A disturbance the organs share

This collection has put noise into the placement rule three ways. placement displaces an organ after the rule has chosen where it goes; field perturbs the energy landscape before the minimum is taken; jostle places the organ exactly and then moves it, so the disturbance is invisible in what a botanist measures and present in every decision the rule makes afterwards.

The measurement phase compared the first two and the mechanism-02 phase added the third, and the standing result is that the lattice fails at about the same recorded scatter — a degree and a half — whichever way the disturbance got in.

Every one of those disturbances was independent from organ to organ. Nobody asked what happens if it is not.

What “the same disturbance” has to mean #

The breadth-03 phase built three correlated disturbances — a memory, a periodicity, and a transport between contact neighbours — and drove them into a lattice with no rule in it. This essay drives the same streams, the same functions, through the rule.

Doing that fairly needs one piece of care. A stream’s deviates are meant to have unit variance so that an amplitude in degrees means a displacement in degrees, and one of the three does not: the transport recursion inherits from two lagged terms and normalises as though it inherited from one, so at a coupling of 0.7 its deviates have a standard deviation of 1.84.

So every stream here is divided by its own measured standard deviation, and an amplitude of half a degree is half a degree of displacement per organ in every row of every table.

That is a correction to the previous phase as well as a precaution for this one, and it is taken up at the end.

The table #

Seven disturbances, six displacements, five stems each, at a rise of 0.005 on a converged azimuth grid.

Independent errors survive half a degree of displacement per organ on all five stems and none at one degree.

A memory with a coefficient of 0.9 survives one degree on all five and 1.5° on three. A memory of 0.97 survives 1.5° on all five, 2° on four, and 3° on one.

So a lattice takes three times the displacement when the organs share it, and the range from the most fragile disturbance to the most survivable is a factor of six.

The middle of the table is worth reading too, because it is where the effect turns on.

A memory with a coefficient of 0.5 behaves exactly like independent noise: five stems at 0.25°, five at 0.5°, none at 1°. Its deviates are correlated at lag one with a coefficient of 0.49 and at lag five with 0.005 — which is to say consecutive organs share half their displacement and organs a few apart share none.

A memory of 0.9 correlates 0.89 at lag one, 0.53 at lag five and 0.36 at lag eight. A memory of 0.97 correlates 0.96, 0.83 and 0.74.

So the tolerance turns on between a correlation length of about one organ and about five, and it is still rising at the longest correlation measured. The quantity that matters is not whether the disturbance is correlated at all — half a correlation at lag one buys nothing — but whether it is correlated across the neighbourhood the rule sums over, which at this rise is some eighty-five organs deep and a spacing wide.

The scatter is the relative part #

The explanation is in the third column of the same table, and it is the reason this is not a curiosity about correlated noise.

At 1.5° of displacement per organ, the memory of 0.97 leaves stems whose recorded divergence scatter is 0.62°. At 0.5°, a third the displacement, independent errors leave stems recording 1.03°.

The rule places each organ against its neighbours’ positions. A displacement all the neighbours share moves the whole neighbourhood without changing its shape, and the argmin moves with it: the organ goes to a slightly different place, and so does the next one, and the divergence between them is nearly what it would have been.

What destroys a lattice is not how far an organ moves. It is how far it moves relative to the organs it is placed against.

This does not contradict the standing result. It explains it. The boundary is at a degree and a half of recorded scatter whichever mechanism produced it, because recorded scatter is the relative part; the amplitude that produces that scatter is not the same number for different disturbances, and nobody had varied the one thing that separates them.

There is an arithmetic version of this that is worth stating because it predicts the numbers rather than describing them.

The divergence between two consecutive organs is the difference of their azimuth errors. If those errors have standard deviation σ and correlation ρ at lag one, the difference has standard deviation σ√(2(1 − ρ)). At ρ = 0 that is 1.41σ; at ρ = 0.9 it is 0.45σ; at ρ = 0.97 it is 0.24σ.

So a memory of 0.97 delivering 1.5° of displacement should record about 0.37° of scatter from that term alone, and it records 0.62° — the remainder being the rule’s own answer moving in response, which the arithmetic does not include. A memory of 0.9 at 1° should record 0.45° and records 0.61°.

Two predictions with nothing fitted, right to within a factor of 1.4, and both in the same direction: the rule’s response adds to the recorded scatter rather than subtracting from it. That is the self-correction visible as a cost — the pattern pulls back, and pulling back is itself a divergence.

What a plant’s disturbance is likely to be #

The reason to care is that a real apex’s disturbances are almost certainly not independent.

An organ is displaced because the tissue under it grew unevenly, because the meristem’s surface expanded, because a neighbouring primordium pushed it. Every one of those acts over a region rather than on a point, and a region contains several organs. The natural expectation for a plant is a disturbance with a correlation length of a few organs — which on this evidence buys the pattern a great deal of tolerance.

Put the other way round: a plant recording half a degree of divergence scatter may be carrying displacements several times larger than that, and there is no way to tell from the finished stem. The site has spent three phases establishing that the recorded scatter is what a protractor gets and that the amplitude is not recoverable; this puts a number on how far apart the two can be.

The edge of the table is stochastic, and that is a measurement too #

Several cells are partial — three stems of five, four of five, one of five — and it is worth saying what that means rather than reading the table as a set of thresholds.

Whether a particular stem keeps its lattice at a displacement near the boundary depends on its seed. The disturbance is a draw, and a draw that happens to put two large displacements on adjacent organs early in the run can start a cascade the rule does not recover from, while another draw of the same amplitude does not.

So the boundary is a probability rather than a value, and the table’s partial cells are where that probability is between zero and one. For independent errors the transition is sharp — five of five at 0.5° and none of five at 1°, with nothing in between measured — and for the long memory it is broad, running from five of five at 1.5° through four at 2° to one at 3°.

A broader transition is what a longer correlation length should give: a correlated draw has fewer effective independent chances to produce a damaging run, so the outcome depends more on which draw it was. That is consistent and it is not measured carefully here; five stems per cell locates a probability to about a fifth.

What it means for the rate results #

One earlier finding is worth re-reading in this light, and it survives.

The scale phase found that a growing shoot loses a Lucas seed above about ninety nodes per rung, and asked whether that threshold was a statement about the rate or about the noise amplitude — because noise explores a metastable state too, and would knock a pattern out of one at any rate. The phase after it established that the threshold is about the rate: at the amplitudes involved, noise does not do the knocking.

Nothing here disturbs that, but it adds a caveat with a size. The amplitudes that argument was run at were independent ones. A plant carrying the same recorded scatter with a correlated disturbance is carrying a larger displacement, and whether that is enough to knock a metastable pattern out is not the same question. The number to check is the displacement rather than the scatter, and the check has not been done.

The correction to the previous phase’s control #

The previous phase describes its four arrangements as “normalised so the recorded scatter is the same half degree as every other arrangement in this thread”.

Measured, the recorded scatters are 0.71°, 0.22°, 0.73° and 1.08°. A factor of five, and the transported forgery — the arrangement the phase’s whole retraction rests on — is the loudest of the four.

Two causes, and they are different. The memory’s low figure is arithmetic and harmless: a recorded divergence is the difference of two consecutive azimuth errors, and differencing a correlated series gives a smaller spread than differencing an independent one. Half a degree of azimuth error with a memory of 0.9 is a quarter of a degree of divergence scatter.

The transport’s high figure is the normalisation bug: its stream carries 1.84 times the variance it is supposed to.

So the forgery ran at 1.08° of recorded scatter against a control at 0.71°, and the phase described them as equal.

The forgery survives the repair, and it has to be shown rather than argued. Driven at a stream normalised to unit variance, so that its recorded scatter is 0.54° against the control’s 0.70°, the transported disturbance still returns 8/13 on every seed and still puts its second comb above its main one — 0.671 against 0.553, the identical numbers, because a comb is a correlation and a correlation does not care about scale.

That is the point worth carrying: the result was right and the control was described wrongly, and the reason both are true is that the quantity in question happens to be scale-free. A control that is not what it says it is will sometimes be harmless. The way to find out is to measure it, and this collection had not.

Why this was not found earlier #

Three phases have put noise into this rule and none of them varied its correlation structure. The reason is worth recording because it is not an oversight so much as a category.

An amplitude is an obvious parameter: it has a unit, it appears in the code as a number, and sweeping it is the first thing anybody does. A correlation structure has no natural scalar and does not appear in the code at all — the disturbance was gaussian(seed), a function that returns independent draws, and independence is not a setting there. It is the absence of one.

The three kinds of noise this site does distinguish are three answers to where the disturbance enters, which was the question the measurement phase asked because it was the question that separated its two candidate models. Nobody asked what shape the disturbance has until the previous phase needed a correlated one to attack its own control, and then it built the streams and drove them into a kinematic lattice rather than through the rule.

So the machinery existed for a phase before it was pointed at the rule, and the phase plan said so and predicted the outcome. The prediction was that a self-correcting rule shortens whatever correlation length it is given, so the previous phase’s negative result would hold a fortiori. Half of that is right — the rule does damp a shared displacement, by a factor of three in what it records — and the a-fortiori half is wrong, which is the subject of the third essay in this thread.

One more comparison is worth making, because it is the cheapest test of the account and it is already in the table.

If what matters is the relative displacement, then two disturbances that produce the same recorded scatter should destroy the lattice at the same place, whatever their amplitude. Independent errors at 0.5° record 1.03°; a memory of 0.9 at 1.5° records 0.98°. Those are the same scatter to within a twentieth, at three times the displacement — and both are marginal in the same way, five stems of five standing for the first and three of five for the second.

Two disturbances, three times apart in amplitude, at the same recorded scatter, failing at the same place. That is the account stated as a prediction and confirmed within the table it came from, which is weaker than an independent test and better than nothing.

What this does not say #

It does not say correlated disturbances are benign. One of them — the transported kind, correlated at the contact offsets — is harsher than independent noise rather than gentler, which is the subject of the next essay and is the result that stops this one from being a rule of thumb.

It does not measure a correlation length in a plant. No such measurement exists. What is established is that the quantity matters, which is a reason to want it rather than a substitute for having it.

And it does not move the boundary. Every stem in the table that still has a lattice records under 1.2° of divergence scatter and every stem that does not records over 30°. The measurement phase’s degree and a half is exactly where it was; what has moved is the amplitude that reaches it.