Like with like

The strongest objection available against a three-hundred-organ endpoint is that it names the wrong place. A run still moving at organ three hundred reports where it happened to be, and the number goes into a table looking exactly like a number that means something.

Regrowing the sixty-three wrecked runs to twelve hundred organs tests that objection directly, because the endpoint can be read again at the new length by the same rule. If the short reading names the wrong place, the two readings disagree.

They do not disagree. The median absolute drift over four times the run is 0.0000 degrees and the worst is 0.0039.

Sixty of sixty-three do not move at all #

Not to the precision the endpoint is reported at. One run drifts upwards and two drift downwards; the mean signed drift over the sixty-three is about minus three ten-millionths of a degree, which is a number with no direction in it.

Among the seventeen that settle the picture is the same: median 0.0000, worst 0.0039, fifteen of the seventeen unmoved, one up and one down. The drift is four orders of magnitude below the half a degree at which two settled values are called one place, so it could be an order of magnitude worse and change nothing.

The two readings are the same quantity #

The endpoint at twelve hundred organs is taken by the same rule as the endpoint at three hundred: a mean over a whole number of the orbit’s periods at the top of the run. Reading one with one rule and the other with another would have produced a difference that was about the rules.

That is the sort of care that is invisible when it works. It is also the reason the drift can be quoted as a property of the run rather than as the residue of two instruments disagreeing, and it is the thing a length check on a different reading had to get right too.

And the growths are the same growth #

A twelve-hundred-organ continuation truncated at three hundred reproduces the three-hundred-organ continuation to the last digit — a largest difference of 0.0 degrees over three hundred organs, checked on three cells at three rises.

So this is a comparison of lengths and not of growths. Nothing about restarting, reseeding or recomputing enters, and a difference between the two readings would have had to come from the extra nine hundred organs.

Every settler settles where it already was #

All seventeen settle at the divergence their three-hundred-organ run already reported. None settles anywhere else.

That is a null result and it is the more important half of this essay. The interesting outcome — a run that looked as though it finished at 151 degrees and in fact settles at 139 — happens zero times in seventeen. The short reading was never pointing at the wrong place.

Which locates the defect exactly #

A three-hundred-organ endpoint is precise about where and silent about whether. It says correctly what divergence the stem’s top is arranged around; it says nothing at all about whether the stem is going to stay there.

Both halves are needed to state the defect honestly. The comparison that read twenty-nine agreements off the endpoint column was not reading a noisy quantity, and no amount of extra run length would have changed a single one of its numbers. It was reading a quantity that does not answer the question it was asked.

What a refusal is #

Not an approach that ran out of room. The forty-six refused runs have a tail standard deviation between 20.85 and 44.07 degrees; the seventeen settled ones lie between 0 and 0.27.

Between those two populations there is an empty gap 20.58 degrees wide. Not one run in the table sits in it. Whatever the criterion is doing on these sixty-three, it is not splitting a continuum at a threshold; the two groups are separated by a factor of nearly a hundred.

Why the gap is the important number #

Because it makes the verdict robust to the criterion. The settling test uses 1.5 degrees as its tolerance, and a threshold is always open to the complaint that it was chosen.

Here the complaint has nowhere to go. Any tolerance between about 0.3 degrees and about 20 degrees returns exactly the same seventeen, which is a range of nearly two orders of magnitude. The number 1.5 is doing no work: it could be divided by five or multiplied by thirteen and every verdict in the table would stand.

The counts, without the gap drawn #

Reading the same spreads as two counts rather than as two ranges makes the shape plainer: every settled run is under a third of a degree and every refused one is over twenty, and the two lists have no member in common at any width between them.

That is worth one drawing on its own because a range quoted as 0 to 0.27 against 20.85 to 44.07 invites the reader to imagine the two distributions overlapping somewhere in their tails. They do not have tails that reach.

Eleven values for seventeen runs #

The seventeen settle at eleven distinct divergences: 65.25, 79.41, 79.55, 101.48, 101.78, 101.80, 138.75, 140.78, 140.86, 150.43 and 150.44 degrees.

Seventeen runs at eleven values means six of them land on a divergence another run also reaches, and the eleven are not scattered either: they fall into five groups, three of which hold values within a third of a degree of each other. That is what a lattice reached by more than one route looks like when the routes are cut differently, and it means a wrecked stem that settles is not settling at an arbitrary place.

Which is a claim the endpoint column could not make #

Forty-three distinct endpoints were counted over the sixty-three runs. Eleven distinct values over the seventeen that settle is a much tighter set, and it is tighter for a reason: the refused runs contribute an arbitrary number each, drawn from an orbit tens of degrees wide.

So the apparent variety of the endpoint column is mostly the refusals. Restricting to the runs that have actually stopped moving removes about three-quarters of the distinct values and leaves clusters, and the earlier count of forty-three was a count of the column rather than of the arrangements.

Forty-four of forty-six repeat a block #

A refused run is not wandering. Forty-four of the forty-six cycle through an exact repeating block of divergences: seventeen at period four, nine at seven, seven at five, seven at eight, three at six and one at fifteen.

An exact block is a strong statement. It is not a near-repetition or a slow spiral; the sequence of divergences returns to a value it has already taken and then takes the same path again, for as long as the run is grown.

The two that do not #

Both are the same lattice: the Lucas rung counted 4 and 7 at a rise of 0.01096, cut at the smaller wall and cut at both walls. Their tails hold twenty-seven and twenty-five distinct divergences and do not close.

Two aperiodic runs out of forty-six is not enough to say what they are. They might be blocks with a period longer than the window, or they might be genuinely aperiodic, and nothing here separates those. They are recorded because a claim that every refusal is periodic would be wrong by two, and because both of them sit on one lattice.

That last point is the one worth following. The pair is two different cuts of one lattice, so whichever wall is removed the run behaves the same way, and whatever prevents the block is more likely a property of the rise than of the cut. One lattice is one lattice, and the cheap test is the rises either side of it: if they close, the aperiodic pair is a feature of a single rise and the sweep’s resolution has found its edge again.

A block is not the damage profile #

Periodicity has been found on a wrecked stem before, and it is a different periodicity. The displacement profile above a hole repeats at the lag the stem kept, which is a statement about how far each organ moved relative to an uncut control.

The block here is in the divergence sequence itself, at the top of a run nine hundred organs past anything the control could be compared to. The two need not have the same period and nothing here checks whether they do. What they have in common is that a cut stem is more orderly than disturbed suggests.

Where the refusals sit #

On four rungs, entirely. The rungs that settle nothing settle nothing at any length — six of six, eight of eight, five of five and six of six — and every run on them repeats a block or, in two cases, does not close.

So the orbit population is not scattered across the ladder either. Twenty-five of the forty-six refusals come from rungs where refusal is unanimous, which is a stronger regularity than the settling side of the table has anywhere.

An orbit, drawn as an orbit #

A refusing run and a settling run can have endpoints a degree apart and be doing completely different things. The endpoint column cannot show that, because it holds one number per run and both runs have one.

Drawn against the organ index the difference is not subtle. One trace is a flat line; the other is a band tens of degrees wide with a repeating figure in it.

What a settler looks like #

Flat, and flat from early on. A settled run’s tail has a width between 0.000 and 0.703 degrees and holds between one and four distinct divergence values.

One distinct value is a stem placing every organ at the same divergence to the last digit. Four is a stem alternating over a short block whose spread is still under a degree. Either way the arrangement is a lattice and it is not changing.

What a refuser looks like #

Wide. The refused tails span between 51.3 and 110.9 degrees, median 81.9, and hold between two and twenty-seven distinct values with a median of six.

A stem whose divergence takes six values spread over eighty degrees, in a fixed repeating order, is a real arrangement. It is not a lattice with a settled divergence and it is not noise either, and the collection has no name for it beyond the block it repeats.

What this rules out #

That a refusal is an incomplete approach. A run on its way to a destination would narrow as it went, and its tail spread would fall with run length; these do not narrow, they close.

So a wrecked stem that does not settle here is not going to settle later. It is in an orbit and it will stay in it, and the four hundred, six hundred, eight hundred, one thousand and twelve hundred organ readings all say the same thing about the same runs.

And what it does not rule out #

That the orbits are themselves attractors. Nothing here tests whether a run perturbed off its block returns to it, which is the question that would decide whether a repeating block is a stable arrangement of the rule or merely where a particular stem happens to be.

That is a different sweep — a second cut, high above the first, on a run already in an orbit — and it is the obvious next thing. The rule’s stable arrangements have been read off intact stems only, and forty-four exact blocks are a second population that has never been asked.

Why the null is not a formality #

Because the alternative was live and cheap to believe. An endpoint read once on one rung was called unusable precisely because it moved between rises, and moving between rises and moving with run length are easy to conflate.

They are different. The endpoint is exquisitely sensitive to which lattice is being cut and completely insensitive to how long the cut stem is grown. A quantity can be unstable in one argument and rigid in another, and reporting only the instability would have made the wrong case here.

What the settling table would say #

That most runs do not settle, and it would be right. About thirty per cent of its own intact runs settle and 27.0 per cent of these wrecked ones do.

The rates are close enough that no story about damage is needed to explain the refusals. A criterion that asks whether a run has stopped changing refuses most runs it is given, and the sixty-three are not being refused for having been cut.

The comparison this makes possible #

Like against like. Before the regrowth, one table held destinations tested for stability and the other held endpoints tested for nothing, and the comparison between them was between two different kinds of number.

Now both sides have been through the same criterion. Twelve of the sixty-three are stems that settled at a place the intact table also reaches, and that is a sentence about two tables rather than about one table and one column.

The window the criterion looks through #

Sixty organs. A run is settled when the last sixty divergences hold a standard deviation of at most 1.5 degrees and there is an organ past which nothing leaves that band.

Sixty is short against a run of twelve hundred and long against a repeating block of four, so a refusing run’s window always contains several complete turns of its orbit and its spread is the orbit’s spread rather than a sample of it. That is why the refused spreads are stable across lengths, and it is why the empty gap does not close as the runs get longer.

Why twelve hundred and not more #

Because twelve hundred is the settling table’s own length, and matching it is the whole point. A criterion applied at a length the criterion was not calibrated at would be a different criterion.

Growing further would answer a different question — whether a block ever breaks — and this sweep cannot answer it. What it can say is that no block breaks inside four times the original run, on any of the forty-six.

The cost of finding out #

The longest single regrown run took 27.3 seconds and the whole sweep of sixty-three ran in 320 seconds on a quiet machine. Five minutes.

Both readings above — the drift and the tail spread — fall out of the same sweep, because a run grown once can be truncated and re-read at any length rather than regrown. That is the only reason six lengths cost the same as one.

What is not claimed #

That the endpoint is a good measurement. It is a precise one, which is a different thing, and the next reading takes most of what precision buys back again.

Nor is it claimed that three hundred organs is enough for every purpose. It is enough for the endpoint, on these sixty-three, to four thousandths of a degree. A different quantity read off the same runs — an onset, a period, a level — has its own length requirement and one of them has been measured and does move.

The check that would break this #

Regrow a run twice from the same lattice and read the endpoint from each. If the two disagreed by more than the drift reported here, the drift would be measuring the arithmetic rather than the stem.

The truncation check does that job from the other side and returns zero difference over three hundred organs on three cells at three rises. It is the cheapest assertion in the sweep and it is the one holding up every number above.

What is claimed #

That the endpoint of a wrecked run does not move between three hundred organs and twelve hundred: median drift 0.0000 degrees, worst 0.0039, sixty of sixty-three unmoved, and no sign to the three that move.

That all seventeen settling runs settle at the divergence their short run already reported, and none settles anywhere else — so the short reading is precise about where and silent about whether.

And that the forty-six refusals are orbits rather than approaches: separated from the settlers by an empty twenty-degree gap in tail spread, and forty-four of them repeating an exact block at one of six periods.