A disturbance with a memory

The previous phase ended with the strongest claim this site has made about mechanism, and it was a claim built on a control. Take a stem the placement rule grew, read the correlation between one divergence angle and the one eight organs later, and there is a comb in it — a run of lags at multiples of eight standing well above what a sample of that length wanders by on its own. Then build the same lattice kinematically: put node i at exactly i times the divergence, at the same rise, displace every azimuth by an independent draw, and read it the same way. Nothing. Largest comb mean 0.02 against a sampling band of 0.07, refused on every seed.

From that the phase concluded that the comb is not a property of the arrangement but of the rule that made it — the first observable on this site that separates a plant computing a pattern from a plant that merely has one.

The phase plan then wrote down the assumption that conclusion depends on, and said it was the first thing the next phase should do. Every error in the control is drawn independently of every other one, and no plant’s errors are. An organ that comes out a little late affects the one after it; a shoot that grows through a cold night makes a run of organs that are all a little wrong together. If a disturbance with a structure of its own could put correlation into the angle sequence, the comb would be a fact about the weather rather than about the meristem, and the conclusion would be worthless.

This essay is the first half of that test, and the news is good. The obvious form of the worry — a disturbance that remembers — manufactures nothing at all.

What “correlated” has to mean #

The phrase covers two different things and the difference is the whole essay.

A disturbance has a memory if today’s error resembles yesterday’s: the error at organ i is some fraction ρ of the error at organ i − 1 plus a fresh draw. That is the standard model of a quantity that drifts, and its correlation falls smoothly with separation — strong between neighbours, weaker two apart, gone by twenty. It has a length.

A disturbance is periodic if the error returns to the same value at a fixed separation: every eighth organ gets the same displacement, whatever happens in between. Its correlation does not fall with separation at all; it comes back. It has a period.

Everyday usage calls both of them correlated noise, and the readout this site built is built to find one of them and is blind to the other. A comb is a set of teeth at multiples of a spacing. A memory has no teeth.

The algebra, which is one line #

The reason a memory cannot make a comb is available before any stem is grown, and writing it down first is what makes the measurement a test rather than an exploration.

What a botanist writes down is not the error at each organ. It is the divergence, the angle from one organ to the next, and that is the difference of consecutive azimuth errors. If the errors have covariance γ(k) at separation k, then the differences have 2γ(k) − γ(k−1) − γ(k+1) at the same separation — a second difference, with a minus sign. For an AR(1) error, γ(k) = ρ^k, and that expression collapses to −ρ^(k−1)(1 − ρ)². Divide by the variance of the differences, which is 2(1 − ρ), and the correlation at lag k is −ρ^(k−1)(1 − ρ)/2.

Negative, monotone, decaying. At ρ = 0.9 it is −0.05 at lag one and −0.02 at lag four and −0.005 at lag ten, and at every lag in between it is on the same smooth curve. There is no lag at which it stands above its neighbours, which is the only thing a comb search can see. The spacing search looks for a residue class whose members average higher than the classes around it, and a curve that is the same shape everywhere gives every class the same average.

Notice also which way the coefficient pushes. As ρ rises the factor (1 − ρ) falls, so a longer memory produces a weaker signature in the differences. In the limit of a perfect memory the error is a constant offset, every divergence is the same, and the disturbance has vanished from the measurement entirely. The threat gets smaller as the thing feared gets bigger.

What the measurement adds #

If the algebra is right, why measure at all? Because the algebra is about a model of the disturbance and the figure is about a stem. Three things could have gone wrong between them, and each is worth naming.

The readout is not the correlation. It is a search over spacings, over residue classes, over thresholds that themselves depend on how many members a comb has. A search with several free choices in it can find structure in a smooth curve, and a search that finds a spacing in an AR(1) sequence would be a defect in the instrument rather than a fact about plants. It does not: the spacing it returns wanders from seed to seed and never clears its own band.

The scatter is not held fixed by the algebra. Every shape here is normalised to unit variance before the amplitude is applied, so a lattice with a memory has the same recorded scatter as the control. Without that the comparison would be between disturbances of different sizes and any difference in the readout would follow the size.

And the arrangement could have stopped being a lattice. A correlated error accumulates differently from an independent one, and an arrangement whose points have wandered off the lattice would refuse for the wrong reason. It has not: at ρ = 0.97 the recorded scatter is the half degree it was asked for, and the position counter reads the same parastichy pair it reads from the control.

The memory this essay did not put anywhere #

There is a third place a memory could live and it is not the place tested above, which is worth saying plainly rather than leaving for a reader to notice.

The disturbance here is in the arrangement. Node i goes where the divergence says and is then displaced, and the displacements are correlated with one another. That is the right experiment for the question asked, because the previous phase’s control was an arrangement and a control is contradicted on its own ground or not at all.

A real plant’s disturbance is more likely to be in the rule: the meristem’s next organ is placed against a field that is a little wrong, or against neighbours that have moved since they were put there, and the error at one organ therefore feeds into where the rule puts the next. This site has three names for those — field, placement and jostle — and each of them was given an amplitude two phases ago and none has ever been given a correlation length. A jostle whose displacements were correlated from organ to organ is a different object from the lattice measured here, because the rule responds to it: the placement rule is self-correcting, so a run of errors pushing the same way meets a restoring push that an independent run does not.

Whether that changes the comb is open. The algebra above does not settle it, because the algebra assumes the errors go into the azimuths and come out in the differences with nothing in between, and a rule is exactly a something in between. The prediction worth writing down before anyone measures it is that a self-correcting rule shortens whatever correlation length it is given — the correction acts on the accumulated error, so a memory in the input becomes a shorter memory in the output — and therefore that this essay’s negative result holds a fortiori. A prediction that a later phase can refute is worth more than a hedge, and this is one.

What it would take is one option on grow: a stream of disturbances with a stated structure rather than a stream of independent draws. It is half an hour of work and it is not done here, because the phase’s budget went to the shape that turned out to forge everything, and doing it badly at the end of a phase is how a number nobody checks gets into a plan file.

Where this leaves the previous phase’s claim #

Exactly where it was, and no further. The control has survived the first correlated disturbance put to it, which means the comb is not an artefact of assuming independence in the way that was worried about. It does not mean the comb is evidence of a placement rule, because a memory is only one of the shapes a disturbance can have.

It is worth being precise about what has and has not been shown, because this site has a standing habit of stating what a result does not cover and this is a case where the uncovered part is larger than the covered one.

Shown: a disturbance whose correlation falls with separation cannot produce a comb in the differences, for any correlation length, and the reason is arithmetic rather than empirical.

Not shown: that no disturbance can. The class of disturbances is much larger than the class of AR(1) processes, and the next essay puts a periodic one into the same lattice and gets a comb out of it. The one after that puts in the disturbance a plant would actually have — errors passed between the organs that touch each other — and gets the comb and the pair.

Two things this thread has been careful about since it began #

The distinction between a length and a period is not new here. It is the same distinction the noise thread had to make when it asked whether noise is a slow rate, and the answer had the same shape: two quantities that are described with one word in the literature and behave differently when each is given a knob of its own.

The other half of the care is about what the sequence statistic already knew. The lag-one statistic reads the correlation between one divergence and the next — lag one, deliberately excluded from the comb search because a single misplaced organ enters two consecutive divergences with opposite signs and puts a large negative value there whatever else is true.

That statistic is not blind to a memory. It is the one place where the AR(1) signature shows up, and it shows up exactly where the algebra puts it: the lag-one correlation of a lattice with a memory runs from −0.55 at ρ = 0.3 to −0.10 at ρ = 0.97, climbing towards zero as the memory lengthens, while the rule’s own stems sit near −0.6 regardless.

So a botanist who could measure the correlation length of a plant’s disturbances would learn something — but not from the comb, which does not respond to it.

The test as it will be handed to a plant #

The specification this thread has been assembling for four phases gains nothing from this essay, which is the honest way to report a negative result: it removes a term rather than adding one.

Before this measurement, a survey that found a comb in a real stem’s angles would have had to argue that the plant’s disturbances were not correlated — an argument nobody could make, because nobody has measured a plant’s disturbance autocorrelation and the measurement is harder than the one the comb was supposed to support. After it, that argument is not needed. A memory of any length is consistent with a clean comb and with no comb at all, so finding one says nothing either way about correlation length, and the survey does not have to control for it.

What the survey does have to control for is the other shape, and the next two essays are about how expensive that turns out to be.

It is worth putting the two results side by side in the form a specification uses, because they pull in opposite directions and the pair of them is the honest summary of this thread’s standing.

A correlation length costs nothing. No length, from a tenth of an organ to thirty of them, produces a comb, and none destroys one either — the rule’s own stems keep their comb whatever the disturbance’s autocorrelation, because the comb comes from the neighbour structure and not from the disturbance’s shape. So a survey need not measure it, need not report it, and need not exclude plants that have it.

A periodicity costs everything. A disturbance that returns to the same value every m organs puts a comb at m on an arrangement with no rule in it, and a survey that found a comb without excluding that possibility would have found nothing at all. Excluding it is the subject of the two essays after this one, and the short version is that it cannot be excluded from a single stem.