A limit is a length, and a short day is a share

A count that is not exact found the one failure that biases a growth-line reading badly: a section resolves lines only down to some distance, a limit eats the whorls where lines crowd, and every law is pulled towards a length clock without any sign that something went wrong. Whether a section can see its own limit then found that the section can usually catch itself, by comparing how irregular successive spacings are from one whorl to the next — except in a band of limits on a steep shell from a steady animal, where the whorl the limit binds is left exactly as irregular as the animal leaves the others.

That essay ended on a harder version of the question. A section is never read alone; it is read as evidence about the animal. A limit binding the outer whorls of a steep shell makes the reading fall outward, and so does an animal whose law drifted from steep to shallow, or one that changed it partway through its life. Both shells have been read here before. The question is whether anything in a section separates them.

Two shells built to name the same law #

The test needs pairs a person could actually confuse. For each steep law — an area clock, holding the square of the radius constant per unit time, and a volume clock, holding its cube — each animal and each limit, forty sections of the limited shell are read, and the law their outermost pair of whorls names is noted — the reading the growth lines carry, the ratio of two whorls’ counts as a power of the growth factor. A second shell is then built with no limit at all, whose law drifts in a straight line from the same start at the apex to whatever end makes its counts name that same law; a third holds the start until the last whorl and changes there. Every shell is four whorls of four thousand lines on a spiral growing 3.2 times a turn, laid by an animal whose days vary by a tenth, a fifth or three tenths, exactly as in the essay before.

On the volume clock read through a limit of three thousandths of the outer radius, the forty limited sections name 1.87 at the median. The drift that matches it runs from 3 at the apex to 1.57 at the rim and names 1.87; the step that matches it changes from 3 to 1.36 at the last whorl and names 1.87. Across all sixty cells of laws, animals and limits, the matched shells name the limited sections’ law to within seven thousandths.

The counts are not the same. The limited section keeps 1,159 lines and the drifting one all 4,001, and its inner whorls hold 1 and 3 lines where the drift’s hold 2 and 31. But a person with a section does not know how many lines the animal laid, and inner whorls this empty are not a reading. What the counts can say, they say identically.

The spreads test is blind on both #

The instrument from the essay before compares, across the whorls, the spread of the logarithm of each spacing over the one before — how irregular successive spacings are, whatever smooth trend runs along the whorl — by a permutation test at the one per cent level. On the drifting and stepped shells it flags at most four and five sections of forty at any cell, which is right: nothing is wrong with them.

On the limited volume clock from the steadiest animal it flags 36 sections of forty at the first limit that binds, 19 at a thousandth, and then 3, 1, 0 and 0 as the limit rises to four thousandths, before every section is flagged again from six. Across that band the reading falls from 2.34 to 1.65. So inside it the spreads test cannot tell a limited section from a changed law, because it flags neither, and outside it a flag says “limited” and silence says nothing.

What a limit is and an animal is not #

The way out is to ask what the two things are, rather than what they do to a summary. A resolution limit is a length. Two lines closer than it are read as one, so every spacing a limited section keeps is at least that length — 0.315 of the spiral’s own arc units at three thousandths of the outer radius on this shell — and that floor is the same at the start of the outer whorl and at the end, wherever the spacing around it is wide or narrow.

An animal’s irregularity is a share. A short day lays a short spacing, and how short is in proportion to the spacing the animal’s law lays there: where lines are a millimetre apart a short day leaves eight tenths of a millimetre, and where they are two millimetres apart, sixteen tenths. A drifting law, a stepped law or no change at all moves the spacing along the whorl, and the shortest spacings move with it.

So along a whorl whose spacing changes, the two make opposite predictions about the shortest spacings: the animal’s follow the local spacing, and the limit’s stay put.

The floor test #

The outer whorl’s spacings are cut into sixteen runs of equal count. Each run gets a scale — its median spacing — and a floor: the smallest of its spacings, each first divided by the median of the forty-one spacings around it. Divided that way an animal’s floors are alike from run to run whatever its law, because a short day is a share; a limit’s floor is one length, so where the run’s scale is wide its divided floor is low.

The test is the rank correlation between scale and floor across the sixteen runs, against the same correlation with the floors dealt back among the runs a thousand times. A section is flagged when fewer than one shuffle in a hundred is as negative as the real one.

On one section of each, at a limit of three thousandths, the difference is visible before it is computed. The drifting shell’s divided floors lie flat at about 0.72 across a doubling of scale, a correlation of 0.12. The limited section’s sit higher and fall as the scale widens, from 0.93 to 0.57, a correlation of −0.74, which a thousand shuffles never matched. At a limit of a thousandth, the dial’s first stop, the limited section’s floors are still falling, at −0.61, but less cleanly, and that is where the test will turn out to be weakest.

It never accuses a changed law #

Across all sixty cells — two steep laws, three animals, ten limits — the floor test flags at most two of forty drifting sections and at most two of forty stepped ones. At a test of one per cent that is what chance alone should do, and it is the property that matters most: a check that accuses a real change of being an artefact is worse than none, because it would talk a person out of the finding.

On the limited sections it does the opposite of the spreads test. From the steadiest animal it flags the volume clock on 33, 38 and 40 of forty at limits of two, three and four thousandths — the whole of the band the spreads test is blind in — and on every section beyond. The area clock from the same animal is flagged on 35 to 38 of forty from one and a half thousandths up.

Where it is weak #

It is weak at the first limits to bind, which is where the spreads test was strong. At five ten-thousandths the volume clock from the steadiest animal is flagged on 4 sections of forty by the floor test and 36 by the spreads test; at a thousandth, 16 and 19; at one and a half thousandths, 16 and 3.

The reason is in what a light limit does. Binding lightly, it removes a line here and there, and most surviving spacings are the animal’s own; the floor the limit sets is reached in only a few runs, and sixteen runs are not enough to see a trend in a handful. Binding harder, every run’s shortest spacing is the limit’s, and the trend is plain. The spreads test sees the opposite — a light limit makes successive spacings more irregular, and a heavy one evens them out until they match the animal.

So the two tests are not rivals. Like dividers set against a shell’s opening, each is exact over one range and blind over another, and what matters is where each range ends. One sees a limit as it starts to bind and the other once it has, and the band in which a limit moves the reading without either seeing it is narrow.

Which test catches what #

Read together — a section counted as caught when either test flags it — the limited sections are caught in at least 34 of forty at every cell where the reading has moved by more than a tenth, with three exceptions, all from the steadiest animal and all at the first limits to bind. The area clock at a thousandth, read at 1.68, is caught 19 times; the volume clock at one and a half thousandths, read at 2.34, 17 times; and the volume clock at a thousandth, read at 2.57, 27 times by the two together and fewer than twenty by either alone.

Reading two tests at one per cent each doubles the chance of a false flag, and on the matched shells the two together flag at most five sections of forty. That is a little above the two per cent the levels promise, from the spreads test’s own slight excess, which the earlier essay traced to successive log ratios being negatively correlated.

Every animal, and the animals that help #

The steady animal is the hard case, and the irregular ones are easy. From an animal whose days vary by a fifth or three tenths, the floor test flags every steep limited section on 35 to 40 of forty from the first limit that moves the reading, the volume clock at five ten-thousandths included, which moved only from 3 to 2.85.

That is the opposite of what an animal’s irregularity does to the spreads test, where an irregular animal pushes the blind band to the edge of the range. Here an irregular animal helps directly: its spacings scatter widely below the local median, so a limit cutting that scatter at one length stands out from the first runs it reaches.

How wide a window to divide by #

The width of the window each spacing is divided by is a setting, and a setting nobody varies is a number nobody has checked; here it is the one thing in the test that can hide what it is meant to find. Divide by the median of only eleven spacings and, in a run the limit has evened out, the median is the limit too, so the divided floor sits near one in every run and the trend disappears. At the area clock and one and a half thousandths, eleven spacings flag 15 sections of forty, twenty-one flag 26, forty-one flag 35 and eighty-one flag 39.

A wider window costs nothing in honesty here. At every width from eleven to eighty-one the test flags at most one of forty unlimited sections of any law from the steadiest animal. The numbers above use forty-one, which was chosen after seeing eleven and twenty-one and before seeing eighty-one; the wider window would have closed part of the weak band, and a section read for real should be read with the widest window its outer whorl allows.

How honest the test is #

On shells with no limit and no change — the angular, length, area and volume clocks from all three animals — the floor test flags between none and two sections of forty, and the spreads test the same. Neither depends on the law, because both compare a whorl with itself after a smooth trend is taken out.

The floor test has one case it can say nothing about, and it is worth naming. On a length clock the spacing is the same all along every whorl, so the runs have no range of scales to correlate a floor against. It flags nothing there, rightly; but a limited section whose outer whorl happened to have an even spacing would be invisible to it too. A limit on a steep shell does not produce that: its surviving spacing still follows the crowding beneath it, and in the blind band the limited outer whorls span a factor of 1.4 to 1.6 in scale.

What this says to a person holding a section #

If the reading falls outward and the spreads test is silent, that silence is not evidence of a changed law, any more than a residual that accepts a fit is evidence the fit is right. Run the floor test on the outer whorl. A strong negative correlation between a run’s scale and its divided floor says the whorl is limited, and the law it names is pulled towards one by an amount the section cannot state; a flat one says the shortest spacings follow the animal, and a change in the reading is a change in the animal.

What a flat result cannot rule out is a limit just starting to bind, from a steady animal, where the reading has moved by a few tenths. That is the residual the two tests leave between them, and it is a better position than the earlier budget of a section’s errors had: a known, narrow band rather than every steep reading below its law.

Limits that are not lengths, days that are not independent #

That a real section’s limit is one sharp length. A section worn more on one side than the other, or a contrast that fades along the whorl, would have a limit that changes, and a limit that grew in proportion to the spacing would look to this test exactly like an animal.

That an animal’s days are independent. A rhythm — lines laid tidally rather than daily — that lengthens every fourteenth day, as tides do, puts a structure into the spacings that is a share of the local spacing, so the test would leave it alone; but a rhythm whose short days had a fixed minimum length — a minimum time to lay a line — would put a floor in length into an unlimited shell, and the test would read it as a limit. Nothing here models either.

Nor that four thousand lines over four whorls is typical. The test needs an outer whorl long enough to cut into sixteen runs of eight, and a shell with fewer lines there could not be read at all.

Where the floor test would be wrong #

A drifting, stepped or unlimited shell flagged by the floor test on more than three sections of forty at any cell. A limited volume clock from the steadiest animal, at two, three or four thousandths, flagged on fewer than thirty. A matched drift or step naming a law more than a twentieth from the limited sections it was matched to. A narrower window flagging more limited sections than a wider one.

A floor in length against a floor in share #

A section whose limit pulls a steep law towards a length clock, and a shell whose animal changed its law, name the same law from the same whorls, and the test on the whorls’ spreads is blind on both across a band of limits. They separate on what a limit is: a floor in length, where an animal’s short days are a share of the spacing around them. A test of whether the outer whorl’s shortest spacings follow its local spacing flags at most two of forty drifting or stepped sections and catches the limited ones throughout the band the spreads test cannot see. Read together, the two tests leave a limited section uncaught in only two cells of sixty, both at the first limits to bind on a steady animal.

Still open: a limit that is not one length #

Every limit here is a sharp length, the same everywhere on the section. A real section’s resolution depends on contrast and wear, which vary along a whorl and between whorls, so its limit is closer to a length that changes slowly. The next measurement gives the limit a gradient — tighter at the rim than at the apex, or the reverse — and asks how steep a gradient has to be before the floor test’s correlation, which assumes one length, stops separating a limited section from an animal, and whether reading the floor against position along the whorl as well as against the local spacing recovers it.