What one angle says about the next

The previous essay established that a divergence sequence has a memory: the correlation between one angle and the next is 0.54 in a noiseless run, because the rule corrects itself and a correction spans two placements.

This essay is what happens to that number when the rule is disturbed, and the answer is the sharpest thing in the phase.

Three kinds, one statistic #

The three disturbances the model carries are placement noise, which displaces a primordium after the rule has chosen; field noise, which perturbs the energy profile before the minimum is taken; and a jostle, which displaces the neighbours so that the profile is built from the wrong positions.

Two phases of work have established that a finished pattern cannot tell them apart. Matched at the same divergence scatter, they give the same spread, the same counted pair, the same tolerance, and lattices that fail at the same place. Everything a plant offers as a summary is blind.

Sweep each one’s amplitude and read the lag-one correlation instead.

Placement noise erases the memory, immediately. A tenth of a degree — which moves the scatter from 0.50° to 0.62°, a change nobody would report off a plant — takes the correlation from 0.54 to −0.09. Inside the sampling band, which is ±0.13. And it stays there: at every amplitude from a tenth of a degree to eight tenths, the correlation is between −0.13 and +0.10, which is to say it is zero.

A jostle keeps it. 0.67 at a tenth of a degree, 0.66 at two tenths, decaying through 0.29 and 0.18 to 0.12 at the amplitude where the lattice is failing.

Field noise keeps it too. 0.61, 0.59, 0.39, 0.31, 0.11, over its own amplitude range.

Matched at three quarters of a degree of scatter — a quiet, healthy pattern by every other measure — the three sit at −0.09, 0.67 and 0.51. The band is ±0.13. Placement noise is separated from the other two by five bands.

Why placement noise erases it, exactly #

The mechanism is arithmetic and it is worth doing, because it explains both the sign and roughly the size.

Let $x_i$ be the azimuth of node i and $d_i = x_i - x_{i-1}$ the recorded divergence. Placement noise adds an independent displacement $\varepsilon_i$ to each $x_i$. So

$$d_i = (m_i - m_{i-1}) + \varepsilon_i - \varepsilon_{i-1}$$

where $m_i$ is where the rule actually put the node. The noise contributes a term that appears in two consecutive divergences with opposite signs — a moving-average process of order one with coefficient −1, whose own autocorrelation at lag one is exactly −½ and zero at every longer lag.

The recorded sequence is therefore the pattern’s own signal, correlated at +0.54, plus a noise term correlated at −0.5, in proportion to how much of the variance each owns. As the noise fraction rises the sum crosses zero and then goes mildly negative, which is exactly the measured behaviour: −0.09 at a tenth of a degree, −0.13 at two tenths, hovering near zero thereafter.

That the crossing happens so early is the surprising part, and it is a consequence of the two effects pulling in opposite directions. A statistic that merely diluted — that went from 0.54 towards 0 as the noise grew — would take a large amount of noise to become uninformative. This one is being actively pushed the other way, so a small noise fraction is enough to cancel the signal outright.

Why the other two do not #

Neither field noise nor a jostle adds anything to the recorded angle. The rule reads a perturbed profile, or reads a correct profile built from perturbed neighbours, and then places the node at the minimum it finds. Where the node ends up is a choice, not a choice plus an error — and a chosen position is part of the pattern, so it arrives carrying whatever correlation the pattern has.

The disturbance shows up in the recorded sequence only through the choices it changed, and a changed choice is a node in a slightly different place, which the rule then corrects for exactly as it corrects for anything else. The memory survives because the mechanism that produces it is still operating on every node.

That is the whole of the distinction, and it is worth putting in one line: placement noise is added to the record, and the other two are added to the process. A record can be corrupted; a process corrects.

How the prediction was wrong, and it is worth being exact #

The previous phase left a specific prediction and this measurement refutes it while confirming the thing it was reaching for. Both halves are worth setting out, because the way it was wrong is instructive.

The prediction: placement noise displaces each node independently, so successive divergences should be uncorrelated beyond lag one; field noise moves whole gaps, so the correlation should extend further.

The first half is right and it is right for the reason given. An independent displacement produces a moving-average term, which correlates at lag one and nowhere else, and the measured placement-noise sequences do have almost nothing past lag one.

The second half assumed something that is false: that the sequence has no correlation of its own, so that whatever correlation is measured must have been put there by the noise. Under that assumption, more correlation means the noise reaches further, and the two kinds would be told apart by how many lags they occupy.

The sequence has a correlation of its own, and a large one. So the comparison is not between a short-range signature and a long-range one; it is between a kind of noise that subtracts from the existing correlation and two kinds that leave it alone. The discriminating quantity is lag one, not the number of lags, and its sign is the informative part.

The prediction would have been right about a rule with no memory. Choosing to measure the whole autocorrelation function rather than only lag one is what made the mistake visible, and it is the reason the figures plot six lags instead of the one the argument ends up using.

The ceiling, which is the awkward part #

The separation is not available everywhere, and the limit runs the wrong way round from most measurements.

At three quarters of a degree of scatter the gap between placement noise and a jostle is 0.76, six times the band. At nine tenths it is 0.30, still clear. At 1.2° it has fallen to about 0.26 against a combined band of the same size — a factor of three down from where it started, and no longer several bands wide.

The reason is structural rather than statistical. What the other two kinds preserve is the correlation of a lattice, and a lattice at one and a half degrees of scatter is coming apart — its self-correction is losing to the disturbance, which is exactly what being near the tolerance boundary means. So the signal the statistic reads is going away at the same time as the noise it is trying to characterise is getting large.

That is an uncomfortable property and it is the opposite of the usual one. Most measurements get easier as the effect gets bigger. This one needs a quiet plant — a stem whose angles are tight, whose pattern is comfortably inside its tolerance, and which by every other measure has the least interesting noise. The noisier and more obviously disturbed a specimen is, the less this can say about it.

It is asserted rather than mentioned, because a reader who took the statistic as an instrument for any stem would be using it a third of the way past where it works.

The analysis that was tried first and does not work #

Before the autocorrelation, the obvious analysis was attempted and it is worth recording why it fails, because it is what most readers would reach for.

Take a noisy sequence, compute what the same rule produces with the noise switched off, and subtract. What is left should be the noise, and its structure should be readable directly.

It fails for a reason that is not the expected one. The expected failure was chaos — that a tiny perturbation would send the run onto a wholly different trajectory, so the residual would be the difference of two unrelated sequences. Measured, that is not what happens: a twentieth of a degree injected comes back as about half a degree of residual and stays bounded, where two unrelated angle sequences would differ by 104°. The rule is sensitive, amplifying by roughly an order of magnitude, and then it holds.

The analysis fails for a duller and more final reason. There is no noiseless run to subtract on a plant. Computing one requires knowing the rule, its exponent, its neighbourhood, its rate and its initial condition, which is to say it requires knowing the answer. Any residual computed that way would be a statement about the model’s parameters as much as about the plant’s noise, and a disagreement would be unattributable.

So the statistic has to be read off the sequence’s own structure, with no reference to any computed ideal. That is a constraint, and it is also a relief: it makes the measurement independent of every model parameter this collection has argued about for four phases.

What it does not separate #

Field noise and a jostle are within the band of each other at every scatter tried.

That is not a failure of the measurement; it is the measurement being exactly as informative as its construction allows. The statistic reads when a disturbance arrives relative to the choice, and those two arrive on the same side of it. Where the disturbance lives — in the signalling field, or in the tissue that generates it — leaves no mark on the order of the angles.

The basin statistic from the jostle thread draws the same line and no second one. Two independent instruments, one distinction, and it is worth recording as a limit rather than hoping that a third instrument would do better. If the two are to be separated, it will be by something other than the sequence a stem hands over.

The one place the statistic could be fooled #

There is a scenario in which the reading would be wrong, and it should be on the record.

Suppose a plant has both kinds of disturbance — a genuine jostle from tissue growth, and some placement-like error from whatever specifies a primordium’s position. Then the recorded sequence carries a positive correlation from the first and a negative contribution from the second, and the measured value is a mixture.

A mixture that happened to land near zero would be read as pure placement noise, and a mixture landing at 0.3 would be read as mostly-jostle. The statistic reports one number and there are two unknowns, so it cannot separate a small amount of one from a large amount of both.

What rescues it partly is the scatter. The scatter gives the total amplitude, and the correlation gives the mixture, so two measurements on the same stem constrain two unknowns — with the caveat that the conversion from amplitude to scatter differs between the kinds by about a quarter, which is not enough to make the system well conditioned.

The clean version would need a third measurement, and this thread does not have one. What it has is a statistic that goes from clearly positive to clearly negative as the mixture moves from one pure case to the other, which is enough to distinguish the extremes and not enough to quantify the middle. That is a normal position for a first instrument and it is worth being explicit that it is where this one stands.

What it is measuring, in plain terms #

Stripped of the machinery, the statistic asks a simple question of a stem: does a large divergence tend to be followed by another large one, or by a small one?

If large follows large, the deviations are shared between consecutive angles — they have a common cause that persists for a placement or two — and the disturbance was part of the process that placed them.

If large follows small, the deviations alternate — each angle’s error is its own, appearing once positively and once negatively — and the disturbance was applied to the positions after they were decided.

That is a question a person could ask of a list of numbers without any of the apparatus in this collection, and the apparatus is only there to say what the answer means and how many numbers are needed to trust it.

What the plant would have to be like #

Three conditions, and the last is the one that would sink a careless survey.

Quiet. Under about nine tenths of a degree of scatter along the stem, which is tighter than most published spreads and is a real restriction.

Long. Enough internodes to get the sampling band below the effect, which the next essay prices.

Measured precisely. Angular error in the observation enters exactly as placement noise does — it displaces each recorded position independently — so an instrument with half a degree of error would report the placement-noise signature from a perfectly quiet plant. The measurement error and the hypothesis are the same operation on the same numbers, and no amount of care in the analysis separates them. What separates them is remeasuring the same stem and reporting the difference.

That is the honest specification, and it is more demanding than the rest of this thread makes it sound.

Why this is worth more than another model comparison #

It is worth saying what makes this different from the rest of the phase.

The cut-off thread and the jostle thread both end in the same place: a distinction that is real inside the model and has no observable counterpart. A short-ranged inverse first-power rule cannot be told from an inverse-cube one by anything a finished stem shows. A jostle cannot be told from placement noise by any summary statistic. Both threads produce taxonomy.

This one produces an instrument. The lag-one correlation is computed from data a botanist already collects, on a specimen a botanist already has, and it separates two hypotheses by five sampling bands. Whatever else is true of it, it is the first thing in this collection that turns a modelling distinction into a number a plant could supply.

Its limits are real — a quiet stem, a long stem, a precise instrument, and no ability to separate a field perturbation from a jostle — and they are the limits of a measurement rather than the limits of a definition. That is a different kind of thing to have, and the last essay in the thread is about what it would cost to use it.