What the rule does to a drift

There is a sentence in this collection that reads, of a placement rule: its errors are corrections rather than inheritances, and its divergence sequence is anticorrelated at lag one rather than drifting. It was written to say that a rule produces neither of the two signatures a transported disturbance produces, which would make a stem carrying one of them evidence against the rule.

The first half is right and measured. The second half is a claim about a rule under a disturbance, and which disturbance was not specified. Under independent noise the rule produces no drift. Under a drifting disturbance it produces more drift than the same disturbance produces with no rule anywhere.

The measurement #

The same coloured disturbances that this collection drives into kinematic lattices are driven through the rule instead, as a jostle: each organ is placed exactly where the rule says and then moves, so the disturbance is invisible in what is measured and present in every decision the rule makes afterwards. Every stream is normalised by its own standard deviation, so a quarter of a degree means a quarter of a degree of displacement per organ in every row.

Independent noise gives 0.91 at a block of sixty-four, which is the flat line the prediction wanted. A memory with coefficient 0.97 gives 46.47. On a kinematic lattice — the same disturbance, added to positions with no rule involved — the same block gives 30.03.

So the rule does not remove a drift. It multiplies the drift’s visibility by about half again.

The scatter goes the other way #

The same runs, read as a scatter, look like the rule doing exactly what it is supposed to do. Under independent noise the divergence sequence scatters by 0.553°. Under the drifting disturbance it scatters by 0.246° — less than half — at the same per-organ displacement.

That is not a contradiction; it is the two halves of one mechanism.

The rule places each organ at the least of a repulsion profile summed over its neighbours. A displacement that all the neighbours share moves the whole neighbourhood without changing its shape, and the least of a translated profile is the same place relative to the neighbourhood. What a placement rule responds to is the part of a disturbance that is relative between neighbours — which is its high-frequency part.

A drift is nearly the same displacement for every organ in a window of a few dozen. It is almost entirely non-relative, so the rule absorbs almost none of it and corrects almost none of it. Independent noise is entirely relative, so the rule corrects a large share of it.

Why that makes the wander bigger and not merely undiminished #

The statistic is a ratio. Its numerator is the variance of the block means, which the drift dominates and the rule barely touches; its denominator is the variance of the sequence, which the rule cuts by more than half.

A ratio whose numerator is untouched and whose denominator is halved goes up. That is the whole of it: 30.03 becomes 46.47 because the sequence it is measured against got quieter, not because the drift got louder.

This is worth stating carefully because it inverts the intuition that a self-correcting system is harder to read. A self-correcting placement rule is an excellent high-pass filter, and a high-pass filter improves the visibility of anything slow that survives it. The rule makes a drift easier to detect, not harder.

The filter, written out #

It is worth being precise about “high-pass filter”, because the phrase is doing explanatory work and could be doing it loosely.

Take the rule’s response to a disturbance that is a pure oscillation in the organ index — every organ displaced by an amount that goes round a cycle every L organs. If L is short compared with the neighbourhood the rule sums over, the neighbours’ displacements at any moment are spread over a whole cycle, they push in different directions, and the shape of the neighbourhood is genuinely distorted: the rule sees it and responds. If L is long compared with the neighbourhood, every organ in the window is displaced by nearly the same amount, the neighbourhood translates, and the rule sees nothing.

The neighbourhood at these rises is about ninety organs deep. So the crossover is somewhere in the tens of organs, which is exactly where the wander statistic’s block sizes are, and exactly where a memory with a coefficient of 0.97 puts most of its power: its correlation falls to a half at about twenty-three organs.

That gives the two effects a common cause and a common scale. The scatter falls because the fast part is removed; the block-mean variance survives because the slow part is not. Both are consequences of the neighbourhood having a depth, and a rule with a much shorter reach would show neither.

The inherited disturbance, through the rule #

The other half of the table is the shape that has no drift to begin with, and it behaves as the arithmetic says it should.

An inherited disturbance at a coupling of 0.7 gives 1.90 at a block of sixty-four through the rule — near one, as on a kinematic lattice — and at the two contact offsets it gives 0.06 and 0.12. The hole is deeper than the kinematic version’s 0.22 and 0.21, for the same reason the memory’s plateau is higher: the denominator has shrunk.

So the two signatures separate more sharply through the rule than without it. A sequence from a rule-grown stem shows a drift as a larger number and a comb as a deeper hole, both because the rule has removed the part of the disturbance that is neither.

The rule’s own dip at eight #

One detail in the table is not about any disturbance. Under independent noise the rule’s stems give 0.20 at a block of eight, against 0.91 at sixty-four and 0.75 at two.

Eight is the smaller parastichy number of the arrangement these stems carry. The rule’s own placements are correlated at the contact offsets, because that is where its neighbours are; an organ placed against a neighbourhood inherits a little of the arrangement of that neighbourhood, and the arrangement repeats at eight and thirteen.

That is a small, real signal, and it is the reason the hole cannot be read as proof of a transported disturbance on its own. A stem with no inheritance at all still digs a shallow hole at its own contact offsets. The inherited disturbance digs a much deeper one — 0.06 against 0.20 — but the null is not one and a test built on the assumption that it is would be a test with an inflated rate of false positives.

What this does to the observable #

The proposal this collection had made was that a rule produces no drift, so a drifting sequence is evidence against a rule. That is now wrong in a specific way, and the corrected version is more useful.

A drift in a divergence sequence says the disturbance drifts. It says nothing about whether a placement rule is operating, because a placement rule passes a drift through — and it does not even weaken the reading, since a rule-grown stem shows the drift more clearly.

What separates a rule from a transport is still the thing this collection has been using: the anticorrelation at lag one, which is a restoring force and cannot be produced by an error inherited from neighbours. The wander is orthogonal to that question and belongs to a different one.

A prediction with a shape #

The measurement gives an experiment something to look for that is not a single number.

If a shoot is placed by a local rule and its errors are dominated by a drifting environment, its divergence sequence should be quiet at short scales and loud at long ones — a small scatter organ to organ, and block means over a hundred organs varying much more than that scatter would allow. A shoot whose errors are independent should be loud at short scales and quiet at long ones, in the same sequence.

That is a shape rather than a threshold, and shapes are cheaper to trust. It also comes with a warning attached: a shoot with a small organ-to-organ scatter is usually read as a well-behaved shoot, and this says the opposite can be true — a small scatter may be a rule doing a great deal of correcting against a large disturbance, and the size of that disturbance is only visible at long scales.

The plateau names the memory, and the rule makes it lie #

The reading proposed below turns on a plateau naming a memory length, so it is worth checking that it does and by how much the rule spoils it.

A disturbance whose errors decay with coefficient ρ has block means whose variance levels at 1 ÷ (1 − ρ) times a white stream’s. At ρ = 0.97 that is 33.3, and the measurement on a kinematic lattice — the same disturbance, no rule anywhere — is 30.03. The formula holds to a tenth, which is what makes the inversion worth proposing at all: read a plateau, subtract its reciprocal from one, and the answer is the memory’s coefficient.

Through the rule the same disturbance gives 46.47, and inverting that gives ρ = 0.978 rather than 0.970. In the units a reader cares about — the separation at which the disturbance’s correlation has fallen by half — that is 31 organs against 23, an overestimate of a third.

So the inversion works and it is biased, and the bias has a sign and a cause. Any sequence read off a shoot whose organs were placed by a rule will report a longer memory than the shoot’s environment actually has, because the rule has removed part of the denominator and not part of the numerator. A deeper neighbourhood removes more of it, which is the same mechanism read across the rule’s own reach. A third is not fatal for a quantity nobody has measured at all, and it is far too large to ignore in a comparison between two species or two seasons.

The correction is available in principle and needs the one number this essay supplies: the ratio between the plateau through a rule and the plateau without one, which is 1.55 at this coefficient and this displacement. Whether that ratio is itself constant across coefficients has not been measured, and it is the thing to measure before anybody quotes a memory length off a plant.

Reading a real sequence with this in mind #

Suppose a sequence of divergences from a real shoot, several hundred organs long, measured to a fraction of a degree. What the two statistics would say together:

A small organ-to-organ scatter and a flat variance-time curve is a quiet shoot with a quiet environment, and it says nothing about mechanism at all.

A small scatter and a climbing curve is the interesting case. It says the shoot’s placements are being corrected against something large and slow, and the size of that something can be read off the plateau: for a disturbance with a memory, the curve levels at one over one minus its own correlation coefficient, so the plateau names the memory’s length.

A large scatter with a flat curve and holes at the contact offsets is the inherited case this collection has spent three essays on, and the holes are what would say so.

A large scatter with neither is independent error, which is the null.

And there is a fifth case the four-way split hides, which is the one a real shoot is most likely to be in. A plant has a drifting environment and organ-to-organ error, so its sequence should be large-scattered and climbing — the two statistics are not alternatives and nothing forces a shoot into one corner. What the pair then reports is the ratio of the two disturbances rather than which one is present, and the useful reading is quantitative instead of categorical: the plateau gives the slow part’s size, the scatter gives the fast part’s, and neither number is a diagnosis on its own.

That is worth saying because the four cases above read as a key, and a key invites a reader to pick the row their shoot matches. The honest instrument here is two numbers with a ratio between them, and the four corners are what it looks like when one of the two disturbances is negligible.

Four readings from two statistics, and the point of setting them out is that three of the four were unavailable while the wander was believed to be the comb’s other half. Getting the observable wrong cost a hypothesis; getting it right supplies three.

What this does not say #

It does not say the rule amplifies a drift. The drift’s own size in the sequence is unchanged; what changes is everything else. The ratio goes up because the denominator goes down, and the essay is careful to give both numbers — which is the reading a later sweep had to undo the normalisation to make.

It does not say the rule fails to correct. It corrects a great deal — more than half the scatter, at this displacement — and what it corrects is precisely the part that would destroy the lattice. A drift does not destroy a lattice, which is the same fact from the other side and is measured elsewhere here.

It does not say a drifting disturbance is realistic. An autoregressive disturbance with a coefficient of 0.97 has a memory about thirty organs long, and nothing here says a meristem has one. It is a shape chosen to be maximally different from the inherited one, and its value is as a contrast.

It does not say the two disturbances are equally plausible. Nothing here argues that a meristem is more likely to face a drift than an inheritance. The comparison is between two shapes with the same per-organ size, chosen because they are opposite, and its value is in what a measurement could distinguish rather than in what a plant is likely to have.

And it does not say the hole at eight is negligible. Under independent noise the rule’s own stems give 0.20 there. Any test using the hole as evidence has to be calibrated against that and not against one.

What was actually claimed, and what replaces it #

The sentence at the top of this essay has to be replaced rather than qualified, so here is both halves side by side.

Claimed: a placement rule produces neither a comb nor a drift, so a sequence carrying either is evidence against a rule.

True: a placement rule produces no comb and no drift of its own, and a sequence carrying either is evidence about the disturbance rather than about the placement. A rule passes a drift through with its visibility increased; a rule does not manufacture the correlations at the contact offsets that make a comb, and that half of the claim is unaffected and is measured elsewhere here.

The difference between the two versions is the phrase “of its own”, and it is the difference between a statement about a mechanism and a statement about a mechanism under a stated input. Every claim in this collection about what a rule produces has that ambiguity available to it, and this one is the second to fall into it.

The check that would refuse it #

Three assertions, taken on the same rule at the same displacement.

A stem grown under independent noise has to carry no wander — under two at the widest block. That is the half of the original prediction that survives, and it is the calibration for the rest.

A stem grown under the drifting disturbance has to carry at least eight times that. The factor is large because the effect is: forty-six against nine tenths.

And the drifting stem’s per-organ scatter has to be the smaller of the two. That is the assertion that makes the essay’s explanation testable rather than decorative: if the rule were simply passing everything through, the quieter sequence would be the one with less wander, and it is the one with more.