A shoot too fast to remember

The previous essay established that the divergence sequence’s lag-one correlation is a property of a stem in motion. This one sweeps the motion.

The natural unit is nodes per rung — how many internodes a shoot lays down while the ladder’s preferred parastichy pair changes once. It is the site’s existing measure of how fast a pattern is being asked to reorganise, it is dimensionless, and it is a quantity a botanist can count off a specimen without knowing anything about the model.

The sweep #

Thirteen rates, no noise, and the lag-one correlation of the resulting sequence:

The sign changes between fifty-two and fifty-eight nodes per rung, inside one step of the grid, and it changes once — there is no wandering back and forth. That is what makes this a threshold rather than a slope.

What it is not #

Above the crossing the values run from +0.24 to +0.74 with no visible trend, and it is worth saying plainly that this collection cannot resolve their structure. An earlier and coarser version of this sweep used four rates, found +0.54, +0.58 and +0.58, and read that as a plateau — a quantity that saturates and then holds.

On thirteen rates the plateau is not there. What is there is a sign, consistent on every rate on each side of the crossing, and a spread above it that is as large as the effect a careless reading would have called a trend.

The flatness was the grid. It is exactly the failure the site’s own notes record about maxWindow — a parameter swept over a range where it was not binding, giving the same answer every time and reading as robustness — arriving here as three points that happened to agree. What survives a finer grid is the sign, and the sign is what the claim is now stated in.

What sets the threshold #

Two timescales, and the crossing is where they meet.

The first is how fast the equilibrium moves. A rung is a stretch of rise over which one parastichy pair is shortest, and the ladder is geometric with ratio $1/\varphi^2$, so the number of nodes a shoot spends on a rung is $2T\ln\varphi$ for a rise declining at rate $T$. That is the rate at which the target the pattern is chasing moves.

The second is how fast the rule corrects. A node placed to one side leaves a gap the next falls into, and the site’s existing measurement of that is the residual after a nudge: a twentieth of a degree injected comes back as about half a degree and stays bounded, over a handful of nodes. The rule’s correction time is a few placements.

When the equilibrium moves slowly compared with the correction time, the pattern tracks it, and consecutive angles deviate together because they are at nearly the same point on the same drift. That is the positive correlation.

When the equilibrium moves fast, the pattern cannot track it. Each node is placed against a neighbourhood that is already the wrong shape for where the ladder has got to, the corrections do not have time to accumulate into a shared drift, and what is left in the sequence is the alternation — a wide gap followed by a narrow one — which is negative at lag one.

So the threshold is a ratio of a chase to a correction, and fifty-five nodes per rung is where the rule this site uses puts it. Nothing about that number is fundamental; a rule that corrected faster would put it lower.

Why the negative side is not −0.5 #

If the fast end were pure alternation the correlation would be exactly $-\tfrac{1}{2}$, because a deviation appearing in two consecutive divergences with opposite signs is a moving-average term whose autocorrelation at lag one is that number and nothing else.

Measured, the fast end runs −0.28 to −0.43. Close, and consistently short of it.

The shortfall is the tracking that a fast shoot still manages. Even at thirty-five nodes per rung the pattern is not ignoring the ladder — it is following it badly, and badly is not the same as not at all. What is in the sequence is a mixture: a large alternating term at $-\tfrac{1}{2}$ and a small shared drift at something positive, and the measured value is the variance-weighted sum of the two.

That reading makes a prediction and the sweep bears it out. The most negative value is not at the fastest rate: it is at forty to forty-six nodes per rung, with thirty-five sitting slightly higher at −0.34. At the very fast end the run is short — a hundred and thirty-nine internodes against twelve hundred at the slow end — so the sampling band is ±0.26 and the number is worth less. The most negative reliable value is in the middle of the fast regime, which is where the drift has been suppressed and the run is still long enough to measure.

None of that is a strong result and it is not asserted as one. It is the reason the claim in this essay is about a sign rather than about a value: the value is a mixture of two known terms in an unknown proportion, and the proportion is what the rate controls.

A fast shoot is noisier with no noise in it #

There is a confound in the sweep and it deserves to be pointed at rather than buried, because it matters more for fieldwork than the headline does.

The divergence scatter of these runs is not constant across the rates. With no disturbance in the model at all, it runs:

• 35 nodes per rung → 0.76°
• 52 → 0.53°
• 96 → 0.56°
• 250 → 0.47°

A fast shoot scatters its angles by half again as much as a slow one, and every one of those runs is noiseless. The scatter is the pattern’s own tracking error — the distance between where the rule puts a node and where the moving equilibrium would have it — and a pattern that cannot keep up has more of it.

The consequence for the previous phase’s gate is direct and unwelcome. That gate reads the scatter and treats it as a measure of disturbance: above nine tenths of a degree, the lattice is near its tolerance and the test is off. But a stem at forty nodes per rung arrives at three quarters of a degree with nothing disturbing it at all, so a scatter measurement on an unknown stem is a measurement of disturbance plus rate, and the two are not separated by anything in the previous phase’s procedure.

The clause added at the end of this essay fixes it, and it fixes it by measuring the rate directly rather than by trying to subtract it: two spiral counts settle which regime a stem is in before any angle is measured.

The site already had a threshold in the rate, and this is not it #

This is the second time a sweep of the rate has produced a sharp answer on this site, and the two must not be confused.

The scale phase found that a pattern seeded on the Lucas branch keeps that branch below about ninety nodes per rung and loses it above — a fast shoot abandons a metastable arrangement at the first fork it reaches and walks the Fibonacci ladder instead. That threshold is about which branch a pattern ends up on, it is a statement about the pattern’s history, and the fast side is the side that loses.

This one is about what a sequence of angles remembers, it is a statement about the measurement rather than about the plant, and the fast side is the side where the instrument reports the wrong thing rather than the side where the pattern does anything unusual.

They are at different values — about ninety against about fifty-five — which is convenient, because it means a shoot can be in one regime and not the other. A shoot at seventy nodes per rung has a readable sequence and would still lose a Lucas seed. A shoot at forty has neither.

That two independent thresholds in the same parameter come out within a factor of two of each other is not a coincidence and is not a result either: both are comparisons between the rate of change of the equilibrium and the rule’s own correction time, so both live at the same scale. What differs is what is being asked to keep up.

What a fast shoot actually looks like #

It is worth converting the number, because “nodes per rung” is not how a specimen presents itself.

A shoot at fifty nodes per rung transitions from one counted pair to the next every fifty internodes. Over a stretch of a hundred and fifty internodes — a long specimen — it would show three different pairs. That is a visibly rising pattern, the kind that gets remarked on in a description, and it is uncommon.

At ninety nodes per rung, one transition per specimen. At two hundred, a stem that holds one pair from top to bottom and would be described as having a fixed phyllotaxis.

So the regime where the statistic works is the ordinary one, and the regime where it fails is the conspicuous one. That is the right way round for fieldwork and it is the opposite of what the previous essay’s other constraint does, which excludes the ordinary steady stem entirely.

The second clause of the eligibility gate #

The previous phase specified a field test with one gate in front of it: measure the divergence scatter first, and if it is above about nine tenths of a degree, stop. The reason is that what the informative kinds of noise preserve is the correlation of a lattice, and a lattice near its tolerance is losing it.

That gate is necessary and it is not sufficient. A stem can be beautifully quiet — half a degree of scatter, every angle tight — and be at forty nodes per rung, or at equilibrium, in which case there is no positive correlation to erase and the reading would be the same one a disturbed stem gives.

The full gate has two clauses and both are cheap:

Count the spirals at the top and the bottom of the stretch to be measured. If the pair is the same at both ends, the stem is at or near equilibrium over that stretch and this statistic is unavailable on it. If the pair changes more than once in sixty internodes, the shoot is too fast.

Then measure the scatter. Above nine tenths of a degree, stop.

Only a stem that passes both is a specimen for this test, and the first clause needs no protractor at all — it is two spiral counts, which is the measurement the whole subject already knows how to make.

What this costs the previous phase’s headline #

The claim was that the sequence test is the one open question in this collection a single specimen could settle, priced at fifty-six internodes on one stem, where everything else is priced in tens of plants.

That survives, and it acquires a qualifier that narrows the population. The specimen must be climbing — not steady, not fast — and how common that is among the plants somebody would reach for is not something this collection can say. It is a question about botany rather than about geometry, and it is answerable by looking, which is more than most of the qualifiers here can claim.

The cost is real and it is worth stating in the form a reader would want: the test needs a stem that changes its mind about how many spirals it has, slowly, once.

The window between the two thresholds #

Putting the two rate results together gives a map of a specimen, and it is worth drawing because the regions are not nested.

Below about fifty-five nodes per rung the sequence has no positive memory. The statistic reads the alternation whatever the plant is doing, and a Lucas seed would have been lost at the first fork.

Between about fifty-five and ninety the sequence is readable and a metastable arrangement is still lost. This is the region the previous phase’s instrument works in, and it is also the region in which a pattern’s history has been erased — so a count taken here describes the ladder rather than the seed.

Above about ninety the sequence is readable and a Lucas seed survives. Both of the site’s rate-dependent results are in their permissive regime, and a specimen here carries information about where it started as well as about where it is.

Far above — a few hundred nodes per rung — the pattern stops transitioning inside a measurable stretch and the first gate closes again for the opposite reason.

So the usable band for the full apparatus is roughly ninety to two hundred nodes per rung: fast enough to transition once in a specimen’s length, slow enough to remember its own history and to have a readable sequence. That is a narrow window and it is the first time this collection has been able to state one, because it is the first time two independent rate thresholds have been measured on the same model.

The lesson, which is the same one twice #

A number measured at one parameter value is a function evaluated once, and a number measured at four is a function sampled coarsely enough to look flat.

Both halves of that have now bitten this thread within one phase. The correlation was read as a constant of the rule when it is a function of the rate; then the function of the rate was read as a plateau when the plateau was three points on a coarse grid. The second error was made while correcting the first, which is the ordinary way this happens.

What is left is a sign, a crossing located to within one step, and an explanation that predicts a crossing at that scale. That is less than either of the previous readings claimed and it is the part that has been tested.