A destination or a refusal

A wrecked run’s endpoint is the divergence its stem holds at the last organ of a three-hundred-organ run above a hole. Twenty-nine of the slot table’s sixty-three finish within a degree of a destination the settling table reaches, and the two tables share the placement rule and nothing else.

That comparison put two quantities side by side that are not the same kind of thing. A destination is a divergence a run arrived at and then held, tested for; an endpoint is where a run happened to be when the sweep stopped, tested for nothing at all.

The essay that reported the agreement said so, and wrote down the sweep that would fix it: regrow the sixty-three at the settling table’s own length, put them to the settling table’s own criterion, and find out which of them settle. It wrote the prediction down first.

What was regrown #

Sixty-three runs, each one a wrecked cell of the slot table: a settled lattice with one or two of the growing tip’s chain-neighbours removed, regrown to 1,200 organs above the hole instead of three hundred.

They come from thirty lattices spread over eight rungs and both branches, which is what makes the set worth a sweep rather than a check. Nothing about the design changed except the length, and the endpoint at the new length is taken by the same rule as before.

The criterion is not changed #

A run has settled when the last sixty divergences have a standard deviation of at most 1.5 degrees and there is an organ after which every later divergence stays within 1.5 degrees of that tail mean. Divergences are folded for handedness first.

Every one of those numbers belongs to the settling table and none of them was touched here. That matters more than it sounds: a criterion loosened to admit wrecked runs would turn the comparison into a comparison with itself, and the whole point of putting one table’s runs to another table’s test is that the test was written for somebody else’s runs.

The prediction, as it stood #

The twenty-nine runs whose endpoint agreed with a destination settle; the other thirty-four mostly do not. Expected settled count: twenty-nine.

The reasoning behind it was clean. A run sitting within a degree of a place the rule is known to hold has, on the face of it, arrived somewhere; a run sitting near a half turn has nowhere obvious to be. Agreement was supposed to be the signature of arrival.

Seventeen #

Seventeen of the sixty-three settle and forty-six refuse. The prediction is short by twelve runs, which is a fifth of the whole set and two-fifths of what it asked for.

A prediction wrong by twelve is a prediction that named the wrong quantity, not one that mis-estimated the right one. But the size of the error is the less interesting half of what went wrong with it.

And wrong on its own side #

Of the twenty-nine agreeing runs, twelve settle and seventeen refuse. Of the thirty-four disagreeing runs, five settle and twenty-nine refuse.

So the prediction is not merely twelve runs high. It is wrong about the majority of the very runs it was about: most of the agreeing runs, the ones it identified as having arrived, do not settle. Seventeen agreements out of twenty-nine are stems that are not anywhere.

The association is real #

It is also there, which is the part that would be easy to lose in the disappointment. 41.4 per cent of the agreeing runs settle against 14.7 per cent of the disagreeing ones — a ratio of 2.81, an odds ratio of 4.09, a phi coefficient of 0.300, a chi-square of 5.65 on one degree of freedom and an approximate p of 0.017.

Whether a short run’s endpoint sat near a destination therefore carries information about whether the long run settles. Not much information, and not none.

And weak #

An agreeing run is still more likely to refuse than to settle. Forty-one per cent is a minority, and a rule that predicts the opposite of what happens on three cases in five is not a rule.

That is the honest statement of the finding and it is unusual only in being stated. A significant association and a useful prediction are different things, and a two-by-two whose larger cell is on the wrong side of its own hypothesis says so at a glance.

What a phi of 0.300 buys #

A little under a tenth of the variation in whether a run settles, if that quantity is read the way a squared correlation is usually read. The remaining nine-tenths is somewhere else, and this sweep does not say where.

Sixty-three runs is a small table for an effect of that size. The p of 0.017 is the probability of seeing an association this strong by chance in a table of this shape, and it is small enough to report and far too large to build on. A second sixty-three would be the thing that settled it, and there is not a second sixty-three.

Why the prediction was reasonable #

Because it followed from a reading everyone was already making. If the destination list is a property of the placement rule rather than of the settling design — which is what the agreement was evidence for — then a stem at a destination is a stem in one of the rule’s stable arrangements, and a stem in a stable arrangement stays.

The step that fails is the first one: being within a degree of a destination is not the same as being at one. That gap is where the twelve missing runs went, and it turns out to be a gap with a mechanism in it.

Where the settling is #

Not spread. Four of the eight rungs settle nothing at all: the Lucas rungs counted 1 and 3 and counted 7 and 11, and the golden rungs counted 2 and 3 and counted 8 and 13. Between them they contribute twenty-five runs and no settlers.

The other four carry all seventeen. The golden rung counted 3 and 5 settles seven of its eight, the golden rung counted 5 and 8 settles five of fourteen, the Lucas rung counted 3 and 4 settles three of six, and the Lucas rung counted 4 and 7 settles two of ten.

Four rungs with nothing #

That is a stronger structure than the two-by-two has and it was not predicted by anything. Twenty-five runs on four rungs, every one of them refused, and the rungs are not adjacent on the ladder or confined to one branch.

It also means the settling rate is not a property of the sixty-three taken together. Quoting 27.0 per cent as the rate for a wrecked run averages over a set that contains four rungs at zero, and any statement made about a typical wrecked run inherits that.

The rung that carries it #

Seven of the eight runs on one golden rung settle, which is 41 per cent of all the settling in the table from 13 per cent of its runs. Drop that rung and the remaining fifty-five runs settle ten.

Nothing here explains it. The rung is the third and fifth of its branch, it is neither the coarsest nor the finest swept, and the runs on it were cut the same way as everything else. It is recorded as a concentration rather than as a finding, and it is the first thing a second sweep should look at.

Agreement does not redistribute it #

The twenty-nine agreeing runs come from five rungs rather than eight, and three of those five hold all twelve settlers: five of five on the golden rung counted 3 and 5, five of twelve on the golden rung counted 5 and 8, and two of three on the Lucas rung counted 4 and 7. The other two settle none — eight agreeing runs on one and a single run on the other.

So agreement and rung are not two competing explanations of the same pattern; the rung structure is there inside the agreeing runs as well. Whatever decides that a wrecked run on one rung can settle is decided before the endpoint is read, and the eight agreeing runs on the Lucas rung counted 7 and 11 are the clearest case: every one of them names a destination and not one of them holds it.

One rung that settles disappears from the restriction entirely. Every run on the Lucas rung counted 3 and 4 disagreed with the destination list at three hundred organs, so all three of its settlers are invisible here — which is the first sign that the runs which settle and the runs which agree are not the same population.

Reading the table the other way #

Twelve of the seventeen settlers agreed with a destination at three hundred organs, which is 71 per cent of the settlers against 46 per cent of the sixty-three. So a settling run is likelier than a random one to have agreed, and that is the same association read from the column instead of the row.

It is worth reading both ways because the two readings answer different questions. Does agreement predict settling is the question the prediction asked and the answer is barely. Did the settlers agree is the question a reader of the earlier comparison is really asking, and the answer there is mostly, on twelve cases.

Why handedness is folded first #

A divergence of 209.56 degrees and one of 150.44 are the same lattice wound the other way, and the settling table folds the two together before it counts anything. The slot table reports the raw value.

Folding the wrecked runs the same way is not a tidying step; without it the criterion would be applied to a quantity the criterion was not written for, and a run oscillating across the fold would read as a run with an enormous tail. Every number here is taken after the fold, which is how the endpoint comparison was made in the first place and is the one place the two tables had to be made to agree before anything could be compared.

Five that agreed with nothing #

Five of the thirty-four disagreeing runs settle. A run whose short endpoint sat more than a degree from every destination on the list nonetheless stopped moving, held its value, and passed a criterion written for intact stems.

Those five are the reason the destination list cannot be treated as the set of places a wrecked stem can end. Where they went is a separate reading and it is the one that costs the list three of its members.

By branch #

Golden runs settle twelve of thirty-three and Lucas runs settle five of thirty. That is 36 per cent against 17, on a split that is nearly even in size.

It is the same shape as the rung reading and probably the same fact seen more coarsely, since the rung that carries most of the settling is golden. A branch difference built out of one rung is not a branch difference, and the table cannot separate the two.

By what was removed #

The slot design removes the smaller wall of the tip’s slot, the larger wall, or both. Nine of the twenty-nine doubled cuts settle, six of the twenty-four smaller-wall cuts, and two of the ten larger-wall cuts.

That ordering is the wrong way round for a story in which a bigger disturbance is harder to recover from. Removing both walls together is the expensive cell by every measurement the design was built to make, and it is the cell most likely to end somewhere the criterion accepts.

The share, against the intact table’s own #

Twenty-seven per cent of the wrecked runs settle. About thirty per cent of the settling table’s own intact runs do, at forty starting angles — a share that fell as the sampling of starting angles was refined.

A wrecked run therefore settles at about the rate an intact one does. That is not a small observation. The criterion refuses most of what it is given whatever the run is, so the seventeen are not a poor showing against some standard where most runs settle; they are the ordinary yield of a strict test.

The half turn settles nothing #

Twelve of the sixty-three finish within half a degree of a half turn, and not one of them settles. Twenty-nine finish at 160 degrees or beyond, and not one of those settles either.

A stem at a half turn is in two files rather than on a lattice, and the settling table cannot reach that arrangement because it is not a lattice with a counted pair. The regrowth says the arrangement is not stable under the criterion either, which is a different statement and a stronger one.

What the displacement does not predict #

The first organ placed after a cut moves between 12.0 and 163.6 degrees on the runs that settle, median 40.3, and between 8.9 and 164.1 on the runs that refuse, median 35.6.

Those are the same distribution. How far the cut threw the stem — the quantity the whole slot design was built to measure — says nothing about whether the stem ends up anywhere. The displacement and the endpoint are two halves of one run that behave differently, and this is the sharpest form that has taken.

What this does to the earlier comparison #

Less than it might. The agreement between the two tables is not withdrawn; it is re-read. Twenty-nine endpoints still sit within a degree of a destination, and that number was measured correctly.

What cannot survive is the sentence twenty-nine wrecked stems reached a settling destination. Seventeen of the twenty-nine do not settle by the criterion that defines a destination, so at most twelve of them can be described that way, and there is a further subtraction still to come.

What is not claimed #

That the forty-six are failures of the rule, or that they are still moving. Nothing here says a refusing run is on its way anywhere — that question has its own answer and the answer is no.

Nor is it claimed that the criterion is right. It is a threshold on a tail, and a run sitting 1.6 degrees wide is refused where one sitting 1.4 wide is accepted. What is claimed is that applying somebody else’s threshold unchanged is the only way to make the two tables comparable, and the threshold was not chosen with these runs in view.

The test that would break it #

Run the same regrowth with the settling tolerance moved. If the seventeen were an artefact of where 1.5 degrees falls, the count would climb steeply as the tolerance loosened and the twelve agreeing settlers would be first to arrive.

The two populations are separated by an empty gap twenty degrees wide, so they are not, and the count is flat over any tolerance a reader would find defensible. That check is the one that makes the seventeen a measurement rather than a threshold effect.

What a stronger design would be #

More runs on the four dead rungs, and a second set of lattices at rises between the ones swept here. Sixty-three is enough to find an odds ratio of four and not enough to say whether the rung structure is a property of the rung or of the particular lattices that happened to be wrecked at it.

The cheap version is the ninety cells the slot table actually holds, of which sixty-three are wrecked. Regrowing the twenty-seven that recovered would say whether an unwrecked run settles at a different rate, and it costs about the same again.

It would also close the one comparison this sweep cannot make. Everything here is measured on runs the slot design classified as wrecked, which is a classification about the first organ placed after the cut and not about the rest of the run. A recovered cut and a wrecked one that settles are two descriptions that have never been checked against each other, and the second is now known to happen seventeen times.

What it cost #

The longest single regrown run took 27.3 seconds and the whole sweep 320 seconds on a quiet machine, 552 on a moderately loaded one and 1,002 on a heavily contended one. The eight assertions that check it run in 6.5 seconds against a warm sweep.

Five minutes is worth stating because the comparison it corrects stood for two rounds of work with the check named in its own closing section. Nothing was waiting on the cost.

What is claimed #

That seventeen of the slot table’s sixty-three wrecked runs settle when regrown to twelve hundred organs and put to the settling table’s own criterion with no tolerance altered, and forty-six refuse.

That the prediction of twenty-nine is wrong by twelve runs and wrong on its own side: of the twenty-nine runs whose short endpoint agreed with a destination, twelve settle and seventeen refuse, against five of thirty-four among the rest.

And that the association between agreeing and settling is real at an odds ratio of 4.09 and a p of about 0.017, and weak enough that an agreeing run is still likelier to refuse than to settle. Agreement carries information. It is not a rule, and it was written down as one.