A limit that loosens along the whorl hides from the floor test

A limit is a length, and a short day is a share found a way to tell a shell section that could not resolve its finest growth lines from a shell whose animal had changed its deposition law. Both make a volume clock’s whorl counts — the lines in successive whorls, whose ratio names the law — name a shallower law than 3, and the counts cannot tell them apart. The floor test can. Cut the outer whorl’s spacings into sixteen runs; give each run a scale, its median spacing, and a floor, its shortest spacing divided by the median of the spacings around it. An animal’s short days are a share of the days around them, so its divided floors are alike from run to run. A limit is a length, so where a run’s scale is wide its divided floor is low, and the rank correlation between scale and floor runs negative. The floor test flagged 33 to 40 of forty limited sections across the band where the older spreads test was blind, and at most two of forty drifting or stepped shells.

Every limit there was one length, the same everywhere on the section. That essay ended by saying a real section’s resolution depends on contrast and wear, which vary along a whorl, and asked how steep a gradient in the limit has to be before the floor test, 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.

A limit that changes by a factor a turn #

The shells are the earlier ones: a volume clock — one of four clocks that trace the same curve, the one under which the aperture adds a constant volume a day — laid in four whorls by an animal whose days vary by a tenth, a fifth or three tenths, four thousand lines a section, forty sections a cell. The limit is anchored at the middle of the outer whorl at two, three or four thousandths of the outer radius — the band in which the spreads test was blind — and changes by a factor $G$ for every turn outward. At $G = 1$ it is the fixed limit of the earlier essays. At $G = 2$ it is twice as coarse a turn further out, as a section worn more at the rim would be; at $G = 0.5$ it is twice as fine, as a section whose rim was better preserved. The gradients read run from a quarter to 6.3.

The section is walked from the apex as before, keeping one line for every local limit’s length, and read for what a person holding it can compute.

The test against a limit that loosens #

From the steadiest animal the floor test flags 33 to 40 sections of forty at the fixed limit, and 36 to 39 when the limit tightens outward by half a turn. When it loosens outward by half again, at $G = 1.5$, it flags none at any of the three limits; at two, three and 6.3, none. From the two less steady animals it is the same: none of forty at every gradient from two up, and at 1.5 eight of forty at the lightest limit for the middle animal and 34 for the least steady one, with every other cell at nought.

So the gradient at which the test stops working is not steep. A limit that is one and a half times coarser a turn further out is as invisible to it as a section with no limit at all, and so is every steeper one read.

It is not that the limit has stopped binding #

The obvious explanation would be that a loosening limit binds less, leaving the animal’s own spacings to speak. It is the opposite. On the steadiest animal the limit removes 63, 73 and 79 per cent of the outer whorl’s lines at the three fixed limits, and 66, 75 and 80 per cent at a gradient of two; at 6.3 it removes 70, 78 and 82 per cent. The counts, read from the outermost pair of whorls, name a law of 2.15, 1.87 and 1.65 at the fixed limits and 2.07, 1.80 and 1.59 at a gradient of two — further from the animal’s 3, not nearer. A limit that loosens outward binds the outer whorl harder and moves the reading more, and the floor test, which saw the fixed limit, sees none of it.

Over every animal, limit and gradient read the limit removes between 54 and 83 per cent of the outer whorl and the counts name a law between 1.5 and 2.4. A section read through a limit the floor test cannot see is still a section whose law has been moved by most of a whorl.

Why: the limit makes the scale #

One section shows why. With no limit, the volume clock’s spacings shrink sixfold along the outer whorl — from 0.31 in the apex-end run to 0.05 at the rim, in the section’s own units — because the animal lays lines at a steady rate in time and the whorl’s arc grows more slowly than its time. A fixed limit of three thousandths is 0.32 in those units, coarser than the animal’s median spacing in every run of that whorl. It binds in every run: the shortest spacing is 0.32 in fifteen of sixteen runs. The survivors are, almost all of them, the limit’s spacings, and the run’s scale is set by the limit too — 0.34 to 0.51, widest at the apex end where the animal’s own spacings come closest to the limit and leave a little of their irregularity behind.

That small leftover is what the floor test had been reading. Where the animal’s spacing nears the limit, the scale widens and the divided floor drops, and the rank correlation runs negative: −0.74 on this section. The test assumed a limit binding against a varying spacing. What it found, on the fixed limit, was a limit binding everywhere against a spacing it had mostly replaced, and a trend in the residue.

Turned to a gradient of two, the shortest spacing climbs steadily from 0.28 at the apex end to 0.43 at the rim — the limit itself, loosening — and the scale climbs with it, from 0.42 to 0.46 with the same wobble. Divided by its own surroundings the floor is now flat or rising, and the correlation is +0.41. A limit that changes along the whorl drags the scale with it, so the floor it sets follows the local spacing exactly as an animal’s share would. The test’s premise was that a limit is one length; a limit that is not one length is, to this test, a share.

A limit that sharpens towards the rim #

The tightening side is not symmetrical with the loosening one. At a gradient of a half — the limit twice as fine a turn further out — the floor test still flags 36 to 40 of forty, from every animal. At a quarter, four times finer a turn out, it depends on the animal: from the steadiest it flags 5, none and 3 of forty at the three limits; from the middle animal 14, 39 and 40; from the least steady, all forty at each.

A limit four times finer a turn out still binds along the whole outer whorl — on one section from the steadiest animal its shortest spacings fall from 0.51 at the apex end to 0.16 at the rim, against the animal’s own 0.31 to 0.05 — so the scale again follows the limit, from 0.69 down to 0.20, and the divided floor is nearly flat: a correlation of −0.18, which the shuffles match one time in four. It is the loosening limit’s blindness run the other way. Why the less steady animals restore the test at this gradient, when they do not at any loosening one, is not traced here. None of this changes what the limit does to the reading: at a quarter it still removes 54 to 75 per cent of the outer whorl, and the counts name 2.3 to 2.4.

Against position #

The earlier essay’s suggestion was to read the floor against position along the whorl as well as against the scale. On a single whorl of a volume clock the two are not independent — the animal’s spacing shrinks along the whorl, so position and scale move together, and a test of the floor against position in the same direction as the floor test would see what the floor test sees. What it can do differently is look for a trend in either direction.

An animal’s divided floors are exchangeable along the whorl, whichever way the spacing runs; a limit that changes along the whorl leaves its divided floors trending one way or the other. Tested two-sided against position, the same floors flag 33 to 40 of forty at every gradient from two up, at every animal and limit, and 25 to 40 at 1.5. It is weak where the floor test was strong: at the fixed limit and the steadiest animal it flags 8, 39 and 40 of forty at the three limits, and at a tightening gradient of half a turn 3, 11 and 22. Two tests, each blind where the other sees, is the pattern the floor test and the spreads test already had. It never accuses an unlimited section much: at most two of forty steady, drifting or stepped sections, as a one per cent test should.

The spacings are piled against the limit #

There is a reading that does not depend on the limit being one length, and it is in the picture above. Walking a section from the apex and keeping one line per limit’s length leaves the surviving spacings just above that length — every gap the limit closed was smaller than it, and the line kept next is the first one past it. So inside a run, many spacings sit within a hair of the run’s shortest. An animal’s days do not do this: its shortest spacing is the end of a tail, and only a few spacings lie close to it.

In the middle run of one section, 2.5 per cent of the animal’s spacings lie within five per cent of the run’s shortest; through a limit changing by two a turn, 16.7 per cent do. Call the share of a run’s spacings within five per cent of its shortest, averaged over the sixteen runs, the pile.

The pile, everywhere #

Over every limited section read — three animals, three limits, seven gradients, 2,520 sections — the pile is at least 0.060, and its median in a cell runs from 0.073 to 0.25. Over every section with no limit — steady animals of all three kinds, and the drifting and stepped laws matched to each fixed limit, 840 sections — it is at most 0.021, and its median in a cell from 0.006 to 0.015. The two ranges do not touch.

So a threshold can be set without knowing the animal: twice the largest pile any steady section shows, 0.036. Against it every limited section in every cell is flagged, at every gradient from a quarter to 6.3, and no unlimited section, steady, drifting or stepped. The floor test asked whether the shortest spacings follow the local spacing, which is a question about how a limit varies. The pile asks whether the shortest spacing is a floor at all, which is a question about what a limit is.

What it does to the earlier map #

The earlier essay ended with two tests read together — the spreads test, strong where a limit first binds, and the floor test, strong once it binds hard — leaving a limited section uncaught in two cells of sixty. That map was drawn for fixed limits. On limits that loosen outward, the floor test’s half of it is gone at every cell read here, and the spreads test’s half was never strong in this band: the band itself was chosen because the spreads test is blind there. So on a volume clock read through a limit of two to four thousandths that loosens by half again a turn, the two tests together catch nothing.

The pile is not a third half. It flags every limited section in every cell here, fixed, tightening and loosening, and it does so on the same spacings the other two tests read. Where a person had two tests to read side by side and a stated gap between them, the measurement here leaves one test that covers the band and two that cover parts of it — and no gap, inside the shells read.

What it costs a person with a section #

Reading the pile needs nothing the floor test did not: the outer whorl’s spacings, cut into runs. It needs no knowledge of the animal’s irregularity, since the threshold was set from the steadiest animal and holds for all three, and none of whether the limit is fixed, loosening or tightening. And it tells the person something the floor test could not, because it flags a limit whose gradient has hidden it from the floor and whose effect on the law is largest — the hardest case, from the person’s side, is the one it catches most plainly.

What it does not tell them is how much the limit moved the reading. A section flagged by the pile has had most of its outer whorl removed, and the counts from that whorl are what named the law; the honest response is the one the earlier essays gave, to read the law from the inner whorls where the limit has not bitten, and to say that the outer whorl’s law is not known.

A pile a rhythm could make #

The pile has a weakness the floor test shares, and it is the one the earlier essay named. Every animal here lays its lines at independent gamma intervals. An animal with a rhythm — one day in fourteen longer than the others, as tidal growth lines are thought to be — would leave a floor of its own in the spacings, and if the rhythm’s short days were all alike, they might pile. That is a different shell, and nothing here grows it.

Every limit here also changes by a constant factor a turn. A section worn in patches, coarse in one stretch of the whorl and fine in the next, would give the floor test something different again, and a trend test against position would see a patch only if it were long. The pile does not care how the limit varies, which is the reason to trust it more than either test against position or scale; it has been checked here only on limits that vary smoothly.

Readings that would undo it #

A limit loosening outward by a factor of two a turn or more that the floor test flags on any of forty sections. Such a limit removing under half the outer whorl’s lines, or leaving the counts naming a law above 2.45. A two-sided trend against position flagging fewer than 33 of forty at a gradient of two or more. A limited section whose pile does not exceed twice the largest steady pile, or an unlimited section — steady, drifting or stepped — whose pile does. Each is checked against the measured sections whenever they are read.

A limit is a floor before it is a length #

The floor test treated a resolution limit as a length that stays put while the spacing changes around it. A limit that changes along the whorl by half again a turn breaks that, because a limit that binds sets the spacing the test compares it with; the test goes blind while the limit removes two thirds of the outer whorl and moves a volume clock’s reading to 1.5 to 2.4. A trend in the floor against position, tested either way, recovers it where the floor test fails and misses it where the floor test succeeds. The spacings themselves recover every case: a limit, however it varies, leaves a run’s spacings piled against its shortest, a share of 0.06 or more where no animal’s days pile past 0.021.

Still open: a rhythm and a limit together #

The pile separates a limit from an animal whose days are independent. A growth rhythm makes some days systematically short, and if those days are alike they pile too. The next measurement lays sections from an animal with a fortnightly rhythm — one day in fourteen shortened by a stated share — with and without a limit, and asks whether a rhythm’s pile reaches the limit’s, whether the two can be told apart by where the piled spacings fall in the sequence, a rhythm’s every fourteenth and a limit’s anywhere, and whether the law a changing animal names can still be read from a section carrying both.