The plateau was a prediction

An earlier essay wanted a reference organ. The census’s readings of which family survives a removal are stated in terms of the offset — how many places back the removed organ sat from the growing tip — and an offset is a coordinate chosen by the experimenter. A quantity read off the disturbance itself would be a coordinate the stem chose.

The obvious candidate was the organ that moved furthest. It does not exist, and this essay is about why the way it fails is now a consequence rather than a setback.

What was measured then #

For every wrecked cut, the largest displacement in a window three times the larger counted number wide, and the share of that window within a tenth of it.

The largest displacements ran 105° to 180° against a ceiling of 180°, and at sixteen of thirty rows a fifth of the window sat within a tenth of the maximum. Over the last sixty organs of each run — hundreds of organs above the hole — the displacement was still 70° to 179°.

Three readings, one conclusion: the disturbance has no far edge and no peak, so “the organ that moved furthest” names an arbitrary member of a wide flat.

What it was read as #

A failed instrument. The record was that the repair for a reading answerable on nine rows of thirty did not work, and the thread moved on to a different repair — which side of the tip the removed organ sat on — that is answerable on all thirty and scores 22.

That was the right move and the failure was left as an observation. A disturbance that does not decay is odd, it was reported as odd, and nothing followed from it.

What follows from it now #

The displacement above a hole is constant inside each residue class modulo the lag the stem kept. A profile like that has exactly k values in it, not three hundred.

So its maximum is one of those k values, and the maximum is attained not by an organ but by every organ of one class. In a window of n organs that is about n/k of them, and k across this census is between 4 and 11.

A ninth to a quarter of any window sits exactly at the maximum, and the whole class sits within the spread of a class — between 0.12° and 6.09° — of it. A tenth of a maximum of 140° is 14°, comfortably more than that spread, so the share within a tenth of the maximum is the share of the window in that class.

The arithmetic against the measurement #

A fifth of the window was what sixteen of thirty rows showed. One over k for k between 4 and 11 is between 0.09 and 0.25.

A fifth is 0.20, which is one over five, and five is the commonest surviving lag in the census. So the number reported then as evidence that the maximum was a plateau is, to the precision it was quoted at, one over the surviving lag.

That is not a coincidence explained after the fact; it is what a periodic profile forces. And it is testable in the direction that matters: a row whose surviving lag is 8 should show an eighth of its window at the maximum, and a row whose lag is 11 an eleventh.

The prediction is sharper than that, and the sharper form is what makes it worth checking rather than assuming. The share counted is not the share at the maximum but the share within a tenth of it, and a tenth of a maximum of 140° is fourteen degrees — several times a class’s own spread, so a second class sitting within fourteen degrees of the top one is counted too, whole. So the share is not 1/k. It is an integer over k: one over the lag if one class is at the top, two over it if two are close, and so on.

Which is a quantised prediction, and a quantised prediction is easy to refuse. On a row with a surviving lag of 8 the share must be 0.125, 0.250, 0.375 and nothing in between; on a lag of 5 it must be 0.20, 0.40 or 0.60. A row reporting 0.31 at a lag of 8, or 0.30 at a lag of 5, would refute the whole account — and sixteen of thirty rows reporting “about a fifth” is consistent with it only because a fifth is exactly 1/5 and 2/10.

That check has not been made. It is one column beside a column that already exists, and it converts a plausible reconstruction into either a confirmation with a quantisation in it or a clean failure.

Why “no far edge” follows too #

The other half of that finding was that the disturbance is still tens of degrees at the top of the run. A periodic profile explains that with nothing left over: the levels do not decay, so the displacement three hundred organs above the hole is the same as the displacement a hundred above it.

The smallest such tail in the census is 45° and most are far larger. A profile whose classes sit at 0° and ±134° has a tail of 134° whatever window it is read over, forever, because that is what the arrangement now is.

So “the damage does not decay” is not a statement about the damage spreading. It is a statement that the stem settled somewhere else and stayed there.

What a reference organ would have needed #

A profile with a genuine peak: one organ, or a few, displaced much more than the rest, with the displacement falling away either side.

That shape is what a transient looks like, and there is one — below the pattern, where the stem is still moving. A reference organ read there would be reading a real feature.

The trouble is that the window used was three times the larger counted number, which at a 5/8 stem is twenty-four organs and at an 8/13 stem thirty-nine. The transient’s extent runs from 7 to 303 organs, so the window sometimes sits inside the transient, sometimes straddles the boundary, and sometimes is entirely in the pattern. A statistic taken over a window that is in a different regime on different rows is a statistic measuring different things on different rows, and averaging it across the census averages two quantities rather than thirty readings of one.

Which is the sharper criticism #

The reference organ did not fail because the disturbance is featureless. It failed because the window it was looked for in was not aligned to any feature.

That is a criticism of the instrument rather than of the object, and it is one this collection has made before in a different thread: a mean taken over a partial period of a repeating motif is biased by the angles it counts twice. Same shape of error — a window that does not respect the structure of what it is measuring — found the same way, by finding the structure first.

What could be measured instead #

Three things, and all of them are now available.

The level of each class, which is k numbers rather than one, and which carries the whole of the difference between a wrecked stem and its control.

The transient’s extent, which is a genuine coordinate the stem chose: the first organ from which every class stays at its own level. It runs from 7 to 303 organs and five rows never reach it.

And the size of the exchange, which is one divergence step and is the same number on every row to within eight per cent.

None of those is a reference organ, and none of them needs one.

What the plateau share should be, row by row #

The prediction has a specific form and the library asserts a weak version of it: that on every periodic row at least a thirteenth of the window sits at the maximum, since the largest surviving lag in the census is 11.

A stronger version — that the share is one over the surviving lag, row by row — would need the plateau share recomputed with a window aligned to the pattern rather than to the counted pair, which is a different measurement from the one made and would not be comparable with the number it is meant to explain.

So the strong version is stated and not asserted, which is the honest way round.

What this does to the earlier conclusion #

Nothing. The conclusion was that there is no reference organ and that a different reading was needed, and both remain true.

What changes is the status of the observation. “The maximum is a plateau” was an awkward fact about a measurement that did not work. It is now a consequence of the shape of the damage, and a consequence that could have been predicted from the shape before the search was run.

That is worth recording as an ordering: the shape was measurable then from data that had already been computed, and it was not measured, and the search that failed was run instead. The nine thousand numbers were there.

The general shape of the mistake #

A summary statistic taken from a distribution whose shape has not been looked at.

The maximum of a set is a fine statistic when the set has a peak and a poor one when it has a plateau, and which it is cannot be read off the maximum. The site’s own second-statistic thread is fourteen essays about exactly this in a different subject, and the discipline it arrives at — look at the distribution before choosing a number to reduce it to — is the discipline that was not applied here.

It is not a subtle failure and it is not a rare one. It is what happens when a quantity is computed for a purpose and the computation is never plotted.

What the plateau does not explain #

The five rows whose classes are not constant. Their spreads are 10.3° to 156.9°, so their profiles are not k levels and the argument above does not apply to them.

Two of the five are on the Lucas 0.020 stem and two on the golden 0.010 stem, and on those the maximum might well be a genuine peak. Nothing here has looked.

That is a small hole in the account and it is worth naming, because “the maximum is a plateau because the profile is periodic” is a claim about twenty-five rows and the plateau statistic was computed over all thirty.

The window that was chosen, and why it was reasonable #

Three times the larger counted number. On a 5/8 stem that is twenty-four organs and on an 8/13 stem thirty-nine.

The reasoning behind it was sound at the time: a removal is felt across a front that runs to the larger counted number, so a window three times that is generous enough to contain the disturbance and short enough that the answer is about the cut rather than about the run length.

Both halves of that are about the front, which is a property of which offsets have a lasting effect. Nothing in it is about how far above the hole the effect settles down, because until now nobody had measured that. The window was chosen against the one length scale the thread had.

Why a plateau is not the same as no information #

The maximum is uninformative and the levels are not.

A profile with k levels has k numbers in it, and the largest of them is one. Reporting only the largest throws away k − 1, and among those thrown away is the whole structure: the common level, the two exceptions, their sizes and their adjacency.

So the reference-organ search did not fail because the profile is featureless. It failed because it reduced a structured object to its extreme value, which is the one summary that cannot see structure. That is a stronger criticism than “the disturbance does not decay” and it points at what to do instead.

What the tails were saying #

Over the last sixty organs of each run the displacement was 70° to 179°, and the smallest tail anywhere in the census is 45°.

Read as a decay rate that is a decay rate of zero. Read as a class level it is the size of the exceptional pair, which is one divergence step — 88° to 147° across the census, and 45° is below that only because a tail is the largest displacement over a window that may not contain an exceptional organ.

So the tail measurement, which was reported as showing the disturbance has no far edge, is a measurement of the exceptional level in disguise. It had the size of the exchange in it and nothing said so.

What was available and not used #

Every number in this essay comes from runs that had already been grown. The profiles were computed, thirty of them, three hundred organs each; the surviving lag was measured for every one; and the fold is one line of arithmetic.

That is worth recording plainly rather than as a lesson. The search measured a maximum, measured a plateau share, measured a tail, and did not plot a profile — and every one of those three numbers is a consequence of a shape that was in the data.

The general form is: a summary computed for a purpose, three times, without the distribution being looked at once.

Two ways this could still be wrong #

The first is that the plateau share and the periodicity are measured over different windows. The share was measured over three times the larger counted number and the periodicity over the top hundred and twenty organs, and on a row whose onset is 166 those two windows do not overlap at all.

So “the plateau is one over the surviving lag” is an argument about a window that was not the one measured. It holds where the search window sat inside the pattern and it does not where the window sat inside the transient — and which rows are which is now known.

The second is the five rows that are not periodic, whose maxima the argument says nothing about. Between them those two caveats cover a third of the census, and the claim is therefore that the plateau has an explanation on most rows rather than on all of them.

What replaced the reference organ #

A reading that needs no organ at all. Which side of the tip the removed organ sat on is arithmetic on the settled divergence and the offset — the azimuth of the organ k places back, folded to a half turn either way — so it is answerable on every row rather than on the nine the tip’s chains reach.

It scores 22 of 30 against 18 for naming the commoner family, and it reproduces the published reading on all nine rows that reading covers.

Which is the shape a repair should have: not a better statistic taken from the same profile, but a quantity computed from something else entirely. The profile had been asked for a coordinate it does not supply, and the coordinate was available in the arrangement’s own geometry.

The one line #

A profile constant on each of k residue classes has k levels, so its largest value is attained by a whole class — a ninth to a quarter of any window, forever, with no decay above the hole. The plateau that search found and reported as a failed instrument is what that shape forces, and the shape was computable from data already on disk.