A wall that was never measured

Below some rise, a stem started at an arbitrary angle stops reaching a lattice at all. The rise at which half the starting angles still settle is called the wall, and the question it was built for is whether it moves with the falloff exponent — the power in the sum each organ is placed at the minimum of.

The answer has been no for three rounds, and each round has reported it with different numbers.

Three samplings, three orderings #

At nine angles the four walls sit at 0.01397, 0.00711, 0.00711 and 0.00867 — a factor of 1.97 between the ends, with the exponents ordered 2, 5, 3, 4.

At twenty they sit at 0.01180, 0.01106, 0.01199 and 0.01300 — a factor of 1.18, with the ordering completely reversed to 5, 4, 2, 3.

At forty they sit at 0.01448, 0.01232, 0.01300 and 0.02000 — a factor of 1.62, ordered 5, 2, 4, 3.

No two of the three orderings are the same, over samplings that differ by a factor of four.

And a spread that does not shrink #

1.97, then 1.18, then 1.62. That is not a sequence converging on anything.

Meanwhile every error bar behind it behaves exactly as arithmetic says it should. The standard error on a pooled column goes 5.8 per cent at nine angles, 3.8 at twenty, 2.6 at forty, which is the square-root behaviour a binomial sample has.

So the underlying shares are getting more precise at the expected rate and the quantity read off them is not settling down. That is the finding, and it is a statement about the reading rather than about the exponents.

What went wrong with the reading #

A wall is where a curve crosses a level, and it is located only as well as the curve is steep there. Nobody had asked how steep it is.

At exponent 5 and forty angles, the shares at rises 0.03, 0.02 and 0.013 are 0.475, 0.500 and 0.450. One standard error on a share over forty angles is 7.9 points, so all three of those are within one error of a half.

A crossing consistent with three consecutive rises spanning a factor of 2.3 in the rise is not located between two rises. It is barely located at all.

The bracket #

Written down as a measurement rather than left implicit: the range of rises consistent with the crossing, from the finest rise whose share is confidently above a half to the coarsest whose share is confidently below it, at one standard error.

At forty angles: exponent 2 is bracketed between 0.02 and 0.008, a factor of 2.5. Exponent 3 between 0.03 and 0.008, a factor of 3.75.

Exponents 4 and 5 have no rise whose share is confidently above a half at all. Their brackets are open at the coarse end, which means their walls may sit above 0.03 — above the coarsest rise the table holds.

Which settles the question #

Every bracket is wider than the 1.62 the four walls differ by, by a factor of at least one and a half, and two of them are unbounded.

So a separation of 62 per cent between the largest and smallest wall was never something these runs could have seen. The null was never in danger.

That is a better statement than the one it replaces. The exponents’ walls do not separate sounds like a measurement that came out equal; what is true is that the wall is not determined to better than a factor of two by any sampling here.

Why the error bar did not say so #

Because it was the share’s error and it does not propagate to the wall. A share known to two and a half points is a precise share; the rise at which that share crosses a half is precise only if the share changes quickly with the rise.

Here it does not. Between 0.03 and 0.013 the share at exponent 5 moves from 0.475 to 0.450 — two and a half points across a factor of 2.3 in the rise — so a two-and-a-half-point error on the share is a factor-of-two error on the wall.

That is an ordinary propagation of error and nobody had done it. The rounds that reported the wall quoted the share’s error beside it, which is the error of a different quantity.

What is still true #

That the share falls with the rise, steeply and unambiguously, from around three quarters at 0.03 to around a tenth at 0.003. There is a wall in the sense of a region below which stems stop settling, and it is somewhere around a hundredth.

That is a wall rather than a budget: tripling the run length does not move it, and a steeper rule does not wall elsewhere, so a run that does not settle in twelve hundred organs is not a run that was stopped too early.

What is not available is a wall precise enough to compare between exponents. The existence of the phenomenon and the value of the number are separate claims and only the first is established.

The eight rises #

0.03, 0.02, 0.013, 0.008, 0.005, 0.0045, 0.004 and 0.003. That is the list the settling table has always used, kept unchanged so that the exponent-three column of this table and the table that found the wall are the same measurement.

Its spacing is the other half of the problem. Between 0.02 and 0.013 there is nothing, and between 0.013 and 0.008 there is nothing, so a crossing anywhere in there is located to a factor of 1.5 by the rises alone before any error is considered.

Four rises between 0.008 and 0.003 and four above, on a quantity whose interesting region is around 0.012, is a list chosen for coverage rather than for locating a crossing.

What would locate it #

Rises between 0.02 and 0.008 at a finer spacing, and more angles at each. The two are multiplicative: the bracket’s width comes from the share’s error divided by the curve’s slope, so halving the error and halving the spacing each buy about a factor of two.

Getting the wall to a factor of 1.2 — enough to see a separation of 1.6 — would take something like eight rises between 0.02 and 0.008 at eighty angles, which is 2,560 runs and about forty-five minutes.

That is affordable and it is worth stating what it would buy: a wall per exponent good enough to say whether the four differ, on a question whose answer has been no for three rounds and whose no has never been a measurement.

What the orderings were worth #

Nothing, and they were reported as nothing each time. The round at twenty angles said the ordering had reversed and drew the right conclusion — neither ordering means anything; that is the point — without being able to say why.

This is the why. An ordering of four numbers each known to a factor of two is four coin flips, and three independent sets of four coin flips giving three different orderings is not surprising.

What is new is that the conclusion can now be stated as a measurement: the brackets are wider than the spread, so the ordering is not information. The round that reversed the ordering had the right instinct and no instrument to back it.

The one thing the exponent does move #

Not the wall. The clock — how long a settling run takes — moves with the exponent in a way a share cannot see, and that reading is unaffected by anything here because it is a mean over settled runs rather than a threshold crossing.

And the destinations: a steep falloff reaches places a shallow one does not, which is a list rather than a number and is subject to a correction of its own.

So the exponent is not inert. It is inert in the one quantity that was chosen to measure it, for a reason about the quantity rather than about the exponent.

Why the wall was chosen at all #

Because it is the obvious summary. Below what rise does the rule stop working is one number and it compresses a whole column, and a comparison between four exponents wants one number each.

The trouble with a summary is that it can be much less determined than what it summarises. The column has eight shares in it, each known to 2.6 points at forty angles; the wall is one number known to a factor of two.

That is a general hazard and it has a general remedy: report the column and the summary together, with the summary’s own uncertainty rather than the column’s. Which is what this reading now does.

How the wall is computed #

The share of starting angles that settle is read at each of the eight rises, and the wall is where a straight line through the two rises either side of a half crosses it — a linear interpolation in the logarithm of the rise.

That is a reasonable thing to do and it is the source of one oddity in the numbers. At exponent 5 and forty angles the share reaches exactly 0.500 at a rise of 0.02, so the interpolation returns 0.02000 — a rise from the list rather than a value between two of them.

A summary that occasionally returns one of its own input values is a summary whose inputs are too coarse for it. That is visible in the number itself and nobody had looked.

The level is a half and could be anything #

The share falls from most of the angles to none as the rise falls, so any level between about a third and two thirds names the same ordering of the four exponents. A half is the middle of that and was stated once rather than chosen per column.

That reasoning was sound when it was written and it is undermined by the same flatness. If the curve near a half is flat enough that the crossing is unlocated, then moving the level from a half to a third moves the crossing by a great deal — and the claim that any level names the same ordering is a claim about an ordering that is four coin flips.

Checking would mean recomputing the four walls at several levels and seeing whether the ordering holds. It is arithmetic on the same table and has not been done.

What a steeper curve would look like #

At the coarse end of the table the share does change quickly: between 0.005 and 0.003 it falls from around 0.15 to around 0.09, and between 0.008 and 0.005 from 0.30 to 0.15.

So the curve is steep where the share is small and flat where it is near a half, which is the worst possible arrangement for reading a crossing at a half. A wall defined at a fifth rather than a half would be far better located.

It would also be a different quantity, and one whose comparison to the previous rounds’ numbers would be an argument about two designs. Which is a reason to report both rather than to replace one with the other.

What this does not undermine #

Any claim about the settling share itself. The shares are measured to 2.6 points at forty angles and the comparisons between them are sound — including the finding that the original nine angles were a biased sample, which is a comparison between groups of angles at the same rises.

Nor does it touch the destination list, the arrangements or the basins, none of which is a threshold crossing.

It touches one number per exponent, quoted in three rounds, and the correction is to say how well it is known rather than to change its value.

The same hazard, elsewhere on this site #

A summary much less determined than what it summarises is a shape rather than an incident, and this collection has met it before.

A mean side count throws nearly everything away, where the summary is one number over a whole distribution and the second moment turns out to carry the information. A fitted exponent on a branching tree comes out precise and means little, because the fit is over a range where the two candidate laws barely differ.

The common thread is that a summary’s own uncertainty has to be computed from the summary and not inherited from its inputs. That is one line of arithmetic and it is the line nobody writes.

What a barrier this puts on the exponent question #

The falloff exponent is the rule’s one free parameter, and the whole point of sweeping it from two to five was to find something it decides. Three quantities have been tried: the wall, the clock and the destination list.

The clock moves with it. The destination list moves with it. The wall does not, and that negative has been the headline of two rounds because it is the quantity that sounds most like a physical property.

What this reading says is that the wall is the worst of the three instruments for the job, and that the two which do move are the ones to quote. That reorders the thread’s own findings without changing any of them.

What a wall would need to be worth quoting #

A bracket narrower than the difference it is being used to test. That is the whole requirement, and it is arithmetic rather than a matter of taste: a quantity known to a factor of two cannot adjudicate a difference of a factor of 1.6.

Getting there needs two things multiplied together. The share must be measured finely enough that a standard error is small against the curve’s own change from rise to rise — eighty angles would halve it again, to about 1.8 points. And the rises must be close enough near the crossing that the curve’s slope is resolved; eight rises between 0.02 and 0.008 would put four readings where there are currently two.

Together those bring the bracket to something like a factor of 1.2, which is narrow enough to see a 1.6 spread if there is one. It is 2,560 runs and about forty-five minutes, and it is the only thing that would turn three rounds of they do not separate into a measurement.

The cheaper half, which was not done either #

Recomputing the four walls at levels other than a half. The current reading asserts that any level between about a third and two thirds names the same ordering, and that assertion is arithmetic on the table already in hand — no new runs at all.

If the ordering holds across levels, the null is on firmer ground than this essay gives it credit for. If it does not, then the level is a fourth thing the ordering depends on, and the whole quantity is worse determined than the brackets say.

It is minutes of work and it is not done, which is worth recording as plainly as the expensive half. The reason it was not done is that the brackets settled the question the round was asking, and a check that would only strengthen a negative is easy to leave.

What is claimed #

That the four falloff exponents’ walls come out at 0.01397, 0.00711, 0.00711 and 0.00867 at nine starting angles, 0.01180, 0.01106, 0.01199 and 0.01300 at twenty, and 0.01448, 0.01232, 0.01300 and 0.02000 at forty; that no two of the three samplings order them the same way; and that the spread between the ends goes 1.97, 1.18, 1.62 while every error behind them halves.

That at forty angles the rises consistent with each crossing span a factor of 2.5 and 3.75 for exponents 2 and 3, and are unbounded above for exponents 4 and 5 — every one of them wider than the 1.62 the walls differ by.

And that the conclusion the wall does not separate the exponents therefore stands as a statement about what these runs can see, rather than as a measurement that came out equal.