The corner that does not move

A rule with a deep neighbourhood was supposed to correct a slow drift and let a fast one through; a rule with a shallow one, the reverse. The crossover between them was supposed to sit where the drift’s correlation length matched the depth. Swept across a factor of sixty in the depth, the measurement found no crossover at any depth and the deep rule passing more rather than less.

Two things have changed since. The neighbourhood has been counted four ways and they all give the same ordering, so the result is not an artefact of the definition. And the organs at the top of the rule’s profile turn out to be the same five at every exponent, so a sweep of the exponent was never sweeping the thing a filter argument is about.

This essay changes the quantity being read, and a corner appears.

The ratio has the scatter in its denominator #

The statistic the earlier sweep reported is a ratio: how much a drift moves the mean of a block of divergences, against how much independent draws would. It is the right statistic for asking how much of a drift survives the rule, and it has a denominator.

That denominator is the scatter of the divergences themselves, and the scatter depends on the exponent for reasons that have nothing to do with drift: a shallow rule makes a noisier lattice. Across the sweep the scatter runs from 0.205° at the deepest to 0.321° at the shallowest — a factor of 1.57 — while the drift getting through, measured in degrees, moves by a factor of 1.20.

So the quantity to compare is degrees of drift getting through — the ratio undone, the scatter multiplied back in. That is the quantity a reader of a real stem would measure, and it is the one this essay uses.

What “drift getting through” is #

The quantity deserves a definition, because two are in play and the whole result turns on which is read.

Take the divergences of a stem grown under a disturbance. Split them into blocks of sixty-four and take the mean of each block. If the disturbance had no memory, those block means would scatter by the divergence scatter divided by eight; a drift moves them further. The wander is how much further, as a ratio to what independent draws would give.

Degrees through is the wander’s square root multiplied by the scatter of the divergences themselves — which undoes the normalisation and gives the actual size, in degrees, of the slow movement in the block means.

Both are legitimate. The wander answers “how much of the drift survived, relative to the noise floor”; degrees through answers “how big is the slow movement anybody would measure”. The first has the noise floor in its denominator and the noise floor is a strong function of the exponent, which is what makes it the wrong one for a comparison across exponents.

Paired seeds, because the spread is larger than the effect #

Degrees through is noisy. At one setting of the rule and one colour of disturbance, six random seeds give answers spanning a factor of 1.4 to 2.6. The difference between two rules at one colour is smaller than that.

Comparing two means with that much spread under them compares two numbers whose ranges overlap. So the comparison is paired: the same disturbance — the same seed, the same sequence of displacements — is given to both rules, and what is recorded is which of the two let more of it through.

Then the comparisons are pooled. Every pair of exponents whose neighbourhoods differ by at least a factor of three — eight pairs out of the ten available — is run at six seeds, giving forty-eight paired comparisons at each colour.

The corner #

Against white noise the shallower rule lets more drift through, at thirty-seven comparisons out of forty-eight. At a correlation length of a hundred organs the deeper rule does, at forty-five out of forty-eight. The share rises at every step in between, and it crosses a half between correlation lengths of 1.4 and 2.8 organs.

Six columns of forty-eight comparisons is enough arithmetic to say which of them carry the result, and doing it locates the crossing better than the table does.

Written as counts rather than shares the row is 11, 14, 27, 31, 39, 45 out of forty-eight. A coin gives twenty-four with a standard deviation of 3.5, so the columns sit at −3.8, −2.9, +0.9, +2.0, +4.3 and +6.1 deviations from it. The two ends are decisive in opposite directions; the column at a correlation length of 2.8 organs is one standard deviation from a coin and says nothing.

That is exactly what a crossing point looks like from the inside, and it pins the crossing more tightly than “between 1.4 and 2.8”. A column indistinguishable from a half is a column where neither rule is ahead — so the crossing is not merely somewhere in that interval, it is at or very near 2.8 organs, with 1.4 clearly on the shallow side and 4.5 marginally on the deep one.

The usual caution applies and cuts the right way. The forty-eight are eight pairs of rules crossed with six seeds, so they are not forty-eight independent trials: the same six disturbances are used in every pair, and pairs sharing an exponent share half their runs. The effective count is smaller than forty-eight and the true deviations are therefore smaller than the ones above. That weakens the marginal columns — +2.0 at 4.5 organs would not survive much shrinkage — and it leaves −3.8 and +6.1 intact, which are the two the sign change rests on.

That is a corner. The earlier sweep could not see it because the ratio it was reading has the scatter in its denominator, and the scatter moves the other way with the exponent — enough to hold the ratio’s ordering fixed across the whole range of colours.

Why eight pairs and not ten #

Ten pairs of exponents are available and two of them are excluded, by a rule stated before the comparison rather than after it.

The exclusion is on the ratio between the two neighbourhoods: a pair is compared only if one is at least three times the other. Exponents 1.5 and 2 describe neighbourhoods of 182 and 120 organs, a factor of one and a half, and exponents 4 and 6 describe 8 and 3, a factor of two and two thirds. Those two pairs are comparisons between rules that are very nearly the same rule.

What they do is what nearly-identical rules should do. The 1.5-against-2 comparison gives four, two, four, three, three and four seeds out of six across the colours — a coin toss at every one, with no trend. Including it would have added noise to the pooled share and, more importantly, would have added a pair with no crossing to a claim about where crossings sit.

The exclusion also cuts the other way and is worth checking for that. If the crossing were an artefact of which pairs were chosen, the eight would not agree — and they do: every one has the shallower rule ahead at white noise, every one has the deeper rule ahead at the longest colour, and all eight cross within the first ten organs.

And it does not move with the depth #

This is the claim that separates a corner from a filter. A filter’s corner sits at its own scale, so two rules whose neighbourhoods differ by a factor of sixty should change hands somewhere quite different from two whose neighbourhoods differ by four.

They do not.

Every one of the eight pairs has the shallower rule ahead against white noise and the deeper rule ahead at the longest correlation. Every one of them changes hands inside the first ten organs of correlation. The pair whose neighbourhoods are 182 organs and 3 crosses between 1.4 and 2.8; the pair whose neighbourhoods are 120 and 30 crosses between 2.8 and 4.5.

The crossing is also shorter than every neighbourhood in the comparison. It sits at two or three organs; the neighbourhoods it is comparing are 182, 120, 30, 8 and 3 organs wide by the nine-tenths count, and 68, 45, 21, 14 and 11 by the weighted mean. Only one of the ten numbers is anywhere near the crossing.

The disturbance, and what its correlation length means #

The knob being swept is the memory of the jostle, and it is worth saying exactly what is varied and what is held.

Each organ is displaced by a small amount before the rule places it. The displacements are drawn from a first-order autoregressive process: each one is a fixed fraction of the last plus a fresh draw. That fraction is the knob. Its correlation length — the number of organs over which the disturbance forgets itself — runs from zero at white noise to about a hundred at the largest setting used here.

The amplitude is held fixed at a quarter of a degree per organ, and it is held fixed after the disturbance is divided by its own measured standard deviation. Without that division the correlation coefficient would change the size of the disturbance as well as its colour, and a sweep of the colour would be partly a sweep of the amplitude.

So what varies across the columns of the table is one thing — how far into the past a displacement remembers — with the size of the displacement, the rise, the grid, the run length and the block size all fixed.

What the corner is, if it is not the depth #

The measurement stops here and the reading is short, so it is worth marking where one ends and the other begins.

Measured: the two rules change hands at a correlation length of two or three organs, at every separation of depth tried, and the crossing does not move with the depth.

Read: the scale that does not move with the exponent is the contact scale. The five organs carrying the largest terms of the profile are the same five at every exponent, so whatever the rule is holding on to is held at the same distance throughout. A disturbance correlated over fewer organs than that is something the rule can average away regardless of how far its tail reaches; one correlated over more is something every version of the rule carries.

That reading is consistent with everything measured and it is not established by it. Two organs is also roughly the shortest correlation that is distinguishable from none at these run lengths, so a crossing there is consistent with “any memory at all changes the sign” as well as with “the contact scale sets it”. Separating those two would need a sweep of the rise, which moves the contact scale while leaving everything else alone.

That sweep has a stated prediction attached, which is what makes it worth running rather than merely worth wanting. If the corner sits at the contact scale, then moving the rise from 0.032 to 0.005 — which takes the counted pair from 3/5 to 8/13, and the shortest contact lags from three and five organs to eight and thirteen — should move the crossing by about the same factor, from something near one organ to something near three. If the corner sits at “any memory at all”, it should not move: two or three organs is where a correlation first becomes detectable at these run lengths, and that is a property of the run length rather than of the lattice.

The two readings differ by a factor of three on a quantity the table above already resolves to a single column, so the sweep is a real test rather than a hopeful one. It costs three rises of the same paired design, which is the same work again and no new machinery.

What a sign test can and cannot say #

The statistic is deliberately coarse — it throws away how much more one rule passed and keeps only which — so it is worth being clear about the price.

What it buys is robustness. A ratio of two noisy quantities has a distribution with tails; a count of which of two paired numbers is larger does not care how long the tails are. With six seeds a pair gives six comparisons and the sampling noise is a binomial with a known shape, so a five-one split at one colour is informative and a three-three split is not, without any assumption about how the underlying quantity is distributed.

What it costs is size. The table says which rule passes more at each colour and says nothing about by how much. From the means, the answer is “not much”: at the longest correlation the deep rule passes 1.83° and the shallow one 1.44°, a factor of 1.27, and at white noise 0.39° against 0.58°, a factor of 1.51. Those are real differences and they are small beside the seed-to-seed spread, which is exactly why the pairing was needed.

Those four numbers are worth reading in both directions, because the second reading is the more important one and the table hides it.

Across the rules, the difference is a modulation: 1.83° against 1.44° at the longest correlation, and 0.58° against 0.39° at white noise. Ratios of 1.27 and 1.51 — and the larger of the two is at the white end, which is not what the picture of a filter would suggest. The shallow rule’s advantage over the deep one against a memoryless disturbance is bigger than the deep rule’s advantage over it against a hundred-organ memory. The sign changes at the corner; the size does not peak on the far side of it.

Across the colours, the difference is much larger than either. Whichever rule is used, a hundred-organ memory puts three to four times as much slow movement into the divergences as no memory at all: 1.44° or 1.83° against 0.39° or 0.58°. So the dominant term in the whole table is the disturbance’s colour, and the rule’s depth is a modulation of a quarter to a half on top of it.

That ordering is worth carrying because it settles what a reader of a real stem should conclude from a large drift. It is evidence about the plant’s environment first and about its placement rule a distant second — and a comparison of two species’ drift that did not hold the disturbance fixed would be measuring the weather rather than the meristem, by a factor of three.

So the corner is established as a change of sign and not as a magnitude. Anybody wanting the size of the effect would need many more seeds; anybody wanting to know whether it changes sign, and where, has what they need here.

What has been repaired #

The earlier result said there was no corner. There is one; it was in a quantity with a moving denominator and was invisible in the ratio.

The earlier result also said the deep rule passes more drift than the shallow one. That stands, and now has a boundary on it: it is true past a correlation length of about three organs and false below it.

And the prediction the whole thread started from — that the crossover tracks the neighbourhood — is refuted for a second time and more cleanly than before. It is not that no crossing exists. It is that the crossing exists, is sharp, and sits in the same place for rules whose neighbourhoods differ by a factor of sixty-one. A prediction refuted by an absence can always be rescued by looking harder; one refuted by a presence in the wrong place cannot.