What a run length was hiding

Two explanations were available for why a wrecked run’s endpoint might not mean what it looked like it meant. The first is length: three hundred organs is short by the settling table’s standards, so perhaps the runs simply had not finished.

The second is that the endpoint is a summary statistic — a mean taken over the top of a run — and a mean is a number about a set of divergences rather than one the stem ever takes.

The first explanation is the one everybody reached for, including the sweep that was designed to test it. It is wrong, comprehensively and at every length that was checked.

Six lengths, one verdict #

The criterion was re-read at 300, 400, 600, 800, 1,000 and 1,200 organs above the hole. At every one of them, seventeen runs settle and forty-six refuse.

Not one run changes sides at any length. Runs settling at three hundred but not at twelve hundred: none. Runs settling at twelve hundred but not at three hundred: none. The table of verdicts by length has six identical columns.

Which is a comparison of lengths and not of growths #

The six readings are truncations of one run rather than six separate runs. A twelve-hundred organ continuation cut back to three hundred reproduces the three-hundred-organ continuation to the last digit, checked on three cells at three rises.

That is the assertion the whole reading rests on. Without it, six lengths would be six sweeps, and a difference between them could have been a difference in how they were grown rather than in how far.

Three hundred was enough by a factor of two #

The seventeen settling runs arrive between 15 and 138 organs above the hole. The median is 45, the quartiles are 23 and 57, thirteen of the seventeen are inside sixty organs and fifteen are inside a hundred.

Nothing in the table needed three hundred. The slowest arrival in the whole set uses less than half the length the design already grows, and the defensible run length — the one the measurement would justify writing down — is about 140 organs.

The slowest two #

The golden rung counted 5 and 8, at rises of 0.00795 and 0.008, both walls removed. They arrive at organs 133 and 138.

Two runs on one rung at two adjacent rises, twice as slow as the median and still using less than half the available room. A design set at 140 organs would catch them; one set at 100 would lose both. That is the only place in this table where the choice of length would change an answer, and it changes it for two runs out of sixty-three.

What that rules out #

Run length as an explanation of anything in the slot table’s endpoint column. Nothing there was waiting for more room, no verdict was pending, and a longer slot design would buy nothing.

That is a firm negative and it is worth having, because the same question asked about a different reading off the same runs came back the other way: onsets and levels do move when a run is doubled, and three of thirty cuts changed their answer. Length matters for some readings and not for others, and the way to find out is to check each one.

And then the other explanation #

An endpoint is a mean over a whole number of the orbit’s periods at the top of a run. On a settled run that is a mean over a constant and returns the constant.

On a run in an orbit tens of degrees wide it returns a number in the middle of that orbit, and the middle of an orbit need not be anywhere the stem goes. Forty-four of the forty-six refusals repeat an exact block, so this is not a hypothetical: the refusals are precisely the runs whose endpoint is an average over something wide.

Thirty-four of forty-six #

For each refused run, the nearest any divergence in its last hundred and twenty organs comes to the divergence reported as its endpoint. Thirty-four of the forty-six never come within one degree of it. Only seven come within half a degree.

The median nearest approach is 4.38 degrees and the worst is 38.70. The worst three are the golden rung counted 5 and 8 at rises of 0.00795 and 0.01 with the larger wall removed, and the golden rung counted 8 and 13 at 0.00573 with the smaller wall removed.

An angle the stem never takes #

Thirty-eight degrees is not a rounding. A run whose reported endpoint is 38.7 degrees away from anything it ever does is a run whose entry in the table names an arrangement it has never been in, at any organ of twelve hundred.

The number is not wrong. It is the mean it says it is, computed correctly, over exactly the window it says. What is wrong is reading it as a place.

Nothing ever claimed otherwise #

The machinery that computes an endpoint takes a mean over an orbit and says so. It does not claim the stem is at that value, it has never claimed it, and there is no defect in it to fix.

This is a reader’s error and it should be named as one. A column of numbers headed by a word like endpoint invites the reading that each number is where its run finished, and the invitation was accepted for two rounds of work by everybody who looked at it, including the sweep that set out to check whether the numbers were stable. They were stable. They were also, on thirty-four runs, not describing a position.

One of the worst three, drawn #

The golden rung counted 5 and 8 at a rise of 0.01, with the larger wall removed, is the second worst offender in the table: its stem never comes within 38.55 degrees of the number recorded as its endpoint.

Drawn against the organ index beside a settling run, the two look nothing alike, and the endpoint rule crosses the refusing panel through empty space. That is the whole argument in one picture: a horizontal line at a value the trace does not visit.

Why the seven that do come close are not a reprieve #

Seven refused runs come within half a degree of their own endpoint at some organ. That sounds like a third of a case for the original reading and it is not one, because passing through a value is not being at it.

A run that crosses its own mean once every four organs on its way round a block is not near a destination in any sense the settling table would recognise. The criterion exists to distinguish arriving from passing through, and these runs are refused by it.

Why the width is the whole of it #

The wider the orbit, the further its mean sits from the stem. Refused tails span between 51.3 and 110.9 degrees, and the nearest approach grows with that span.

So the defect is not random and it is not confined to a few odd runs. It is a mechanical consequence of averaging a repeating block, and the runs it bites hardest are exactly the ones with the most interesting orbits.

Eleven of the seventeen #

The runs where this matters most are the ones the earlier comparison counted. Of the seventeen runs whose short endpoint agreed with a destination and which then refuse to settle, eleven never come within a degree of their own endpoint.

So those eleven are not stems near a destination that failed to stop. They are stems in an orbit whose arithmetic mean happens to land within a degree of a destination, and the stem itself is a degree from that mean at best and often tens of degrees from it.

What is left of the agreement #

Twelve runs and five destinations, where there were twenty-nine and eight.

The five that survive are 65.21, 79.18, 101.59, 139.06 and 150.96 degrees, held by one, four, four, one and two runs. The three that go — 99.48, 137.78 and 157.67 are held by nothing at all once the refusals are removed.

The row that falls furthest #

The destination at 150.96 degrees was the most-visited of the eight, with fourteen wrecked runs finishing within a degree of it. Two of those fourteen settle there.

That row carried nearly half the original agreement on its own, and it was the row the earlier reading found most persuasive, because fourteen runs from two rungs on both branches converging on one value is exactly what a real attractor looks like. Twelve of the fourteen are orbits whose mean sits near it.

Which is the sharpest form of the correction #

A count of endpoints near a destination is a count of means, and means pile up in the middle. A destination in the middle of the range the refusals orbit will collect endpoints in proportion to how many refusals there are, whether or not any stem is ever there.

That is a bias with a direction, and it inflates exactly the most-visited entries. The correction is therefore not a uniform shrinkage of the agreement; it takes the largest row hardest, which is the row any reader would have weighted most.

What survives untouched #

Twelve genuine cases, and they are not nothing. Twelve wrecked stems, cut from a settled lattice, settle by the settling table’s own criterion at five divergences that table also reaches — from both branches and from several rungs.

The argument the original comparison was making was that the destination list is a property of the placement rule rather than of the settling design, and twelve independent confirmations still support it. What does not survive is the number twenty-nine and the strength that went with it.

And five that were never on the list #

Five runs settle at 140.78 and 140.86 degrees — three cuts of one Lucas lattice and two of one golden lattice — which is 1.7 to 1.8 degrees from the nearest destination, more than three times the tolerance at which two settled values are called one place.

They pass the settling table’s own test at a divergence the settling table has never reached. That is a settled arrangement of the rule found only by damaging a stem, and a destination list built from intact runs alone can only ever be a lower bound.

Why five is enough to say that #

Because they are not one lattice seen three times. Two lattices on two different branches, cut in four different ways, all land within a tenth of a degree of the same value — and the settling criterion refuses the other fifty-eight of the sixty-three, so it is not a test that accepts things easily.

Five is far too few to say how many such arrangements there are. It is enough to say the list is incomplete, which is a claim about one member and needs one counterexample.

The half turn is untouched by this #

Twelve of the sixty-three finish within half a degree of a half turn and none of them settles. Those runs are not affected by the averaging argument in the way the rest are, because a half turn was never a destination on the list and nothing was ever counted as agreeing there.

What the averaging argument does say about them is that their reported value is a mean too, so twelve runs finish at a half turn is a statement about twelve means and not about twelve stems in two files. Whether a stem is actually in two files is a separate reading and it has been made on some of them.

What a counter would add #

The twelve surviving cases could be counted rather than compared by angle. A settled stem has a parastichy pair, and counting the pair at the top of every settled run in the intact table has been done without ever being done on a wrecked one.

Two stems settling at the same divergence and counting the same pair is a stronger agreement than two numbers matching to a degree, because a count is an integer and cannot be coincidentally near. That is the version of this comparison worth building next, and it needs no new growth at all, since the seventeen settled runs have already been grown.

What this does not rule out #

That the destination list is right about what it contains. Nothing here refutes a single one of the fifteen destinations; three of them are simply unsupported by these sixty-three runs, which is a statement about the wrecked runs and not about the destinations.

Nor does it say the endpoint column should be discarded. It is precise to four thousandths of a degree and it answers the question what is the mean divergence at the top of this run. It should be read as answering that.

The reading that replaces it #

For a settled run, the endpoint is the settled divergence and the two are the same number. For a refused run, the endpoint is the centre of an orbit and the useful quantities are the orbit’s width, its period and its levels.

That is two columns rather than one, and the sweep already computes both. The failure was never in the arithmetic; it was in one column being asked a question that only makes sense for the runs in the other.

What a defensible design would look like #

Cut, grow 140 organs, read. That is the length the arrivals justify with the slowest of them at 138, and it is less than half of what the slot design already spends.

Nothing is gained by shortening it, though. The runs are cheap — the longest single regrowth to twelve hundred organs took 27.3 seconds and the whole sweep of sixty-three ran in about five minutes — so three hundred organs is a margin nobody needs to reclaim. The number is worth stating because the runs were too short is now a closed explanation rather than an open one.

Why this was not found earlier #

Because the endpoint came along free. The slot design was built to measure the first organ placed after a cut, and everything past that organ existed only so the run could be grown far enough to say whether it recovered.

A number that nobody designed a reading for gets read the obvious way when somebody finally reads it, and the obvious way was as a position. The one earlier attempt to use the column called it unusable for a different reason — it moves between rises — and stopped there.

The two objections, scored #

Length: eliminated, on six readings of sixty-three runs, with no verdict changing at any of them. Averaging: confirmed, on thirty-four runs of forty-six, with a median offence of 4.38 degrees.

The one that was tested is the one that was wrong. That is the ordinary return on testing the cheap explanation first, and it is worth recording that the sweep which eliminated it was designed and named for it.

The check that would break this #

Report, for each refused run, the divergence it is nearest to most often rather than the mean over its window. If the two agreed, the averaging objection would be about the choice of statistic rather than about the orbit.

They cannot agree on a block of four levels spread over eighty degrees, because no single level is where the mean is. The measurement that settles it is already in hand: the nearest approach is 4.38 degrees at the median, so on half the refused runs no divergence the stem takes is within four degrees of the reported one.

What is claimed #

That the settling criterion returns an identical verdict on all sixty-three runs at every length from three hundred organs to twelve hundred, that settling happens between 15 and 138 organs above the hole with a median of 45, and that run length therefore explains nothing in the slot table’s endpoint column.

That an endpoint is a mean over an orbit up to 110.9 degrees wide, so thirty-four of the forty-six refusing runs never come within a degree of their own reported endpoint — median nearest approach 4.38 degrees, worst 38.70 — including eleven of the seventeen that agreed with a destination and then refused.

And that the agreement between the two tables therefore survives at twelve runs and five destinations rather than twenty-nine and eight, with three destinations holding nothing and the most-visited falling from fourteen endpoints to two settled runs.