The disturbance that travels

The previous essay ended with a rule of thumb and this one breaks it.

The rule of thumb: a lattice survives three times as much displacement when the organs share it, because the placement rule works on differences and a shared displacement has none. It moves the whole neighbourhood without changing its shape, the minimum moves with it, and the divergence a botanist records is nearly what it would have been.

The obvious extension is that the most survivable disturbance of all would be one correlated at the contact offsets — the organs m and n places back, which are the geometric neighbours the rule is really placing against. Everything else in the neighbourhood matters less; correlate the disturbance where it counts and the rule should barely notice.

That is the arrangement built here to forge a comb, and driven through the rule it is the most destructive disturbance in the table.

The measurement #

At a quarter of a degree of displacement per organ, every stem of every colour keeps its lattice. What differs is the scatter it records: 0.26° for a long memory, 0.57° for independent errors, and 0.97° for a contact-transported disturbance of the same size.

At half a degree, independent errors leave all five stems standing at 1.03° of recorded scatter. A contact transport at a coupling of 0.5 leaves none. At a coupling of 0.7 it leaves one of five, at a scatter of 1.94°, which is over the boundary the pattern fails at.

So the transported disturbance is not merely no better than white noise. It is worse by a factor of two in amplitude, and by a factor of nearly two in the scatter it produces at equal amplitude.

Why the rule of thumb was too simple #

The correlations tell the story, and they have to be read at the right lags.

A memory with a coefficient of 0.97 correlates at 0.96 at lag one, 0.83 at lag five, 0.74 at lag eight and 0.63 at lag thirteen. It correlates everywhere nearby. Every organ in the rule’s neighbourhood is displaced by nearly the same amount, and the neighbourhood’s shape survives.

A contact transport at a coupling of 0.5 correlates at −0.04 at lag one, 0.17 at lag five, 0.42 at lag eight and 0.41 at lag thirteen. It correlates at two lags and nowhere else. Organs one through seven places back are as independent as white noise, and organs eight and thirteen move together.

So the two disturbances are not both “correlated”. One is a common mode across the whole neighbourhood; the other is a structure inside it.

And a structure inside the neighbourhood is exactly what the rule responds to. It sums the repulsion from some eighty-five organs and takes the minimum, and what decides where that minimum is, is the arrangement of those organs rather than their common position. A displacement that moves them all shifts the answer and changes nothing else. A displacement that moves two of them together and leaves the rest alone changes the arrangement — and it changes it coherently, which is worse than changing it randomly.

It is worth putting the same point in the language of the essays on how far the rule reaches, because they measured the quantity this argument needs.

They asked what the repulsion exponent controls, and the answer was contrast: the ratio between the variation the nearest shell of neighbours contributes to the profile and the variation the next shell contributes. It runs from 1.1 at an exponent of a half to 296 at an exponent of three, and a lattice forms reliably once it passes about twenty. A lattice exists exactly when the rule is placing organs with respect to their immediate neighbours rather than with respect to a broad shallow landscape assembled from dozens of them.

At the site’s inverse cube the contrast is high, which is what makes the pattern a lattice — and it is also what makes it vulnerable in the specific way this essay reports. A rule that listens mostly to two organs is a rule that can be misled by displacing two organs together.

Why it is worse than random, not just as bad #

The last step is the one worth being careful about, because it is where the essay could stop at “correlated in the wrong place” and miss the size of the effect.

Two of the eighty-five organs dominate the minimum. The 13-hop is 0.069 circumferences and the 8-hop 0.075, against 0.090 for the next one and 0.383 for the organ one place back — which is the longest hop of all, and the reason a memory’s correlation at lag one buys so little. With an inverse-cube repulsion, the two contact neighbours contribute most of the variation the argmin is working with.

Displace those two independently and their contributions partly cancel: one pushes the minimum one way and the other pushes it back, and the expected displacement of the answer goes as the square root of two errors.

Displace them together — which is precisely what a contact transport does — and they push the same way. The answer moves by the sum rather than by the difference, and the two dominant terms have become one term twice the size.

So the quantity that decides a disturbance’s effect is not how correlated it is. It is how the correlation is distributed over the neighbourhood, weighted by how much each member contributes to the minimum:

a disturbance is harmless in proportion to how much of it is common to the whole neighbourhood, and harmful in proportion to how much of it is shared by the organs that dominate the minimum and not by the rest.

A memory is nearly all common mode. A contact transport is nearly all the other thing. Independent noise is in between, which is why it sits in between.

The arithmetic is worth carrying a step further, because it predicts the factor rather than describing it.

If the two dominant neighbours each contribute a displacement of size d to the answer, then independent displacements give an expected shift of d√2 and perfectly correlated ones give 2d — a factor of √2 between them. Measured, the contact transport at a coupling of 0.5 produces a recorded scatter of 0.97° against white noise’s 0.57° at the same displacement, a factor of 1.7.

That is more than √2 and less than 2, which is about where the arithmetic should land: the transport’s correlation between the two dominant organs is 0.42 rather than 1, so it should be short of the coherent limit; and the rule’s own response adds to the recorded scatter, as the previous essay measured, so the observed factor should exceed the input one. Two corrections in opposite directions, neither computed, bracketing the measurement.

The bracket can be closed further, and closing it is more useful than leaving it open, because the arithmetic turns out not to reach the measurement.

Two contributions of equal size with correlation ρ between them produce a shift whose spread is √(1 + ρ) times the independent one. The measured correlation between the two dominant organs under a contact transport at a coupling of 0.5 is 0.42, so the leverage argument predicts a factor of √1.42 = 1.19. The measured factor is 1.70.

So the argument gets the sign right, gets the mechanism right — the off-contact control settles that — and under-predicts the size by a wide margin. Something beyond the two-organ arithmetic is contributing about a further forty per cent, and there are exactly two candidates already named in this essay. One is the rule’s own response, which adds to the recorded scatter on top of whatever displacement it was handed. The other is the accumulation in the recursion, which the normalisation divides out of the amplitude and cannot divide out of the shape.

Those two are separable and neither has been separated. The rule’s contribution is measurable by reading the same disturbances on a kinematic lattice, where no rule acts, and comparing the scatter with the one the rule’s stems record; the accumulation’s is measurable by rebuilding the disturbance as a moving average over the contact offsets, which has the same correlations and no drift. Both are short pieces of work, and until one of them is done the honest form of the claim is: the direction is derived, the mechanism is established by a control, and the factor of 1.7 is a measurement the derivation accounts for less than half of.

The accumulation, which is a second and smaller effect #

There is a second reason the transported disturbance is harsh, and it is worth separating so that the first is not credited with all of it.

The transport recursion has each error inherit a fraction of two earlier ones and add a fresh draw. At a coupling of 0.7 that process does not settle to unit variance: its deviates have a standard deviation of 1.84, and the variance grows along the stem before the correlations saturate it. The same thing was found from the other side — at a coupling of 0.8 the forged lattice’s recorded scatter reaches 102° and there is no lattice left in it at all.

Every amplitude in this essay’s table is normalised by that measured standard deviation, which is the one arrangement in which the comparison is not a statement about its own denominator, so the comparison is at equal displacement and the accumulation is divided out. What cannot be divided out is its shape: the disturbance still drifts slowly along the stem in a way a plain memory from organ to organ does not.

The evidence that this is the smaller effect is the coupling of 0.5, whose stream has a standard deviation of 1.04 and almost no accumulation, and which is still harsher than white noise. The structure is doing the work; the drift adds to it.

The control, which separates the two accounts #

The account above says the damage is done by correlation at the offsets that dominate the minimum. The rival account is that correlation as such is what costs, regardless of where it sits. They differ on a measurement that costs one option on machinery already built here.

Correlate the disturbance at seven and eleven places back instead of eight and thirteen. At this rise those are ordinary members of the neighbourhood: the 7-hop is 0.15 circumferences and the 11-hop 0.17, against 0.069 and 0.075 for the contacts. Same coupling, same construction, same absence of common mode, same correlation magnitude — the leverage is the only thing that changes.

The off-contact transport is not merely as harmless as white noise. It is quieter than white noise, on both rows, and it keeps its lattice at a displacement where the contact transport has lost every stem.

That is the prediction confirmed and the rival refused. Correlation as such does not cost; correlation between the two organs the argmin is listening to costs a factor of two.

The reason the off-contact version comes out quieter than independent noise is the same argument run at the other end: correlating any two members of the neighbourhood removes some of the independent variation the answer would otherwise have had, and if those two are not the dominant ones, all it does is remove noise.

It is worth saying that this control is not a hypothesis about a plant. Nothing transmits a displacement between organs that are not touching, and a seven-and-eleven transport is not a model of anything. It exists to locate the mechanism, which is the only use a physically impossible arrangement has and a good one.

What it means for a plant #

The physically obvious disturbance on an apex is the transported one. Organs touch; a displacement in one is felt by the ones against it; and the ones against it are the contact neighbours, which is what a parastichy pair is. That was the whole force of the retraction of the comb, and it has not weakened.

This essay says that such a disturbance is the expensive kind. A plant whose errors are passed between touching organs loses its lattice at half the displacement a plant with independent errors would, and records nearly twice the scatter on the way there.

Two consequences, and the second is the one that would show.

A plant with contact-transported errors has to be quieter to hold a pattern. Which is a statement about how much displacement an apex can afford, and it runs the opposite way from the intuition that a mechanically coupled tissue is more robust.

And a plant near its tolerance would fail differently. Independent errors degrade a lattice gradually — the scatter rises and the pattern persists until it does not. A contact-transported disturbance at the same recorded scatter is closer to its own boundary, so the same plant would be nearer to losing the pattern than its scatter suggests. Since recorded scatter is the only thing a protractor gets, this is a systematic error in the direction of optimism.

What the control leaves for the ablation #

There is a connection between this essay and the intervention that the rest of these essays are about, and it is worth drawing because it is the same fact from two directions.

The ablation removes an organ, and it moves the next one if and only if the organ removed is inside the front — with the largest displacements coming from organs in the middle of it. This essay displaces organs without removing any, and finds that the displacement matters in proportion to how much the organ contributes to the minimum.

Both are measurements of the same quantity: how much each organ in the neighbourhood is worth to the answer. The ablation measures it by deleting an organ and reading the shift; the disturbance measures it by moving pairs of organs together and reading the scatter. They agree that the contact neighbours dominate, and the ablation adds a boundary — the front, thirteen organs at this rise — that the disturbance work has no way to see.

That is a fair summary of what an intervention buys over an observation, in one sentence, and it did not need a plant to make it.

The two also disagree about one thing, and the disagreement is worth having. The ablation’s largest displacements come from organs in the middle of the front, not from the contact neighbours at its edges; this essay’s largest effects come from the contact neighbours specifically. Both are true and they are answers to different questions — how much an organ’s absence costs against how much its movement costs — and nothing on this site has yet asked whether the two orderings can be derived from one another.

What this does not say #

It does not say the transport account is refuted. Nothing here is evidence about whether a plant’s errors are transported; it is a statement about what happens to a lattice if they are. The account remains the one that could not be distinguished from a placement rule on any observable available then.

It does not measure a plant’s coupling. The couplings here are 0.5 and 0.7 because those are the values at which the forgery works, and no measurement of the corresponding quantity in a plant exists.

And the accumulation is not entirely separated from the structure. The off-contact control settles which of correlation and leverage does the damage, and it does not settle how much of the contact transport’s harshness is its slow drift, since both versions have the same recursion and therefore the same drift. The evidence that the drift is the smaller part is that the off-contact version has it too and is harmless. Work that wanted the last of it would build a contact-correlated disturbance with no accumulation in it — a moving average over the contact offsets rather than an autoregression — which is half an hour’s work and was done afterwards.