Four walls closer than they looked

The rise at which half a set of starting angles still settle onto a lattice is the wall, and there is one per falloff exponent. Until now, two of the four had no located wall at all: no rise in the table had a settling share confidently above a half, so the bracket of rises consistent with the crossing was open at the coarse end and the wall might have sat above the coarsest rise ever grown.

The other two were bracketed to factors of 2.50 and 3.75, against four walls that differed by a factor of 1.62. Every bracket was wider than the difference it was being used to test.

Nine rises at eighty starting angles close all four. What comes out is a better measurement that settles less than the worse one appeared to.

What was grown #

Nine rises at four exponents at eighty starting angles: 2,880 runs, of which 800 were already on disk and 2,080 were new. Each is a stem grown to twelve hundred organs.

Fifty minutes of machine time in a single process, or 32.1 minutes of wall clock across six of them. That is the whole cost of the answer, and it was budgeted in advance at about forty-five minutes by the reading that showed the wall had never been located.

The design contains the old one #

The eighty starting angles are the published forty, their thirty-nine midpoints, and one angle two and a half degrees below the lowest. Spacing halves from 3.12 to 5.00 degrees down to 1.56 to 2.50.

The nine rises keep five of the published eight — 0.03, 0.02, 0.013, 0.008 and 0.005 — and add four inside the stretch the wall crosses. Seven of the nine now sit between 0.02 and 0.008, where there were two.

So the old table is a sub-table of the new one, and every difference between them is a refinement rather than an argument about two designs. That is the rule this thread has followed at every doubling, and it is what makes the before and after comparable.

The step between rises #

The published list stepped by 1.538 and 1.625 across the interesting region, so a crossing anywhere inside it was located to a factor of one and a half by the rise list alone, before any error on a share was considered.

The refinement steps by 1.155 and 1.176. That is the number that matters, because a bracket’s ends are rises: whatever the share’s error does, a bracket cannot be narrower than the gap between two rises.

The four rises that were added #

They are 0.01732 and 0.01501, which split the empty stretch between 0.02 and 0.013, and 0.01106 and 0.00941, which split the empty stretch between 0.013 and 0.008.

All four located walls land on one of those added rises or between two of them, which is the design working exactly as intended: the rises were placed where the crossing was expected and the crossing turned out to be there. Exponent 2’s wall is 0.01501, an added rise; exponents 3 and 5 sit between 0.01501 and 0.013, and exponent 4 between 0.01732 and 0.01501.

A wall that returns one of its own input rises is a wall whose inputs are too coarse, which is what exponent 5’s 0.02000 was. That it happens again here, at a rise four times more finely placed, is a much weaker version of the same complaint.

Every bracket closed #

Exponent 2’s bracket runs 0.02 to 0.013 rather than 0.02 to 0.008 — from 2.50 times down to 1.54. Exponent 3’s runs 0.01501 to 0.01106 rather than 0.03 to 0.008 — from 3.75 times down to 1.36.

Exponents 4 and 5, which had no coarse end at all, now run 0.01732 to 0.01106: 1.57 times each.

Two of four open before, none after. The widest bracket goes from unbounded to 1.57.

Where the four walls actually are #

Exponent 2 at 0.01501, exponent 3 at 0.01377, exponent 4 at 0.01530, exponent 5 at 0.01397.

Two of them moved a long way. Exponent 5 was reported at 0.02000 and sits at 0.01397, a fall of thirty per cent; exponent 3 was reported at 0.01232 and sits at 0.01377. Exponent 2 moved from 0.01448 to 0.01501 and exponent 4 from 0.01300 to 0.01530.

Every one of those moves is inside the old bracket, which is what a bracket is for and is the one thing here that went entirely as it should.

Closer together, not further apart #

The spread between the largest wall and the smallest falls from 1.624 to 1.111.

That is the result the design was least prepared for. A refinement that locates four quantities can find them further apart, which would have separated them, or closer, which cannot. It found them closer by a factor of nearly one and a half.

The 62 per cent difference the earlier reading printed was mostly the exponent 5 column returning a rise from its own input list, and it went away when the list got finer.

Which makes the answer worse #

Six pairs of exponents, and at a half none of them separates. Not one pair’s brackets fail to touch.

The arithmetic is one line. Each wall is located to about 1.5, the four sit inside 1.111 of one another, and a set of quantities closer together than the width of any one of their error bars cannot be put in order.

The closest and the furthest pair #

Exponents 3 and 5 are a factor of 1.014 apart — one and a half parts in a hundred, on a quantity located to fifty parts in a hundred.

Exponents 3 and 4 are the furthest, at 1.111, which is the spread. Even that pair, the best case available, is separated by a tenth of what its own brackets are wide.

So this is not a near miss. The narrowest bracket in the study is 1.36 times wide and the largest separation between any two walls is 1.111, so even the best-placed pair sits closer together than the best-located of the four walls is uncertain.

The ordering moved a fourth time #

At a half the four exponents came out ordered 5, 2, 4, 3 on the published reading. On this one they come out 4, 2, 5, 3.

That is a fourth sampling of this table and a fourth ordering. Nine angles gave one, twenty gave another, forty gave a third, and eighty on finer rises gives a fourth.

Which is precisely what an ordering read off overlapping brackets does, and the interest of it here is only that it kept doing it after the instrument was improved.

One exponent’s bracket, on its own #

Exponent 3 is the clearest case because it started as the worst. Its bracket ran from 0.03 to 0.008 — the coarsest rise the table held down to the fine end of the interesting region, a factor of 3.75, which is to say the wall was somewhere in the table.

It now runs 0.01501 to 0.01106, a factor of 1.36, and it is the narrowest of the four.

A factor of 2.76 of narrowing on one exponent is the largest single improvement in the study, and it still leaves that exponent indistinguishable from the other three.

The one place anything comes apart #

Not at a half. At a level of a third, exponent 3’s wall at 0.01183 sits above exponent 2’s at 0.00737, a factor of 1.606, and the two brackets do not touch — 0.013 to 0.01106 against 0.00941 to 0.005.

One separated pair, at one level, out of fifty-four comparisons across nine levels. It is the only thing this quantity has ever resolved and it is not at the level anybody read.

Two results, and both have to be said #

The first is that a measurement which had two unbounded intervals in it now has none. That is a real gain, it was bought with half an hour of machine time, and it converts the wall may be anywhere above 0.03 into the wall is at 0.01397, give or take a factor of one and a half.

The second is that the gain settles nothing about the question it was bought for. Locating four quantities precisely enough to see that they are not distinguishable is a result, and it is a disappointing one.

Reporting only the first would be the ordinary temptation, because the first is the part that sounds like progress.

What a factor of 1.5 buys #

It bounds the wall. The four walls sit between 0.01377 and 0.01530 and every one of the four brackets lies inside 0.011 to 0.02, where the statement four essays had been carrying was that stems stop finding lattices somewhere below about a hundredth.

It also removes a live possibility. An open coarse end meant exponent 5 might have walled above 0.03 — above every rise the settling table holds — which would have made every share in that column a share taken below the wall. It did not.

That negative is worth as much as the positive and it could not have been read off the old brackets at all.

The coarse end and the fine end #

The columns themselves are unambiguous either side of the crossing, which is what makes the crossing worth locating at all. At a rise of 0.03 the four exponents settle at 0.787, 0.662, 0.537 and 0.487; at 0.005 they settle at 0.138, 0.113, 0.138 and 0.138.

So the share falls by a factor of four to six across the table, and the four columns are plainly ordered at the coarse end and plainly on top of one another at the fine end. Neither of those is in doubt at any sampling.

What is in doubt is one number in the middle of each column. A curve that is steep at both ends and flat where the level sits is the worst arrangement for reading a crossing, and it is the arrangement these four columns have.

The binomial error was never the wall’s error #

The standard error on a share falls from 0.0791 at forty angles to 0.0559 at eighty: a factor of 0.707, which is exactly the square root of two that arithmetic requires.

That number was quoted beside the wall for three readings and it is the error of a different quantity. A share known to five and a half points is a precise share; the rise at which the share crosses a half is precise only if the share changes quickly with the rise, and here it does not.

So halving the error bar behind the wall did not halve the wall’s own uncertainty, and there was never any reason to expect it to. The essay that measured the brackets said so; this is the confirmation on runs that were grown to test it.

Which half of the refinement did the work #

Almost all of it was the rises. Doubling the angles, on its own, moved the brackets by very little, because a bracket’s ends are rises and the published list floors any bracket at 1.54.

That decomposition is an argument in its own right and it matters here for one reason: the expensive half of this design bought the smaller share of its own result. Two thousand and eighty runs went in, and the runs were mostly paying for the shares rather than for the wall.

What the doubling did not move #

The pooled settling shares. Over the five rises both samplings hold, exponent 2 goes 0.470 to 0.475, exponent 3 goes 0.445 to 0.415, exponent 4 goes 0.405 to 0.393 and exponent 5 goes 0.380 to 0.385 — every one inside its own error.

Over all nine rises the original forty angles and the added forty settle at 0.478 against 0.444, 0.444 against 0.394, 0.422 against 0.406 and 0.417 against 0.444, each pair inside an error of 0.026.

So the settling bias that shrank from nine angles to twenty to forty has converged. Doubling again moves the pooled share by nothing, which is the clean positive this design buys.

The destination list, checked in passing #

The whole table reaches 104 distinct destinations at forty angles and 108 at eighty.

Of the four added, none is more than half a degree from a destination already reached. So the forty new angles found no new place for a stem to end up, which is the same shape a list defined by an absence has shown before: the doubling refines the boundaries and adds no members.

The clock, at the new sampling #

Median settling times come out at 34, 26, 31 and 32 organs across the four exponents, and the slowest settling anywhere in the refined table is 591 organs at exponent 3.

Settling remains fast or absent. Nothing in two thousand new runs settles late enough to make the twelve-hundred-organ budget look tight, which is the other half of the wall being a wall rather than a budget.

What the slowest runs did #

The slowest settling in each column goes 406 organs to 441 at exponent 2, 591 to 591 at exponent 3, 264 to 397 at exponent 4 and 350 to 423 at exponent 5.

Three of the four rise, which is what doubling a sample does to a maximum and is not a finding. The one that does not is exponent 3, whose slowest run at forty angles was already the slowest run anywhere and stays so with forty more angles beside it.

A maximum over a larger sample can only go up, so the interesting reading here is the medians above, which did not.

What the negative now is #

Before, the four exponents’ walls did not separate and nobody could say whether that was because they are equal or because the instrument was blunt. The brackets said the instrument was blunt.

Now the instrument is nearly three times sharper on the worst exponent and the four walls have moved towards one another. That is the reading under which the exponent does not move the wall stops being a statement about an instrument and starts being a statement about the exponents — weakly, and at one level.

The honest form is that the four walls agree to within a factor of 1.111, and that brackets between 1.36 and 1.57 times wide could not have seen a disagreement much smaller than half.

What would still be needed #

Rises closer together than the difference being tested, because a bracket’s ends are rises and no bracket can be narrower than the step between two of them. To separate the furthest pair the step would have to fall below 1.111, from the 1.155 and 1.176 this list uses. To separate the closest it would have to fall below 1.014, which is about ten times as many rises across the same stretch.

Which is the arithmetic that says to stop. The quantity is not badly measured any more; it is a quantity whose four values are the same to within what any affordable design could see, and that is a property of the wall rather than a shortfall of the sweep.

What is not ruled out #

That the exponent changes the shape of the settling curve rather than a threshold on it. Two of the four columns carry a maximum at a rise the published list stepped straight over, and that maximum is the first quantity in this thread that tells the exponents apart.

It is also the reason two brackets were open at the coarse end: those columns were being read on the far side of a turn the rise list could not see.

So the refinement’s most informative product is not the located walls. It is the four rises that were added between them.

What this does not undermine #

Any claim about the settling share itself, which is measured over 720 runs an exponent at an error of 0.0185 and is the most precise thing in the table.

Nor the finding that the exponent moves the clock, nor that a steep rule reaches destinations a shallow one does not. Neither is a threshold crossing and neither inherits any of this.

It touches one number per exponent, quoted four times, and it replaces a number that was not located with one that is.

What is claimed #

That eighty starting angles on nine rises, 2,880 runs of which 2,080 were newly grown, close all four brackets on the wall at a half: exponent 2 from 2.50 times to 1.54, exponent 3 from 3.75 to 1.36, and exponents 4 and 5 from open at the coarse end to 1.57 each.

That the four walls then sit at 0.01501, 0.01377, 0.01530 and 0.01397 — a spread of 1.111 where the reading that could not locate them said 1.624 — so the refinement brought them closer together.

That none of the six pairs separates at a half, the closest being exponents 3 and 5 at 1.014 and the furthest exponents 3 and 4 at 1.111; and that the ordering changes for a fourth time, from 5, 2, 4, 3 to 4, 2, 5, 3.

And that a measurement can be a great deal better and settle nothing: what is now established is not that the four walls are equal but that they are equal to within a tenth, and that no sampling this collection can afford would see a difference that small.