A wall that stopped moving

Below a certain rise a stem stops reaching a lattice at all, whatever angle it is started from. That is the wall, and it is read as the rise at which the share of starting angles that settle falls through a half.

The four falloff exponents were swept to ask whether a steeper rule walls somewhere else, and the answer was that it does not. The numbers underneath that answer looked less settled than the answer did.

What the nine angles gave #

Walls at 0.01397, 0.00711, 0.00711 and 0.00867 for exponents two, three, four and five. A factor of 1.97 between the ends, and an ordering of 2, 5, 3, 4.

That is a two-fold spread on a quantity reported not to move. The file said so plainly — it computed a binomial standard error of about 0.17 on each cell and declined to read a spread of 0.056 against it — and reported the conclusion rather than the spread.

What twenty give #

Walls at 0.01180, 0.01106, 0.01199 and 0.01300. A factor of 1.18 between the ends, and an ordering of 5, 4, 2, 3.

The spread collapses from two-fold to a fifth, and the ordering reverses completely: the exponent that had the coarsest wall now has the third, and the one that had the finest now has the second.

Which is what a noisy ordering does #

An ordering that reverses when the sample doubles is not an ordering. Had the four exponents genuinely separated, doubling the sample would have tightened the same order rather than producing a different one.

So the two samplings agree about the conclusion and disagree about every number that might have been read as evidence against it. That is the pleasanter of the two ways a re-sampling can go and it is worth naming as such.

The error, halved #

The binomial standard error on a pooled column falls from 5.8 per cent to 3.7, and on an individual cell from about 17 to about 11.

That is exactly what doubling a sample buys and it is worth stating that it is not more. A share over twenty runs is still a coarse instrument, and the readings this table supports are orderings and presences rather than precise shares.

Why the wall survives a bias #

The extra angles settle less often than the original nine, so every share in the table falls by about fifteen points. A level crossing read at a half might have been expected to move a long way.

It does not, because the shares fall by roughly the same amount everywhere and the curves are steep where they cross. At the working exponent the share falls from 60 to 30 per cent across one step of the rise, so a fifteen-point shift moves the crossing by well under one rise.

What a level does to a curve #

Worth a paragraph, because the wall is defined by a threshold and this collection distrusts thresholds. The level is a half, chosen because the share falls from most of the angles to none of them as the rise falls, so any level between about a third and two thirds names the same ordering of the four exponents.

That was asserted when the level was chosen. At twenty angles it can be checked: the shares still run from 70 per cent at the coarse end to 5 at the fine one, so any level in the middle third crosses in the same place to within a rise.

What the pooled shares say #

Another reading, and it does not use a level at all. Pool every cell of an exponent’s column and the shares are 41.7, 43.1, 40.3 and 37.5 per cent at nine angles and 36.9, 32.5, 31.9 and 28.1 at twenty.

The ordering there is nearly stable — 3, 2, 4, 5 becomes 2, 3, 4, 5, which is one swap at the top — and the spread is about four points against an error of four. So the pooled reading says the same thing the wall does and says it without a threshold.

Two readings that agree #

That agreement matters more than either reading. A quantity that is stable under a threshold and stable without one is stable; a quantity that needs the threshold is a quantity whose result is partly the threshold’s.

The wall needed a level and the pooled share does not, and both say the four exponents do not separate. So the conclusion does not rest on the half.

Where the exponents do separate #

They separate on the clock. A steeper falloff settles a stem faster — the slowest settling run at exponent two takes longer than the slowest at exponent five — and that is a time rather than a share, so it carries no binomial error.

They also separate on destinations: the steep exponents reach places the shallow ones do not. Both of those readings are unaffected by the sampling in the way a share is, and both are where the file’s own conclusion put the difference.

The destination list does grow with the sampling — from three steep-only entries to five — but growing is the only thing it can do, since an angle added can reveal a destination and cannot conceal one.

What the wall is, if it is not exponent-dependent #

A property of the rise. Below about 0.011 in the rise, most starting angles fail to reach a lattice, and that is true whether the falloff goes as the square, the cube, the fourth or the fifth power of the distance.

Which is a statement about the geometry rather than about the rule’s shape, and it is the statement the wall being a wall rather than a budget already pointed at: a longer run does not rescue those stems, so what stops them is not time.

What happens instead is that the stem never settles onto a branch — its divergence wanders rather than arriving, and a counter pointed at it still returns a pair, which is why the test has to be made on the divergence.

The finest rises are unchanged #

At the bottom of the table almost nothing settles at any starting angle: one of twenty at a rise of 0.004, one of twenty at 0.003 at the working exponent.

The extra angles leave that exactly as it was, which is the other thing worth checking. A sampling change that moved the fine end as well as the middle would have been a change in the whole table rather than a correction to a share.

What a factor of 1.18 is #

It is 0.01106 to 0.01300, which is less than one step of the rises the table is grown at. The eight rises are 0.030, 0.020, 0.013, 0.008, 0.005, 0.0045, 0.004 and 0.003, and the gap between 0.013 and 0.008 is a factor of 1.6.

So all four walls now sit inside one interval of the table’s own grid, and the interpolation that places them within it is an interpolation between two cells rather than a measurement. That is the honest description of a factor of 1.18.

Which means the wall is a coarse number #

It always was, and the two-fold spread at nine angles was the first sign. A wall computed by interpolating a share between two rises five parts in a thousand apart cannot be quoted to five digits and mean it.

The right form is between 0.013 and 0.008, at every exponent, at both samplings. That is what the table resolves and it is what all four columns agree on.

Why it was quoted more finely #

Because the interpolation returns a number and a number invites its digits to be read. That is the same shape as a position quoted in sampled steps — a quantity computed inside an instrument’s own resolution and reported as though it were outside it.

Both are easy to make and both are visible the moment a second sampling exists. Neither is a mistake in the arithmetic.

What twenty angles did for the conclusion #

Confirmed it, and removed the awkwardness. Before, the file said the exponents do not separate while printing four walls spanning a factor of two, and a reader had to take the error bar on trust.

Now the four walls span a factor of 1.18, which is inside one step of the table’s grid, and the sentence and the numbers agree. That is worth twelve minutes on its own.

What did not need re-checking #

The claim that the wall is a wall rather than a budget. That was established by growing the same runs nearly three times as long and finding the table unchanged, and a run length has nothing to do with how many starting angles are sampled.

Similarly the destinations, which are presences rather than shares: a destination the table reaches is reached whatever else is sampled, and adding angles can only add to that list.

What a re-sampling can do to a conclusion #

There are three outcomes and this is the third. A re-sampling can break a conclusion, which is the case everyone plans for. It can confirm it and tighten its numbers, which is the case everyone hopes for.

Or it can confirm the conclusion while destroying the numbers that were reported alongside it — which is what happened here, and which is the case nobody has a name for. The file’s sentence survives and four of its printed values do not.

That is not a failure of the earlier work. The earlier work printed the values and then declined to read them, which is exactly right; what it could not do was say how much of the spread was error, because with nine angles a spread and an error are the same size.

The habit that made it survivable #

The file computed its own error bar before reading its own numbers. That single step is why this reading corrects an arithmetic rather than a claim.

It is worth naming because the alternative is common and cheap: report four values, notice they differ, and write a sentence about why the steeper exponents wall coarser. Nothing would have caught it, because every one of the four values is a correct value of what it measures.

What a third sampling would say #

Forty angles would halve the error again, to about eight per cent on a cell, and would put the four walls somewhere inside a factor of about 1.1 if the current reading is right.

That is not worth twenty-five minutes for its own sake. It would be worth it as a by-product of something else — the basins want a finer spacing, and the destination list has grown at every refinement so far — and the wall would come along with it.

What the four exponents are #

Two to five, and the range is chosen rather than swept. Three is the falloff every other measurement on this site uses, so it sits in the middle of the range rather than at an end of it — a sweep that started at the working value could only ever say the wall moves one way.

Below two the neighbourhood stops being local at these run lengths, so the sum an organ is placed at the minimum of reaches most of the stem. Above five the sum is decided by one organ, which is a different rule rather than a steeper one.

So the four are the range over which steeper falloff means something, and the finding is that across the whole of it the wall does not move.

And what the wall is not #

It is not the finest rise at which anything settles. That is a different number — some angle reaches a lattice at 0.003 at three of the four exponents — and it is a maximum over twenty runs rather than a level crossing.

Quoting the two together is worth doing, because they answer different questions. The wall says where most starting angles stop working; the finest says where the last one does.

What a reader should carry #

That a spread the file’s own error bar said not to read has turned out not to be readable: at twenty starting angles the four exponents’ walls span a fifth of what they spanned at nine, and their ordering reverses.

And that the wall should be quoted as between 0.013 and 0.008 in the rise, which is what the table’s grid resolves, rather than to five digits — at every exponent and at both samplings.

What the picture at the top shows #

Four curves per sampling: the share of starting angles that reach a lattice, against the rise, at each of the four falloff exponents. The small dashed marks are nine angles and the large solid ones are twenty.

The horizontal rule is the half share the wall is read at. At nine angles the four curves cross it at rises spread over a factor of two; at twenty they cross within one step of the table’s own grid, and in a different order.

The one line #

At nine starting angles the four falloff exponents’ walls sit at 0.01397, 0.00711, 0.00711 and 0.00867 — a factor of 1.97 apart, ordered 2, 5, 3, 4. At twenty they sit at 0.01180, 0.01106, 0.01199 and 0.01300 — a factor of 1.18, ordered 5, 4, 2, 3.

An ordering that reverses when the sample doubles is not an ordering, so the conclusion that the exponents do not separate is confirmed and the numbers that had looked like evidence against it were the coarser sampling’s own error.