Three rows a window moves

Twenty-eight of the census’s thirty wrecked cuts are periodic when sixty organs are read. Twenty-five are when a hundred and twenty are, and twenty-five when a hundred and eighty are. The three that move are the subject here.

They are g026 cut at offset 4, g010 cut at offset 4, and g010 cut at offset 5. All three are periodic at the narrow window and not at the working one.

Three of thirty is a tenth of the census, which is small enough to be easy to dismiss and large enough that any count quoted from the classification should carry it.

Their spreads do not scale #

On the rows that keep one verdict at every window, the spread is proportional to the window: 0.8, 1.6 and 2.4 degrees at sixty, a hundred and twenty and a hundred and eighty organs.

These three read 5.1, 10.3 and 105.5 degrees; 3.9, 69.6 and 163.3; and 4.0, 69.5 and 162.9. Those are ratios of twenty-one, forty-two and forty-one against a proportional prediction of three.

So they are a different effect #

A quantity that scales with the window is measuring a drift. A quantity that grows by forty times when the window triples is measuring something that is only present in part of the window.

That is the shape of a transient: a stretch at the start of the reading where the quantity is doing something else entirely, whose contribution to a spread grows enormously once it is included at all.

The distinction is available for nothing once three windows have been read, and it is not available at all from one. A single spread of 69.6 degrees says the row is not periodic and says nothing about why.

Where the transient is #

The window is the tail of the run. A window of a hundred and eighty organs on a run of three hundred begins at organ 120, and a window of sixty begins at organ 240.

So a wide window reaches further back into the run, and what it reaches back into is the part where the disturbance from the cut is still healing rather than the part where the stem has settled into the arrangement it keeps.

Which is exactly what these three rows have #

The onset is the organ from which every class stays within ten degrees of its own tail mean for the rest of the run. These three rows have the census’s latest onsets: 229, 230 and 239 organs at the working window.

A window of a hundred and eighty begins at organ 120, which is over a hundred organs below every one of them. So the widest window on these rows is more than half disturbance.

The check that would have caught it #

It is arithmetic and costs nothing. A window of N organs on a run of M begins at organ M − N; a row whose onset is above that is being read partly on its transient.

Nothing in the thread had been making that comparison, because with one window and one run length the two numbers were both constants and the comparison never came up. It comes up the moment a second window exists.

What happens on a longer run #

The repair for a window that reaches too far back is a longer run, and it works. At six hundred organs the same three rows read 5.1, 10.3 and 15.4; 3.9, 7.9 and 11.8; and 4.0, 8.1 and 12.2 degrees.

Those scale. Ratios of 3.0, 3.0 and 3.1 against a prediction of three, which puts all three rows back into the drifting population.

So the effect is an interaction #

Not between the window and the stem, but between the window and the run length together. A window of a hundred and eighty on a run of three hundred reaches into the transient; the same window on a run of six hundred does not.

That is a more comfortable finding than the alternative. It means nothing about these three stems is peculiar; what is peculiar is a reading whose start falls in the wrong place.

What the three rows have in common besides that #

All three keep a lag of 5, all three are on the golden branch, and all three sit at rises of 0.026 and 0.010.

That coherence is worth noting and worth not over-reading. Late onsets and a lag of five go together in this census for reasons the census does not explain, and three rows is three rows.

It is also not independent of which cuts wreck at all, since the census’s rows at a lag of five come from a particular stretch of the ladder rather than from across it.

The direction is always the same #

Every one of the three loses its periodicity as the window widens. None gains it, which is not true of what a longer run does — there rows move in both directions.

That is what both effects predict, which is why they do not fight each other on any row: a wider window includes more drift, and a wider window may also include transient. Both push a spread up. Nothing in the census gets steadier when more of it is read.

Which makes the narrow window the generous one #

At sixty organs twenty-eight of thirty rows are periodic. That is the largest count any window gives and it is the least informative, because a window of sixty organs on a run of three hundred is reading a fifth of it.

The prediction the setting was chosen against was exactly this — that a short window would make every class look constant. It is right, and it is right by three rows rather than by the whole table.

What the working window is doing #

A hundred and twenty organs on a run of three hundred begins at organ 180. Of the thirty rows, twenty-seven have onsets below that and three do not.

So the working setting is reading the pattern on twenty-seven rows and part of the disturbance on three. That is a defensible place to be and it is not what the thread had been claiming, which was that it was reading the pattern.

The same distinction — between what a setting does and what it was assumed to do — is what a residual turned out to be when its instrument was changed for one with no level in it.

Whether the three should be reclassified #

They should not, and it is worth saying why rather than leaving it implied. A row whose pattern begins at organ 229 on a run of three hundred has less pattern in it than a row whose pattern begins at organ 7.

That is a real property of the row and calling it periodic on the strength of a window narrow enough to miss the disturbance would be hiding it. The right move is to say what the verdict is a verdict about, which is what these readings make possible.

The same failure at the other end #

There is a matching defect at the far end of the run and it has already been caught. An onset is the first index from which a condition holds for the rest of the run, and that reading is satisfied trivially at the last index — so it returns a number whether or not the condition ever holds.

The guard is to ask the condition of the tail before searching for a first index. One row of the census needed it. The window’s version of the same problem is at the beginning rather than the end, and its guard is to compare the window’s start against the onset.

Three rows out of thirty #

The size of the effect matters as much as its existence. Three of thirty is ten per cent, and the other twenty-seven rows keep their verdict at every window at both run lengths.

So the classification is mostly robust to the setting and specifically not robust on the rows whose onsets are latest. That is a better description than either the classification is stable or the classification depends on the window.

And three rows the run length moves #

Doubling the run length also changed the periodicity verdict on three rows, in both directions: two joined as their spreads fell from about 70 degrees to about 8, and one left as its spread rose from 6.1 to 171.2.

One row appears in both sets. So of thirty rows, five have a verdict that one instrument setting or the other decides, and twenty-five are stable under both.

What five unstable rows out of thirty is #

It is a sixth of the census, and it is the honest figure to attach to any statement of the form N of the census’s cuts are periodic. That number is twenty-five at the working settings and it is twenty-three to twenty-eight across the settings that have been tried.

Quoting the range rather than the point is what this reading makes possible, and it is cheaper than it sounds: the readings exist, so the range is a maximum and a minimum over numbers already computed.

The row that is worst on both #

g010 cut at offset 5 is the most sensitive row in the census. Its spread goes 4.0, 69.5, 162.9 across the windows and it moves at both run lengths.

It is worth naming because a reader tracing a claim through the thread will meet it several times, and because it is the row most likely to be reported differently by any two readings of the same table. Nothing about it is broken; it is simply the row whose pattern begins latest and whose transient is therefore the largest fraction of any window.

What a transient is, on a cut stem #

Worth naming, because the word is doing work. Remove an organ and the organs placed after it are placed against a neighbourhood that is missing one member. For a stretch the arrangement is unlike anything the control has and unlike what the cut stem will end up with.

Then it settles. The displacements fall into a pattern that repeats at the lag the stem kept, and from there on the profile is that pattern rather than the healing. The onset is the organ where the second begins.

So a transient is not noise and it is not error. It is a different regime of the same run, and a reading that spans both is averaging two things.

Why the onset is itself window-dependent #

There is a circularity worth stating. The onset is found by comparing each organ’s displacement against its own class’s tail mean, and the tail mean is taken over the reading window.

So moving the window moves the onset as well as the spread. The onsets quoted here are at the working window, and at sixty organs they differ — which means the check comparing a window’s start against an onset is comparing two quantities that both depend on the setting being checked.

That does not make the check useless. It makes it a consistency reading rather than an independent one, and the honest form of it is that at the working window three rows have onsets above where the widest window starts.

What would break the circle #

A definition of the onset that does not use the window. Comparing each organ’s displacement against the final value rather than against a windowed mean would do it, at the cost of using information from the end of the run to describe its beginning.

That is a trade rather than a fix, and the collection has made it in the other direction elsewhere: a settling time is measured against a tail mean for exactly the same reason, and the wandering runs are excluded first so that a mean of a wandering tail is never taken.

A third way out, which is to let the circle close #

There is an option between the windowed onset and the final-value one, and it is worth setting down because it costs no stems and turns the circularity into a procedure.

Read at some window and take the onsets it gives. Then give each row a window of its own — one that begins above that row’s onset rather than at a fixed depth for the whole census — and read again. New onsets come out, the windows move, and the reading repeats. What is being sought is a fixed point: a per-row window that starts above the onset that window itself reports.

Two properties make that better than either alternative. It is window-free in the sense that matters, since the answer is the one the row’s own settling picks rather than one a global setting picks for it. And it uses no information from the end of the run to describe the beginning, which is the whole cost of the final-value definition.

Its failure mode is the useful part. A row whose window keeps chasing its onset upward, never settling, is a row whose pattern does not begin inside the run at all — and that is the honest verdict for it. At present such a row is classified not periodic, which is the same word given to a row that settles and then drifts, and the two are different objects. An iteration that fails to converge names the difference for nothing.

The three rows here are the candidates, since their onsets at 229, 230 and 239 organs sit inside a run of three hundred by a margin thinner than a window.

What the three rows cost the classification #

Nothing, in the sense that the classification is unchanged at the settings everything else in the thread uses. Twenty-five of thirty rows are periodic and the same twenty-five as before.

What they cost is a sentence. Any statement of the form the damage falls into a pattern at the surviving lag on twenty-five of thirty cuts now needs at a hundred and twenty organs of a three-hundred-organ run, because three of the thirty are decided by that pair of numbers and not by the stems.

What the same rows do to the exchange #

Nothing at all, and that is worth checking rather than assuming. The exchange is defined on rows with a balanced pair of exceptional chains, and whether a row is periodic is a separate reading of the same profile.

Two of the three movers are unpaired and go to the set the exchange sets aside; the third carries a balanced pair and is in the exchange table at every window. So the window moves a periodicity verdict and moves no row in or out of the exchange.

What a reader should carry #

That three of the census’s thirty cuts have a periodicity verdict decided by how many organs are read, and that they are the three whose pattern begins latest in the run.

And that the two effects a window has are separable by arithmetic: a spread that triples when the window triples is drift, and a spread that grows forty times is a transient reached into. Nothing needs a new measurement to tell them apart.

What the picture at the top shows #

Three bars, one per row, each drawn as the run from left to right. The pale stretch is the disturbance still healing and the dark stretch is the pattern repeating, with the boundary at that row’s onset.

The vertical marks are where each of the three windows begins. On all three rows the widest window’s mark falls inside the pale stretch, and on two of them it falls a hundred organs inside it.

The one line #

Three of thirty cuts are periodic at sixty organs and not at a hundred and twenty, and their spreads grow by twenty to forty times across the windows against a proportional three — because a window is the tail of a run, and on these three the widest window begins more than a hundred organs below where their pattern starts.

At six hundred organs all three scale like everything else, so the effect is an interaction between the window and the run length rather than anything about the stems.