; this one does not.

It also says something about the instrument that produced the first half of this essay. A quantity that moves by eleven between two reasonable conventions and a quantity that moves by 1.14 between the same two are not the same kind of quantity, and only the second is a measurement of the shells.

But the errors are not independent #

Everything above treats the two errors as separate error bars. They are not separate. There is one section and one guessed centre, and both numbers are read from it.

So the set a specimen could really occupy is not the rectangle two independent margins describe. It is the one-dimensional locus traced out as the guessed centre swings round, and the rectangle is that curve’s bounding box.

From a centre displaced a quarter of a whorl, the expansion reads between 2.144 and 2.794 and the distance from the axis between 0.3521 and 0.3932 — but not in every combination, which is what the curve says and the rectangle does not.

The first thing to report is a negative result. The rectangle is wider than the curve by a factor of 1.195: the independence assumption overstates the uncertainty in the product by about a fifth, and no more. That is a modest overstatement, and it means the arithmetic in the first half of this essay is a fair approximation rather than a construction to be discarded. It is worth saying plainly because the usual reason to compute a joint distribution is that the independent one is badly wrong, and here it is not.

The second thing is not negative at all.

One guessed centre flips the verdict #

The shell drawn throughout the shells figures sits at W = 2.4 and D = 0.42, so its product is 1.008 — a thousandth of the way outside the boundary, on the evolute side.

The products a quarter-radius centre error can produce run from 0.8022 to 1.0897. They straddle one. The same shell, read from one guessed centre swung through the directions that centre could have been guessed in, is reported as free of contact from some of them and in contact from others, with nothing on the section to say which.

And the error has a direction. The mean product over the forty-eight directions is 0.9057 against a true 1.008 — 10.15 per cent low. The two errors compound over most of the circle rather than cancelling: a centre guessed too far out reads the expansion low along that bearing and reads the axis distance low as well, and the product takes both.

A scattered error is fixed by more specimens. A displaced one is not. A survey of shells read from imperfectly located centres does not converge on the truth as it grows; it converges on a point ten per cent toward contact, and every conclusion drawn about how much of the morphospace is occupied on each side of the line inherits that shift.

That is the strongest practical statement in this essay, and it is about a census rather than about a shell. Raup’s famous result was a census — large regions of the space are constructible and empty — and a systematic displacement toward one side of the most interesting line in that space is exactly the kind of error a census cannot detect from inside itself.

There is no third route #

One escape remains to be closed, and the band essay named it: the angle criterion. Seen from the apex, every whorl of a Raup shell subtends the same half-angle, and two whorls touch exactly when the sine of that angle exceeds (W − 1)/(W + 1).

That criterion is already known to carry the expansion’s error differently — the expansion enters its threshold rather than its measurement, which is a different place and a smaller derivative, and it does better above an expansion of $1 + \sqrt{2}$ and worse below it, exactly. The hope was that it might do something similar for the second number. A half-angle read from the two tangent lines to a whorl’s cross-section is an angular measurement rather than a ratio of two lengths, and there is no reason in advance why a wrong centre should cost the two the same.

It costs them exactly the same. Over the same displaced centres, the same directions and the same azimuths, the distance from the axis implied by the tangent half-angle and the distance read straight off the two walls differ by 3.3 × 10⁻¹⁶ — the last bit a double holds. Both give a worst ray of 79.34 per cent and both give a mean of 9.335 per cent low.

The reason is algebra rather than luck. The whorl’s cross-section on a ray is a disc whose radius is half the difference of the two wall distances and whose centre sits at half their sum, so the sine of the half-angle from that point is (dOut − dInn)/(dOut + dInn) — a function of the ratio and of nothing else. Reading the angle and reading the ratio are one act of measurement written two ways.

So the angle criterion is a genuine second route against the expansion and is not a route at all against the distance from the axis. The floor stands whichever criterion is used, and a worker choosing between them is choosing how to handle the error that is no longer the limiting one.

And half the space has no second number at all #

There is a limit under all of this, and it is not a matter of precision.

The distance from the axis is the ratio of a whorl’s inner wall distance to its outer. On an involute shell the inner wall lies beneath the previous whorl and is not on the section. No care with the centre, the camera or the saw recovers it, because the length is not there to be measured.

So everything in this essay is a statement about the evolute half of the space. The band drawn here is a band approached from above the line; the floor is a floor on shells whose whorls run free; and a contact criterion leaning on the distance from the axis has nothing to lean on for most gastropods, which are involute.

That asymmetry is new here and it should change how the boundary is talked about. It has been treated throughout as a line with two sides that are alike apart from their verdict. They are not alike. One side has two measurable coordinates and the other has one.

What this does not establish #

That any of these errors is the size a real worker makes. A quarter of the innermost radius is the displacement treated throughout as a gross error, and it is used here because it is the one condition at which both numbers have been priced — not because anybody’s centre is that bad. Halve it and both entries roughly halve; the shape of every conclusion above survives and the sizes do not.

That the guessed centre is the only coupling. It is the one this essay carries because it is the one both budgets share. An oblique view couples nothing — it costs the axis distance exactly nothing — but a partial section, a worn aperture and a shell whose expansion changed with age are each shared conditions that could correlate the two readings in ways nothing has measured.

And nothing here touches the third number. Every band above is drawn at T = 0, and translation along the axis enters the boundary as its square, so a spired shell’s band is a different band. Whether the axis distance’s error behaves the same way out of the coiling plane is not known, and the reading used here was defined in the plane.

What would withdraw it #

A reading from the true centre that did not return both of the shell’s numbers. It returns the expansion to fifteen digits and the distance from the axis exactly.

A locus whose bounding box was narrower than the locus, which would mean the joint reading had been computed wrongly rather than that the errors conspire. The box is wider, by 1.195.

An undecidable share that went on falling as the expansion’s error was driven to zero. It flattens onto 12.6 per cent, and the same computation with the second error set to a hundredth flattens onto 0.8 — so the floor is the second error and not an artefact of the grid the share is counted on.

Or a product read round the circle that straddled its true value rather than sitting below it. It runs 0.8022 to 1.0897 about a true 1.008, with a mean 10.15 per cent low.

Still open: whether a census can see its own displacement #

The finding with consequences is the last one: a mislocated centre does not blur a specimen’s position, it moves it, and it moves every specimen the same way.

What follows from that depends on a quantity nobody here has measured — whether the displacement leaves a signature in the survey itself. A scattered error is invisible in one specimen and obvious in a hundred, because the spread is the evidence. A systematic one is invisible in both, unless something about the shape of the cloud gives it away.

There is a reason to think something might. The size of the shift depends on the shell: the axis distance is read twenty-two times better far from the boundary than on it, so a displaced census is compressed toward the line rather than translated along it, and a cloud of real shells with a real spread should show a density that rises toward the boundary by an amount the measurement predicts. Whether that rise is distinguishable from a genuine biological preference for shells near the contact line — which is a claim people have actually made — is the question.

The measurement is a synthetic census: a population scattered across the space at a stated density, each member read from a centre guessed with a stated error, and the resulting distribution compared with the one it came from. If the pile-up near the line is large enough to see at plausible sample sizes, then any published occupancy map of this space has a testable artefact in it, and the test is arithmetic rather than a re-measurement of anybody’s shells.