The residual is not the test

Least squares on log r against unwrapped angle returns two things: a growth factor, and the worst departure of the points from the line it fitted. The second is the residual, and the essay that introduced the machinery said what it was for — the same fit answers what the parameter is and whether the parameter means anything.

Pointed at twelve curves, it does not.

What the residual was supposed to do #

The argument is short and it is a good one. A logarithmic spiral fitted to a logarithmic spiral is exact, because the model is the curve; the residual comes back at arithmetic noise, two parts in ten thousand million million. A curve of some other kind cannot be straightened by taking its logarithm, so the fit returns a factor with a departure that is not noise, and the size of that departure says how badly the model applies.

So the instrument was supposed to be self-policing. A reported growth factor arrives with its own warrant attached, and a number quoted for a curve that is not a logarithmic spiral is a number the residual would have flagged.

Twelve curves and one instrument #

The test is to point the fit at curves whose kind is known in advance: three logarithmic spirals at the factors this collection draws, four readings of an Archimedean spiral, two circles and three ellipses. Every one is handed to the same recovery from its own true centre with no noise added anywhere, so nothing below is a measurement error.

Every one of the twelve returns a growth factor. That is the first thing to notice and it is not a defect: least squares always returns a slope. What matters is which of them the residual then declines, and the answer is not the four the reader would expect. The comparison to keep in mind throughout is a fit whose residual at the minimum really does report whether the shape is right, which is the behaviour this one was assumed to share.

A circle is fitted exactly #

The clearest case is the simplest. A circle read about its own centre comes back at a growth factor of 1.0000 with a residual of zero — not small, zero, because a circle about its own centre has constant radius and a constant is a perfectly straight line in log r against angle.

So the curve with the least in common with a logarithmic spiral is the one the residual likes best. Nothing about this is a bug in the arithmetic: the fit is a straight-line fit, and a horizontal line is straight.

Every ellipse returns exactly one #

Squash the circle and the exactness goes, but the factor does not move at all. An ellipse of ten to nine comes back at 1.0000 with a residual of 0.0542; one of two to one at 1.0000 with 0.4062; one of four to one at 1.0000 with 0.9175.

The factor is 1.0000 to nine decimal places in every case, whatever the aspect. That is a consequence of closure rather than of shape. The radius at the end of a full turn is the radius at the start of it, so the slope of log r against angle over a whole number of turns is zero however the radius wandered in between, and the fit reports the only number it can.

What actually refuses a closed curve #

Not the residual. A circle read from its own centre has a residual of exactly zero and an ellipse of ten to nine has 0.0542, which is comfortably inside the threshold of 0.15.

What declines them is a separate rule: a recovered factor of one is not growth, so the recovery refuses to call the curve a spiral at all. That rule is doing the work the residual was credited with, and it works only because a closed curve closes. It says nothing about an open curve of the wrong kind, which is the case the residual was actually needed for.

An Archimedean spiral read from its second turn #

Which brings the test to the curve that matters. An Archimedean spiral adds a fixed amount per turn rather than a fixed factor — it is what a coiled rope makes, and it is the curve Bernoulli’s stonemason carved by mistake. It is not self-similar, so no growth factor describes it and the fit has no business accepting one.

Read over three and a half turns starting from its second turn, it comes back at a growth factor of 1.3062 with a residual of 0.0903, and the recovery accepts it.

And from its fourth #

Start further out and it gets worse, in the sense that matters. Read over the same three and a half turns from its fourth turn, the same curve returns 1.1881 at a residual of 0.0345 — a quarter of the threshold, and a smaller departure than a genuine nautilus spiral leaves when its points are located to one part in a thousand.

So the instrument’s confidence in a curve it should refuse rises the further out the arc is read. There is nothing subtle about why. An Archimedean spiral flattens as it goes; over a stretch far from its origin it is nearly a straight radial ramp, and a straight ramp over a short enough range of log r is nearly a straight line.

The one reading that is refused #

The fit does decline the Archimedean spiral once. Read over three and a half turns from its first turn — its tightest, most curved stretch — it returns 1.4531 with a residual of 0.1976, which is over the threshold, and it is refused.

That is the only refusal on shape anywhere in the twelve, and it is one reading of one curve.

The residual is a fact about the arc #

Which reading of the curve is chosen decides the verdict, and it decides it across a factor of twenty-six. Holding the span at three and a half turns and moving only where the arc starts, the same Archimedean spiral leaves 0.912 from a twentieth of a turn out, 0.553 from a quarter turn, 0.198 from one turn, and 0.035 from four.

Nothing about the curve changed between those four numbers. What changed is which part of it was handed over — the same freedom that lets a photograph of one turn of a shell be compatible with almost any growth factor, which is the whole difficulty the nautilus claim lives in.

And a fact about how much of it is read #

Fixing the start at the first turn and lengthening the arc instead moves it the other way: 0.036 over one turn, 0.098 over two, 0.165 over three, 0.198 over three and a half, 0.292 over five, 0.459 over eight.

The threshold is crossed between two turns and three, so the same curve read the same way from the same point is accepted at one length and refused at another. A test whose verdict is set by the crop is a test of the crop.

Two turns is the only guard, and it is not this one #

The recovery does impose a floor on how much arc it will read — two turns — and it is worth being clear that the floor is not doing this job either. The Archimedean spiral read over a single turn from its origin returns 1.8481 at a residual of 0.0359, and is refused for span rather than for shape: below two turns the recovery declines to say anything at all.

That floor is a good rule for an unrelated reason, since the cost of a misplaced centre falls as the inverse square of the span. It removes the shortest readings from consideration, and every reading long enough to pass it is a reading the residual has already been shown to accept. It also means a refusal carries no diagnosis: span, growth and residual are three separate rules with one word for their outcome, which is the accounting problem a null reading always has.

What a genuine fit leaves behind #

The other half of the failure is on the honest side of the ledger, and it is the half that makes tuning the threshold hopeless. A genuine logarithmic spiral fitted with no noise gives exactly zero. A genuine one whose points are located imperfectly does not, and how much it gives is worth measuring rather than guessing.

Twelve of them, across six growth factors and two spans, with the points located to a ten-thousandth of the outer radius.

Two and a bit times the noise #

They lie on one line of slope one. Express the location error as a fraction of the innermost whorl’s own radius rather than of the picture, and the measured residual is 2.03 times that number — mean 2.030, running from 1.818 to 2.388 across the whole set, a spread of 1.313 from end to end.

Nothing in that is about the curve. The factor varies by a factor of four and a half across those rows and the span by nearly two, and the ratio barely moves.

Why the innermost whorl decides it #

The reason is geometric and it is the whole of the result. The residual is measured in log r, and a fixed error in position is a fractional error in radius that is largest where the radius is smallest. The smallest radius on the arc is the innermost point, so the worst departure is the innermost departure, and its size is the location error divided by the innermost radius.

That radius is set by the growth factor and the span and by nothing else: a spiral at 3.2 over three and a half turns has an innermost whorl 1.706 per cent of its outer radius, and a golden one 0.119 per cent. A golden spiral is fourteen times worse off before a single point is measured.

The number is available before the curve is #

Put those together and the residual of a genuine measurement can be written down in advance. It is roughly twice the location error divided by the growth factor raised to the power of the span, and every quantity in that sentence is a property of the frame — how much arc was photographed, at what factor, to what precision.

A quantity computable before the curve is looked at cannot carry information about the curve. This is the same shape of finding as a disorder residual that turned out to be the reading window: something that looked like what the data had left over was a property of the instrument.

The band inverts #

Now put the two halves on one axis. A genuine golden spiral over three and a half turns, at a location error of a ten-thousandth, leaves 0.153; a genuine spiral at 7.4 over the same span leaves 0.2006. The smallest a curve that is not a logarithmic spiral gets away with is 0.035, from the Archimedean spiral read from its fourth turn.

The genuine floor sits above the failure by a factor of four and a half, and above the Archimedean spiral’s worst reading as well. The band is the wrong way round.

Widen the noise as well as the factor and it opens further. A genuine 7.4 spiral over the same span, with its points located to three ten-thousandths, leaves 0.670 and recovers its own factor to a tenth of a per cent. That is nineteen times the largest residual any non-spiral in the set is asked to survive.

No threshold separates them #

That is not a badly chosen number to be improved on. A threshold is a single cut, and a single cut cannot put 0.153 on the accept side and 0.035 on the reject side, because 0.035 is the smaller of the two. Any cut that admits the golden spiral admits the Archimedean one read from its fourth turn, and any cut that rejects the Archimedean one rejects the golden.

The current setting of 0.15 — a worst point 16.2 per cent off the fitted curve — is bracketed by two numbers on a much narrower scope: 0.1128 from a nautilus spiral over three and a half turns at a location error of a thousandth, and 0.1976 from the Archimedean spiral read from its first turn. Within that scope it is a defensible choice. Outside it, it is a number between two other numbers.

What the threshold’s own scope costs #

The casualties are worth naming, because they are the readings a shell collection would most want to trust. A genuine golden spiral, over three and a half turns, with its points located to a ten-thousandth of the outer radius, comes back at a growth factor accurate to 0.035 per cent — and is refused, at 0.1533 against a threshold of 0.15.

It is not alone. Five genuine spirals in the noise ladder recover their own factor to better than a tenth of a per cent and are refused on the scatter: the golden one at two precisions, the 7.4 at two, and a 4.5 spiral at a thousandth. Every one of them is a curve whose innermost whorl is under one per cent of its outer radius, which is the same geometry the last section priced.

An accurate reading of a real curve is declined while a curve of the wrong kind entirely is waved through. The instrument’s two errors point in opposite directions at once, which is what makes them impossible to fix by moving one number.

The arc that discriminates is the arc to discard #

There is a way to read all this that looks like a repair, and it fails for a reason worth recording. The residual does discriminate near the origin: the Archimedean spiral leaves 0.912 read from a twentieth of a turn out and 0.035 read from four turns out, so the inner whorls are where the two kinds of curve differ most.

But the innermost whorl is also where a genuine fit’s own residual is manufactured, and trimming a measurement to its outer turns is what rescues it. The arc that carries the shape information is the arc that carries the noise, and this collection currently has one recommendation for each. That conflict is not resolved here.

What the residual does report #

Something, and it is worth stating positively so the number is not simply discarded.

It reports the location error, in units of the innermost radius, on the assumption that the model applies. That is a real and useful quantity — it says how well the points were measured relative to the smallest feature the fit depends on, and a residual far above 2.03 times the expected noise means either the points are worse than claimed or the model is wrong, without saying which.

Read that way the residual is a diagnostic with one number and two unknowns, and reporting it as a verdict is asking one equation to settle two things. The useful version of it is a statement about the instrument rather than about the animal, in the way that a parastichy readout turns out to be a requirement on the protractor.

What a real discriminator would have to be #

Three requirements follow directly from the numbers above, and none of them is met by a threshold on a scalar.

It would have to take the noise as an input. The genuine residual is a known multiple of the location error, so the informative quantity is the measured residual divided by that prediction — a ratio near one for a genuine curve at any factor and any span, and unbounded for a curve with no noise at all that still misses. Without an independent estimate of the location error there is nothing to divide by, and no test.

It would have to read the sign structure rather than the magnitude. An Archimedean spiral’s departures are ordered along the arc, because the model error is systematic; a genuine spiral’s are not. Runs of one sign are the signal, and the worst single departure throws exactly that away.

And it would have to be stated with its scope, because the arc read decides the answer across a factor of twenty-six.

What is not measured here #

The second of those three is a proposal and not a result. Nothing in this work counts runs of sign or tests residual structure of any kind; the whole of what was measured is the worst departure, which is what the recovery reports. A reader should treat the sign-structure suggestion as the obvious next measurement rather than as a finding.

The first is measurable now and is not free either: it needs a location error estimated independently of the fit, which for a photographed shell section means knowing the pixel scale and the edge-finding error, and which nothing in this collection currently supplies.

What this does not show #

It does not show the recovered growth factors are wrong. Every genuine spiral in the set comes back at the factor it was drawn at, exactly with no noise and to a fraction of a per cent with realistic noise, and a factor recovered to fifteen digits is still recovered to fifteen digits. The parameter is fine; its warrant is not.

It does not show a nautilus is anything other than a nautilus. The gap between 3.2 and 6.854 is a comparison of two factors and nothing above touches it. And it does not show any real shell was ever accepted wrongly — the curves that get through here are constructed, and whether a shell section is ever Archimedean enough to matter is not a question a synthetic ladder can answer.

What it does show #

That one sentence in this collection was wrong, and the sentence was the one that made the measurement self-checking. A residual reported without an independent estimate of the noise is not a test of shape, and every place the collection has treated it as one is a place where a curve of the wrong kind would have passed unremarked.

That is a smaller claim than it sounds and a more useful one. The fit still measures. What it does not do is police itself, and an instrument that is trusted to police itself is the one nobody points a second instrument at — which is the habit that found this in the first place.