Whether a section can see its own limit

What the growth lines carry showed that two whorls’ growth-line counts name the law an animal grew by: the ratio of the counts is the growth factor raised to the power the law holds constant, so a logarithm returns the power. A count that is not exact then found the one way that reading fails badly. A section resolves lines only down to some distance, and because lines crowd at one end of a shell or the other according to the law, the limit eats more of some whorls than others. Every law is pulled towards a length clock, $p = 1$, and where the limit binds completely every law reads exactly 1, cleanly, with no sign that anything went wrong.

That essay ended on the question this one answers. The counts cannot see the limit. Can the spacings? A section reading its own limit has surviving lines about one limit apart. A limit truncates from below, and an animal does not. Whether that difference survives at the number of lines a single whorl holds was not measured.

The failure being guarded against #

Drawn over a sweep of limits, the failure is a set of curves converging on one. An angular clock reads 0 until the limit reaches the inner whorls’ spacing and then climbs; an area clock and a volume clock read 2 and 3 until the limit reaches their crowded outer whorls and then fall; a length clock reads 1 throughout, because its lines are evenly spaced and every whorl loses the same share. By a limit of a hundredth of the outer radius every law is somewhere near 1.

Nothing in the counts marks where along that sweep a section sits. So any check has to come from something the counts throw away, and the spacings are the obvious candidate.

An animal, for the first time #

Every line in the earlier readings was laid at exactly equal intervals of time. That made the question of spacing trivial and also unanswerable: a perfectly regular animal and a section at its limit both produce evenly spaced survivors.

So the animal here is given days that vary. Each interval between lines is drawn from a gamma distribution with a mean of one and a coefficient of variation of a tenth, a fifth or three tenths, seeded, and four thousand lines are laid over four whorls of a shell growing 3.2 times per turn under an angular, a length, an area or a volume clock. Forty sections are read at each of eleven limits from none to sixteen thousandths of the outer radius. The person reading each section knows only the section: its counts, and the positions of the lines it resolves.

One whorl cannot tell #

The first instrument is the one the earlier essay proposed: within one whorl, how close the shortest spacing sits to the median. A limit pushes it towards one, because every surviving pair is at least a limit apart and most are only a little more. On a volume clock from an animal whose days vary by a tenth, the outer whorl’s floor rises from 0.47 with no limit to 0.71 at a limit of a thousandth, 0.87 at three thousandths and 0.97 at sixteen.

A length clock with no limit at all shows a floor of 0.70 when the animal’s days vary by a tenth, 0.46 when they vary by a fifth, and 0.28 at three tenths. A steadier animal would show a higher one still. So a floor of 0.87 in one whorl is either a limit or a very steady animal, and nothing in that whorl says which. The within-whorl reading needs to know the animal’s steadiness, which is the one thing a section does not record.

Comparing whorls instead #

The animal is unknown, but it is the same animal in every whorl. Its steadiness does not change from one whorl to the next, and a limit’s effect does: a limit binds where lines crowd and leaves the rest alone. So the useful measurement is not how regular a whorl is but whether the whorls agree.

The quantity compared is the spread of the logarithm of each spacing over the one before it. It measures how irregular successive spacings are and is untouched by the smooth trend in spacing along a whorl that every law but the length clock has. That matters more than it sounds: on an angular clock the spacing grows by the growth factor, 3.2, across every whorl, which on its own gives the raw spacings in a whorl a coefficient of variation of 0.33 — more than three times a steady animal’s. A test on raw spacings would be measuring the law. The ratio of successive spacings removes the trend, since the trend changes each spacing by the same factor, about a thousandth, and leaves only the animal and the limit. On an unlimited length clock it is the animal’s own irregularity times root two in every whorl — 0.140 at a tenth, 0.28 at a fifth, 0.43 at three tenths. The whorls are then compared by a permutation test: their centred log ratios are pooled and dealt back at the observed sizes two hundred times, and a section is flagged when the largest spread over the smallest is exceeded by fewer than one shuffle in a hundred.

On an angular clock, the section always sees it #

For a clock shallower than a length clock the answer is as good as it could be. Lines spread outward on an angular clock, so a limit eats the innermost whorl first and the outer whorls last, and the spreads disagree from the first limit that binds anywhere. At two ten-thousandths the innermost whorl keeps 430 of its thousand lines and its spread rises from 0.141 to 0.168, the next whorl’s to 0.186, while the outer two stay at the animal’s 0.142 and 0.141.

That limit costs the reading nothing, because a person reading the law takes the outermost pair of whorls, and the limit has not reached them. The test is seeing the limit in whorls the reading does not use, which is the best possible place to see it. Every one of forty sections is flagged at every limit from two ten-thousandths upward, for all three animals — including the smallest limit, at which the reading is still 0.00, within a hundredth of the truth. On an angular clock the section reports its limit before the limit has cost anything.

A length clock is the opposite case and behaves as it should. Its lines are evenly spaced along the arc, which is what a constant length added at the opening means. No limit biases its reading, because every whorl loses the same share, and the test flags it at no more than the rate it flags unlimited sections. Read from the steady animal at every limit up to sixteen thousandths, it names 1.00 each time, to within four thousandths, and is flagged on between none and four sections of forty at the one per cent level; from the irregular animal, on at most two. At the largest limit its surviving lines are a limit apart in every whorl, which is exactly the regularity a limit imposes on any law — but a length clock loses the same share of every whorl, so its whorls stay alike and the test has nothing to find. It is never wrong and, near enough, never accused, and a section that reads 1 with agreeing whorls is either a length clock or a limited steep shell in its blind band, which is why a reading of 1 deserves the most suspicion of any.

On the steep clocks, it depends on the animal #

The area and volume clocks crowd their lines outward, so a limit eats the outer whorls first. With an irregular animal the test still works: at three tenths, a volume clock is flagged on 38 of 40 sections at a limit of a thousandth and on every section from 0.0015 upward. Only at the very first limits to bind, five ten-thousandths, where the reading has moved from 3 to 2.85, is it flagged on 11 of 40.

On these clocks the test also has less to work with. A limit that binds the outer whorls empties the inner ones below the twenty lines a whorl needs to be read at all, so at three thousandths a volume clock holds 1, 3, 118 and 1,037 lines in its four whorls and the test is a comparison of two whorls, one limited and one not. An area clock at two thousandths holds 4, 35, 344 and 1,471 and compares three.

With a steady animal the comparison goes blind over a band of limits. At a tenth, a volume clock is flagged on 36 of 40 sections at five ten-thousandths, 19 at a thousandth, then 3, 1, 0 and 0 as the limit rises to four thousandths — while the reading falls from 2.57 to 1.65 — before the test recovers at six thousandths and catches every section. An area clock from the same animal has a similar gap: read at 1.68, 1.44 and 1.24 at limits of one, one and a half and two thousandths, it is flagged on 4, 2 and 0 sections of forty.

Where the blind band is, and why #

Drawn as readings, the blind band is a stretch of the area and volume curves where the law named is badly wrong and the test is silent. Its location has a cause that the dial on the second figure shows, and it is not the one the question assumed.

A limit does not simply make a whorl’s spacings more regular. Binding lightly it makes them less regular: a surviving line sometimes follows the last after one missed line and sometimes after none, and those two gaps differ by a whole spacing. On the steady animal’s volume clock the outer whorl’s spread goes from the animal’s own 0.142 with no limit to 0.222 at five ten-thousandths — and the section is flagged on 36 of 40. Binding harder, the survivors are separated by more and more missed lines and their gaps even out: 0.194 at a thousandth, 0.174, 0.159, then 0.139 at three thousandths, 0.125 at four, 0.074 at eight.

Somewhere on that descent the limit’s own irregularity passes through the animal’s. For this animal it is at three thousandths, where the limited outer whorl reads 0.139 and the unlimited whorl inside it 0.146. The two whorls agree, and a test that looks for disagreement finds none.

The same shell from an animal whose days vary by three tenths starts at 0.436 in the outer whorl, and nothing a limit does to a whorl comes near that: the limited whorl reads 0.392 at five ten-thousandths, 0.318 at a thousandth and 0.190 at three, always below the animal’s own and falling away from it. The crossing happens at the very first limit, which is the one limit — five ten-thousandths, 11 sections of 40 flagged — at which this animal’s volume clock hides anything.

So the blind band sits wherever the limit’s descent crosses the animal’s steadiness. A steady animal meets it in the middle of the range of limits, where the reading is already badly wrong; an irregular one meets it at the edge, where the reading has barely moved. Between them, at a fifth, the band sits at five ten-thousandths and a thousandth, with readings of 2.88 and 2.55 flagged on 6 and 5 sections. The band moves outward and widens as the animal gets steadier, and past six thousandths, when the limit reaches the next whorl in and the whorls disagree again, the test finds every section whatever the animal.

How honest the test is #

A check that accuses clean sections is worse than none, so the rate at which it does has to be measured, not assumed. On sections with no limit, the one per cent test flags at most two of forty for any law or animal. The five per cent test flags between one and six of forty — somewhat more than one in twenty — because successive log ratios within a whorl are negatively correlated, one long day making the next ratio small, and a shuffle does not preserve that. So the one per cent level is the one to use, and every result above is stated at it.

What a person with a section can do #

The practical reading is a short list. If the whorls’ spreads disagree, the section is limited, and the law its counts name is pulled towards 1 by an unknown amount; the earlier budget of what a section can get wrong has one more entry, and it is not small. If they agree and the shell is shallow, the reading can be trusted as far as the limit goes. If they agree and the reading is between 1 and 3, the section might be limited and hiding it, and the only way to tell is to know how steady the animal was — from a living relative, a laboratory record, or a stretch of the same shell sectioned at a finer resolution.

That last case is the unwelcome one, because steep clocks are also the ones the earlier reading found most fragile: a volume clock is already biased when a sixth of its lines are gone. The laws most easily damaged by a limit are the laws on which the damage is hardest to see.

It is the same shape of failure the residual is not the test found in a fit that accepted a circle and refused a golden spiral: an instrument whose check is quiet on exactly the input it handles worst. The difference here is that the quiet has a measurable location, and a reading that lands in it can be reported as undecided rather than as a law — a count that says how it could be wrong rather than one that does not.

What this does not establish #

That real animals lay lines at independent, gamma-distributed intervals. Many do not: tidal and lunar rhythms put a periodic structure into line spacings, and a period that differs between whorls would read to this test as a limit. That a real section’s limit is a sharp distance, rather than a function of contrast and wear. And that four whorls of 4,000 lines are typical; a section with more lines per whorl has a more sensitive test, and one with fewer a blunter one.

It also does not repair the reading. A flagged section is a section whose law is uncertain, not one whose law has been corrected; the essay on changing laws showed how much a sequence of whorl ratios can say, and a limit binding in some whorls and not others will also look like a law that changed.

What would withdraw it #

An angular clock at any limit read whose sections are flagged less than every time. A section with no limit flagged at the one per cent level more than twice in forty. A volume clock from a steady animal at a limit of three thousandths that is flagged — or one from an irregular animal at the same limit that is not.

Still open: whether a limit and a changed law can be told apart #

A limit that binds in the outer whorls of a steep shell and not the inner ones produces a sequence of whorl ratios that falls outward, and so does an animal whose law drifted from steep to shallow. Both readings exist here, and both would be flagged or not by the same test. The measurement is the pair side by side: a drifting law from a steady animal against a limited fixed law, read by the ratio sequence and by the whorls’ spreads together, asking whether any combination of the two separates a shell that changed from a section that could not see it.