The level was doing the ordering

Below some rise a stem started at an arbitrary angle stops reaching a lattice at all, and the rise at which half the starting angles still settle is called the wall. Four falloff exponents — the power in the sum each organ is placed at the minimum of — have had a wall computed for each of them, and the four have been printed in order of size three times.

A half is not a property of the wall. It is a level, stated once and never varied, and the ordering it produces is the only ordering of these four walls this collection has ever published.

Read at nine levels rather than one, the ordering turns out to be a property of the level.

The check that cost nothing #

The runs were already on disk. A wall is read off a column of settling shares by finding where the column crosses a level, so moving the level is arithmetic on a table rather than a reason to grow a stem.

The reading that showed the wall was never located said as much and said why it had not been done: the brackets had already settled the question that was being asked, and a check that could only strengthen a negative is easy to leave. It is done here, and it does not strengthen the negative. It changes what the negative is about.

Nine levels, and the five that resolve #

The levels tried run from a fifth to four fifths — a fifth, a quarter, a third, two fifths, a half, three fifths, two thirds, three quarters, four fifths.

Five of the nine resolve at every one of the three samplings the settling table has been grown at, and it is the same five each time: a fifth, a quarter, a third, two fifths and a half. A half is the coarsest level any of these columns can be read at.

That is not a design decision. It is a fact about the curve, and it is the first thing the level scan reports.

Why the other four have nothing to read #

A level above the top of a column has no crossing. The settling share never rises above 0.778 at nine starting angles, 0.750 at twenty and 0.750 at forty, so three fifths is already near the ceiling of the best column in the table and two thirds is above it in most of them.

At forty angles it is worse than the pooled figures suggest: exponents 4 and 5 never exceed 0.550 and 0.500 at any rise. Their columns touch a half at one point and go nowhere above it.

So a wall on this curve can be read at a half or below and nowhere else, and the level that was chosen sits at the very edge of what the runs support.

Fifteen readings of one table #

Three samplings by five levels is fifteen readings, and every one of them is the same runs read differently. No stem was grown for any of this.

That matters for what the disagreement between them can mean. Two readings that disagree because they sampled different angles are two measurements; fifteen readings that disagree while sharing every run cell for cell are one measurement being reported fifteen ways.

Ten orderings out of twenty-four #

Four exponents admit twenty-four arrangements. The fifteen readings give ten of them.

Ten distinct orderings from fifteen readings of one table is a great deal of disagreement. A quantity being measured rather than manufactured would give one ordering fifteen times, or two if the reading were marginal; ten is what a set of numbers put in order at random looks like.

The picture is drawn as all twenty-four cells rather than as a count, so the ten filled ones sit against fourteen that never occur. What is worth reading off it is not that ten is a large number but that no cell is filled often.

Within one sampling, and across three #

The disagreement is not an artefact of pooling three different samplings together, which would be the obvious objection and is the easy one to check.

Nine starting angles give three different orderings across their five levels. Twenty give three. Forty give four. So each sampling contradicts itself internally before any two of them are compared, and the pooled count of ten is not three samplings each speaking consistently in its own voice.

The nine-angle readings #

At nine angles the five levels order the exponents 5, 3, 2, 4 at a fifth, then 5, 3, 4, 2 at a quarter, at a third and at two fifths, then 2, 5, 3, 4 at a half.

The spreads between largest and smallest wall run 1.455, 1.163, 1.333, 1.487 and 1.964.

Three of the five give exactly the same ordering, a fourth differs from those three by one swap, and a half gives something else altogether. It is also the level reporting much the largest spread. The level that was published is the outlier of its own scan.

The twenty-angle readings #

At twenty the five levels give 2, 3, 5, 4 at a fifth and at a quarter, then 4, 2, 3, 5 at a third and at two fifths, then 5, 4, 2, 3 at a half.

Spreads: 1.036, 1.265, 1.206, 1.084 and 1.176.

Two pairs of adjacent levels agreeing with each other and disagreeing with the pair below is what a quantity drifting smoothly with the level looks like. What sits on the end is again a half, again alone.

The forty-angle readings #

At forty the walls come out 0.00576, 0.00585, 0.00549 and 0.00549 at a fifth; 0.00632, 0.00684, 0.00663 and 0.00663 at a quarter; 0.00740, 0.00860, 0.00867 and 0.00891 at a third; 0.00941, 0.00993, 0.01020 and 0.01106 at two fifths; and 0.01448, 0.01232, 0.01300 and 0.02000 at a half.

The orderings are 3, 2, 4, 5 · 3, 4, 5, 2 · 5, 4, 3, 2 · 5, 4, 3, 2 · 5, 2, 4, 3, and the spreads 1.065, 1.081, 1.205, 1.176 and 1.624.

Two of the five levels here agree with each other, which is as much agreement as forty angles produce. Neither of them is a half, and the spread they report is a third of what a half reports.

A half is unique at every sampling #

That is the finding stated as sharply as it goes. At nine angles the ordering read at a half occurs once in the scan. At twenty it occurs once. At forty it occurs once.

Three samplings, and at each of them the level that has been published is the one level whose answer no other level repeats.

Which is not a coincidence about a half #

It is what happens when a level is pushed to the top of the range the curve supports. Near the ceiling a column is flat, the crossing is unlocated, and a small change in a share moves the wall a long way — so the reading at the highest available level is the reading most exposed to the flatness.

A half was chosen because it is the obvious middle of a share, and the reasoning written down for it was that any level between about a third and two thirds names the same ordering. That reasoning was stated as an assertion and it is the thing the scan tests.

It is false at every sampling. Between a third and a half the ordering changes at all three.

Which exponent comes first #

Across the fifteen readings, exponent 5 has the largest wall eight times. Exponent 2 has it three times, exponent 3 twice and exponent 4 twice.

Eight of fifteen is the only thing in the scan that looks like a signal, and it is worth saying why it is not one. Exponent 5’s column is the one that touches a half at a single rise and never exceeds it, so its crossing at the top of the range is the least located of the four — and the least located quantity is the one that most often comes out extreme.

An ordering that ranks by wall size ranks a badly located number first for the same reason a noisy measurement wins a race it has no business winning.

An ordering is a claim about separation #

Two walls can be put in order only if the ranges of rise consistent with them do not overlap. Where they do, saying which is larger reports the sign of a difference smaller than its own uncertainty, which is a coin flip with a number printed beside it.

That is not a new observation here — it is what the reading that measured the brackets established — but it is what turns a count of orderings from a curiosity into a correction. Ten orderings is what four coin flips give.

Fifty-four pairs, and one that separates #

The nine levels give six pairs of exponents each, which is fifty-four comparisons. Exactly one of them has brackets that do not overlap.

It is exponent 2 against exponent 3, at a level of a third. Exponent 3’s wall sits at 0.01183 with a bracket running from 0.013 to 0.01106, a factor of 1.18; exponent 2’s sits at 0.00737 with a bracket from 0.00941 to 0.005, a factor of 1.88. The two walls differ by a factor of 1.606 and the bars do not touch.

One separation in fifty-four is the whole of what this quantity has ever resolved.

And it runs the other way #

The published reading, at a half and forty angles, puts exponent 2 above exponent 3 — 0.01448 against 0.01232.

The one comparison anywhere in the study that separates puts exponent 3 above exponent 2, by a factor of 1.606.

So the published ordering is not merely unsupported. On the one pair where support exists, it is contradicted.

What a reader who met the old claim should believe #

That the four falloff exponents’ walls have never been ordered by a measurement. The sentence exponent 5 walls highest and exponent 3 lowest was arithmetic on one level, and nine levels give ten answers.

That the one ordered pair available says exponent 3’s wall sits above exponent 2’s, at a level of a third, and that this is a single comparison rather than a new ranking of the four.

And that nothing about the existence of a wall is in question. Stems stop settling at the fine end, longer runs do not reach past it, and the four columns fall from about three quarters to about a tenth as the rise falls. What is in question is one number per exponent and the order of four of them.

The finer reading does not rescue the level #

Eight rises and forty angles is not the last word on this table. Read on a finer list of rises at eighty starting angles, six of the nine levels resolve rather than five, and those six give four different orderings.

The ordering at a half is still one no other level gives.

So the level dependence is not a symptom of a coarse rise list. It survives the refinement that was designed to remove it, and survives it while every bracket closes.

A column that crosses a level twice #

Three of the thirty-six cells of that finer scan have a share that crosses its level more than once. A wall is defined as the crossing, and where there are two the definition is choosing.

That is a small number and it is not small in what it implies. A quantity defined by a crossing is well defined only where the curve is monotone, and the finer rises show the curve is not monotone everywhere. The coarse list could not have seen that, which is a general property of coarse lists rather than a fault of this one.

Where the level enters the arithmetic #

The wall is a linear interpolation in the logarithm of the rise, between the two rises either side of the level. So the level enters twice: it picks which two rises bracket the crossing, and it sets where between them the answer falls.

Moving the level by a tenth can therefore move the wall between two different pairs of rises, which is a jump rather than a slide. On a rise list whose steps are a factor of one and a half apart, a level that moves the crossing one rise along moves the answer by half again.

What the scan does not touch #

The shares themselves. Every share in the table is a count of settled runs over a count of runs, it is the most precisely known thing this thread has, and nothing here moves one of them.

Nor does it touch the two quantities the exponent does decide. How long a settling run takes is a mean over settled runs rather than a threshold crossing, and where a settled run ends up is a list. A level appears in neither.

So the exponent is not inert. It is inert in the one quantity chosen to measure it, and the reason is now two-fold: the wall is badly located, and the ordering it produces belongs to a setting nobody varied.

The same shape, twice on this site #

A statistic whose value belongs to the instrument rather than to the subject is a shape this collection keeps finding. A crowding ordering came out backwards when the window was changed and turned out to be the level inside the instrument rather than a property of the families.

A summary that cannot vary with what it describes is the other half of it. Here the summary can vary — it varies a great deal — but what it varies with is the level.

The diagnostic in both cases is the same and costs nothing: move the setting and see whether the answer moves with it.

Why a level scan is a better instrument than a doubling #

Doubling the sampling halves the error on a share. It does not touch the level, so a quantity whose value is set by the level comes out equally precise and equally wrong at every sampling — which is what three samplings giving three orderings looked like from inside.

That is the ordinary way an instrument setting hides. Every check that was run varied something the answer did not depend on, and the check that would have found it was one line of arithmetic on runs already grown.

What it would take to order these four #

Brackets narrower than the difference being tested, at a level the curve can actually be read at. The second half of that requirement is the new one: at a half two of the four columns never rise above the level, so the reading is being taken where the data runs out.

A level of a third or a quarter is far better placed. Its brackets are narrower — a factor of 1.18 on one of them — and it is the level at which the one separation in the study occurs. That it also gives a different ordering from a half is the point rather than an inconvenience.

The design fault, stated plainly #

One level, chosen for tidiness, applied to four columns with different ceilings. A half is near the top of two of them and comfortably inside the other two, so the four walls are not even being read at comparable places on their own curves.

An ordering of four numbers read at different distances from the tops of four different curves is not a comparison. That would be true even if each wall were perfectly located, and none of them is.

What is claimed #

That the four falloff exponents’ walls, read at nine levels from a fifth to four fifths on the three samplings the settling table holds, resolve at five levels each and give fifteen readings; that those fifteen give ten of the twenty-four orderings four numbers admit; and that each sampling internally gives three, three and four.

That the ordering obtained at a half is, at every one of the three samplings, an ordering no other level produces — and that on the finer reading at eighty angles, where six levels resolve and give four orderings, it is still alone.

That of the fifty-four pairwise comparisons the nine levels afford, exactly one has brackets that do not overlap: exponent 3’s wall at 0.01183 above exponent 2’s at 0.00737, at a level of a third, a factor of 1.606 apart.

And that the published ordering puts exponent 2 above exponent 3, so the one place this quantity resolves anything is a place where it contradicts what was reported.