The sequence has a memory

A stem hands over its divergence angles in an order. Node one to node two is an angle; node two to node three is another; and so on up the plant, one number per internode, in the sequence they were made.

Every measurement this collection has taken of that sequence has been a summary — a mean, a spread, a classification of the counted pair — and every one of them is invariant to shuffling. Reverse the sequence, deal it out at random, sort it: the mean is the same, the scatter is the same, the reported pair is the same. Four phases of work on this model have thrown the order away without ever saying so.

This essay is what is in the order.

The measurement #

Take a grown stem, drop the first sixty nodes as a transient, remove the slow drift with a centred moving average, and compute the correlation between each divergence and the one immediately after it.

With no noise at all, that correlation is 0.54.

That is a large number. The band an uncorrelated sequence of this length would sit in is ±0.13, so 0.54 is four times outside it. And it is not noise: it is the noiseless run. The pattern’s own dynamics put it there.

Why a lattice has a memory #

The reason is the definition of a lattice, and it is worth stating slowly because it makes the rest of the thread predictable.

A lattice is an arrangement in which each element sits between its neighbours. That is what the placement rule computes: it puts the next primordium where the repulsion from the existing ones is least, which is the position furthest from the nearest of them — the middle of the largest available gap, roughly speaking.

Now suppose one node lands a little to one side of where an ideal lattice would put it. It has left a slightly wide gap on one side and a slightly narrow one on the other. The next node is placed in the largest available gap, so it goes towards the wide side. And its divergence from the previous node is therefore also a little large.

Two consecutive divergences, both displaced in the same direction, by the same cause. That is a positive correlation at lag one, and it is what a restoring mechanism looks like when it is read as a time series.

The decay is the same story continued: the correction is not complete in one step, so a residue survives to lag two and less to lag three, and by lag five or six the sequence has forgotten. The correlation is a measure of how many placements the rule takes to pull itself straight.

The independent check #

A correlation can be manufactured by an analysis. The one free choice here is the detrending window, and a moving average is a low-pass filter, which can create exactly the kind of correlation being reported. So the claim needs an independent route.

The route is a direct measurement of the restoring behaviour. Take a run, inject a known displacement into one placement, and compare the resulting sequence against the same run without it — not to compute a residual sequence, but to ask a single question: does the disturbance grow or stay bounded?

A twentieth of a degree injected comes back as about half a degree of residual across the sequence, and stays there. Four times the amplitude gives less than twice the residual. Two unrelated angle sequences would differ by 104°, which is what the root-mean-square difference of two independent uniform angles is.

So a nudge stays a nudge. The rule amplifies it by roughly an order of magnitude — which is itself worth knowing, since it says the model is sensitive without being chaotic — and then holds it. A system with a bounded response to a perturbation is a system with a restoring force, and a restoring force is what produces the positive correlation. Two measurements, one property.

This also settles a question the thread was set up to ask. The first analysis attempted was the obvious one: difference a noisy sequence against the noiseless run and read the noise off the residual. It fails, but not for the reason expected. It fails because there is no useful noiseless run to difference against on a plant, not because the rule runs away — the runs stay close, which is the opposite problem and a much friendlier one.

What the higher lags say #

The lag-one number is the one the rest of the thread uses, and the shape of the whole curve carries something too.

The noiseless run’s correlation runs 0.54, 0.37, 0.12, −0.07, −0.23, −0.19 across lags one to six. Positive and decaying for the first three, then crossing zero and going mildly negative.

The decay is the correction, as above. The negative lobe is worth a sentence, because it is what distinguishes a correction from a drift. A process that simply wandered would have a correlation that decayed to zero and stayed there. One that overshoots — that pulls back slightly too hard, so a wide gap is followed a few nodes later by a narrow one — has a negative lobe at the lag corresponding to the overshoot. The lattice is doing the second thing.

That has a physical reading. The arrangement is not merely relaxing towards a lattice; it is oscillating into one, with a period of about five nodes at this rise. Five is close to the smaller of the two parastichy numbers over most of this run, which is what one would expect if the oscillation is the arrangement passing the correction round a parastichy family — a node’s error is felt most by the node one family-step later, which is five or eight nodes on rather than one.

That last is a suggestion rather than a measurement. The lag at which the negative lobe sits could be measured against the counted pair as the rise falls, and if the two moved together it would be a real result about how a correction propagates through a lattice. Nothing here does that, and it is the cleanest untested thing this essay leaves.

What this means for the site’s earlier work #

Three consequences, in increasing order of how much they matter.

The scatter is not the wrong statistic; it is an incomplete one. Everything measured with it stands. The tolerance results, the noise boundaries, the comparisons between kinds — all of those are about how wide the distribution of angles is, and the width is what they claim to be about.

But a shuffling-invariant statistic cannot see a mechanism that acts in time. The placement rule is a sequential process: each node is placed against the ones before it, and the disturbance to one node propagates to the next. Any statistic that would give the same answer on a shuffled sequence has, by construction, discarded everything about that propagation. The previous phase concluded that its two kinds of noise were indistinguishable, and the conclusion was correct for the statistics it used — which is a weaker statement than it sounded.

And the correlation is a property nobody has measured on a plant. Published divergence data is reported as a mean and a spread, sometimes over specimens rather than along a stem. Nothing in the literature this collection has looked at reports the sequence’s own structure, and reporting it would cost nothing extra: the data required is exactly the data already collected, in the order it was collected in.

The detrending, and why it does not make the result #

The one manipulation between the raw angles and the statistic deserves scrutiny.

A growing stem’s divergence is not constant. It climbs the ladder as the rise falls, and near a transition it moves by a degree or so over a few dozen nodes. That is signal — it is the thing three phases of this site are about — and it is also a slow trend, which contributes positive correlation at every lag and would swamp the statistic if left in.

The trend is removed with a centred moving average over twenty-one nodes. That window has to be long compared with the correlation being measured and short compared with the drift, and both of those are satisfiable here because the two timescales are far apart: the correlation decays over four or five nodes and the drift takes dozens.

The check is that the result survives the choice. At windows of eleven, twenty-one and forty-one nodes, the separation between the noise kinds this thread depends on is unchanged. That is asserted rather than hoped, and it is the assertion that would catch a filter manufacturing its own answer.

There is one thing the detrending genuinely costs, and it is worth stating: the statistic cannot see correlations longer than the window. If the sequence had structure over fifty nodes, this analysis would have removed it and reported nothing. What is measured is the fast structure, which is where a per-placement disturbance lives, and a claim about slow structure would need a different treatment.

The transient, and why sixty nodes go in the bin #

The first sixty nodes of every run are dropped, and it is a large fraction of a three-hundred-node stem, so it wants justifying.

Every run here starts from a seeded stretch of ideal lattice at a stated divergence. That is deliberate and it is what makes “can this pattern change branch” a question with an answer — a pattern has to be on a branch before it can leave one. But it means the run begins in a configuration the rule did not choose, and the first stretch is the rule settling out of it.

A transient of that kind has a correlation structure of its own, and it is a strong one: the sequence is moving systematically from the seeded angle towards whatever the rule prefers, which is a trend, and a trend correlates at every lag. Including it would report the seed rather than the dynamics.

Sixty is chosen as comfortably past where the tracking essays show the seed’s influence gone. It could be checked more carefully — a plot of the statistic against where the window starts would show exactly where the transient ends, and would be a better justification than “comfortably past” — and it is not, which is the kind of thing worth admitting rather than dressing.

The equivalent on a real plant is not obviously the same length. A stem’s first internodes are not seeded by an experimenter; they are whatever the seedling did, and the pattern there is coarse and often not yet Fibonacci. A survey would have to make the same decision and would have the same difficulty defending it.

What comes next #

The number 0.54 is the baseline, and on its own it is a fact about a model rather than a tool. What makes it a tool is what happens to it when noise is added, and the answer is the sharpest measurement in the phase: one kind of noise erases it completely at an amplitude that moves the scatter by a tenth of a degree, and the other two leave it standing.

That gives the thread its instrument. The scatter says how disturbed a stem is; the correlation says where the disturbance got in. And unlike every other discriminator this collection has proposed, it needs one plant.

What a survey would have to record #

Since the whole value of this statistic is that it might be measurable, it is worth being precise about what a field measurement would need — and the answer is encouraging in one respect and demanding in another.

Encouraging: the data is the data. Divergence angles along a stem are what a botanist measures when measuring divergence angles; there is no additional instrument, no imaging, no dissection beyond what a count already requires. What is needed is that the angles be kept in order and reported individually, rather than averaged into a mean and a standard deviation.

Demanding: the angles have to be good. A correlation of 0.5 is measured against a sampling band, and measurement error in the angles enters exactly the way placement noise does — it displaces each recorded position independently, which drags the correlation towards zero. So an instrument with a degree of error would produce, from a perfectly quiet plant, the signature of a plant with placement noise.

That last point is the serious one and it is not a detail. The statistic cannot distinguish placement noise in the plant from measurement error in the observer, because the two are the same operation applied to the same numbers. A survey using it would need its angular precision to be well inside the effect it is looking for — a few tenths of a degree, on a specimen a few centimetres across — and would have to demonstrate that rather than assert it, by remeasuring the same stem and reporting the difference.

None of that is impossible. All of it is the kind of thing that separates a measurement from a number, and it is the reason the last essay in this thread prices the experiment rather than merely recommending it.

A note on what a sequence is #

There is a small conceptual point worth making explicit, because it is the reason this was available all along and nobody took it.

A stem is usually treated as a spatial object: a set of positions, from which counts and angles and lattices are extracted. That treatment is the right one for almost everything in this collection, and the site’s counting machinery is built on it — a blind counter is shown coordinates and nothing else, which is what makes a count evidence rather than a restatement.

But a stem is also a record of a process, laid down in order, with the earliest event at the bottom. Read that way it is a time series, and every tool for time series applies: autocorrelation, spectra, tests for independence, model identification. None of that machinery has ever been pointed at a phyllotactic sequence in this collection, and the first thing it says is that the sequence is strongly non-independent.

The two readings are not in competition. A spatial reading gets the lattice, the counts and the ladder; a temporal reading gets the dynamics of how the arrangement was arrived at. What the phase’s last essays do is use the second to answer a question the first could not.

It is also worth noticing that the temporal reading is available on a plant in a way it is not available in most of physics. A stem is its own record: the internodes are laid down in order and stay where they were laid, so a single specimen carries the whole history of the process that made it. Nothing has to be filmed, and nothing has to be repeated. That is an unusual gift, and this collection has spent four phases reading it as a snapshot.