The test a plant could settle

This collection has spent a phase and a half writing down what it would take to settle its open questions with real plants, and the answers have been discouraging in a consistent way. Whether a divergence angle is systematically off 137.5° needs tens of specimens. Whether multijugate patterns occupy more than the first rung needs a census. Whether the Fibonacci share matches the model’s needs a population, sampled without the selection effect that ruins every published count.

This question is different, and the difference is structural rather than lucky.

Why it is cheaper #

The other questions are about shares. What fraction of sunflowers show a consecutive Fibonacci pair; how often a divergence lands within a degree of the limit; what proportion of a genus is bijugate. A share needs a population, because a share is a property of a population, and the sample size follows from the binomial arithmetic — which the site’s survey library does exactly, and the answers come out in the tens.

This one is about a sequence. The lag-one correlation of a stem’s divergence angles is a property of that stem, computed from the angles it has, and its uncertainty falls with the number of internodes rather than with the number of plants. One long stem carries as many observations as its internodes, and every one of them is a datum.

That is the whole of the difference and it is worth being explicit that it is not a trick. The two designs answer different questions. A share is a statement about plants in general; a sequence is a statement about one plant, and saying anything general would take several. But the first measurement — does this statistic distinguish the hypotheses at all, on any real plant — costs one stem, and it is a measurement nobody has made.

The arithmetic #

An autocorrelation computed from n points has a sampling standard error of about $1/\sqrt{n}$ for a sequence with no correlation, and the conventional band is two of those: $\pm 2/\sqrt{n}$.

Two correlations are distinguishable when they are further apart than their combined bands, so the requirement is

$$\Delta > \frac{2}{\sqrt n} + \frac{2}{\sqrt n} = \frac{4}{\sqrt n} \qquad\Longrightarrow\qquad n > \frac{16}{\Delta^2}$$

with a factor of two for the two sequences being compared.

The difference to be resolved is the one measured in the previous essay: at three quarters of a degree of scatter, placement noise gives −0.09 and a jostle gives 0.67, so $\Delta = 0.76$.

$$n = 2\left(\frac{4}{0.76}\right)^2 \approx 56$$

Fifty-six internodes.

What fifty-six internodes is #

A number is only useful if it can be turned into a plant, so: fifty-six internodes is a stem with fifty-seven leaves on it, counted from wherever the pattern settles into its final parastichy pair.

That is not a small plant and it is not an unusual one. A mature sunflower stem, a tobacco plant, a well-grown Arabidopsis inflorescence, a monocot with many internodes — any of these carries fifty-six comfortably. What would not work is a short rosette, a seedling, or a specimen sampled from the region where the pattern is still climbing the ladder.

Two practical caveats sit on top of the count.

The transient has to be excluded. The model drops sixty nodes at the start of each run because the pattern is settling out of its seeded configuration, and a real stem’s early internodes are coarse and often not yet on the final pair. So a plant needs fifty-six usable internodes past the point where the pattern has settled, not fifty-six in total.

The stem has to be quiet. The separation collapses above about nine tenths of a degree of scatter, so a specimen whose angles scatter by more than that is not eligible however long it is. That eligibility is checkable before any of the rest: compute the scatter, and if it is too large, stop.

Why the count is so forgiving of the details #

Fifty-six is robust to most of the choices behind it, and it is worth showing why, because a sample size that moved by a factor of five with a reasonable change of assumption would not be worth quoting.

The count goes as $1/\Delta^2$, so it is entirely governed by how far apart the two hypotheses sit. At a scatter of three quarters of a degree they are 0.76 apart and the answer is 56. Halve the separation — measure at a noisier scatter, or accept a smaller effect — and the count quadruples to about 220, which is a long stem but not an impossible one. Halve it again and the experiment is off.

What barely matters is the conventional factor of two on the band. Using one standard error instead of two would give 14 rather than 56, and using three would give 126; the answer stays in a range a single plant can supply throughout. That is the sense in which the measurement is cheap: it is not near a threshold in any of its own parameters.

What matters enormously is the eligibility. A stem at 1.2° of scatter has the two hypotheses inside the combined band at any length, so no number of internodes rescues it — the count is infinite rather than large. The experiment is therefore not “measure a stem and see”; it is “measure the scatter, and if it qualifies, measure the sequence”. Getting that order wrong would produce a null result that meant nothing and looked like a finding.

Where fifty-six sits against the other questions #

The comparison is the reason this essay is in the wrong field rather than beside the sequence work.

Three population questions and one that is not. And the one that is not is, on the model’s own account, the most discriminating: the population questions all bear on whether the geometry is right, which four phases of computation have already made a fairly strong case for, while this one bears on what kind of process is doing it, which nothing here has any evidence about at all.

That is an unusual position for a cheap measurement to be in. Cheap measurements are usually cheap because they answer easy questions.

The reason it is not cheap for that reason is worth stating. The geometry questions are hard to survey because they are about how often something happens, and frequency is a property of a population however easy each individual observation is. This question is about what kind of process produced one pattern, and the pattern carries its own history in the order of its parts. A single specimen is a complete record of one run of the process, which is exactly what the geometry questions never have.

What would count as an answer #

Worth stating in advance, so that the result cannot be read after the fact.

A correlation near zero, on a stem quiet and long enough to qualify, would say the disturbance is applied to the primordium after its position is decided — or that the observer’s angular precision is the dominant noise, which is the same operation and has to be excluded separately.

A correlation near 0.5, on the same kind of stem, would say the disturbance is part of the process: either the field the rule reads or the neighbours that generate it, and the statistic cannot say which.

A correlation well above 0.6 would be interesting in a different way, because the noiseless model sits at 0.54 and nothing here produces more. A stem correlating at 0.8 would mean the sequence has structure the model does not, which would be a result about the model rather than about the noise.

And an answer in between would be a mixture, which the statistic reports as one number with two unknowns behind it. That is the outcome most likely on a real plant and it is the one that would need a second measurement to unpick.

What could go wrong that is nobody’s fault #

Three failure modes that would not be errors, and would still make the measurement uninterpretable. A specification that lists only the things a careful experimenter could avoid is not a specification.

The stem might not be on one rung. The count assumes a stretch of stem where the pattern is settled. A stem that crosses a transition mid-sequence is climbing the ladder, which is a real trend and would show up as correlation at every lag — the detrending would remove some of it and there is no guarantee it removes the right amount. The check is to count the parastichy pair at the bottom and the top of the measured stretch and require them to agree, which is a measurement this collection’s machinery already does.

The angles might not be independent of the observer’s method. Measuring a divergence by eye against a protractor introduces error that is correlated with whatever the observer is anchoring on — a previous mark, a memory of the last angle — and correlated observation error is a signature that could be mistaken for either hypothesis depending on its sign. The defence is a method with no memory in it, and its adequacy is itself something to be demonstrated.

The plant might be doing something neither hypothesis covers. A stem whose divergence sequence has periodic structure — every third internode systematically different, say — would give a correlation that is neither zero nor 0.5 and means neither. That would be a genuinely interesting result and it is one this thread’s statistic would report as an ambiguous middle value rather than as the anomaly it is. Plotting the whole autocorrelation function rather than quoting lag one is what would catch it, which is the reason the figures show six lags.

What the exposures are #

The site’s convention is that a specification names what it would have to withdraw on each outcome, and this one has two.

If a quiet, long stem came back at zero, the claim that a growing organ’s deformation reaches the placement rule would be unsupported by the only observable this collection has. The jostle thread’s argument that the substrate is inside the model rather than around it would stand as a modelling point and lose its empirical prospect.

If a stem came back at 0.5 but so did a stem measured twice by an instrument known to be imprecise, the whole statistic would be compromised, because the measurement error and one of the hypotheses are the same operation. The remeasurement is not optional and its absence would invalidate the result rather than merely weaken it.

Both of those are stated now rather than later, which is the only way an exposure means anything.

What a second stem would buy #

The specification is for one plant, and it is worth saying what the second and third would add, because “one is enough” is a claim about the first question rather than about the programme.

A second stem of the same species would say whether the first was typical. The statistic is a property of a stem, and two stems of one species could differ — one grown fast and one slow, one crowded and one not. Two agreeing would be worth much more than one, and the arithmetic is unchanged: each stem is its own measurement with its own band.

A stem of a different species would begin to say whether the answer is a property of plants or of a plant. That is the point at which the design becomes a survey again and the population arithmetic in the site’s survey library applies — how many species, at what effect size, is the same binomial question as everything else here.

A stem measured twice is not optional and is not an extra: it is the control that separates the observer’s precision from one of the two hypotheses, and without it the measurement cannot be interpreted at all. It is listed here rather than under “what a second stem buys” because it is part of the first measurement.

So the honest framing is: one stem answers can this statistic distinguish the hypotheses on a real plant, which nothing currently answers. Several stems answer what do plants do, which is a survey and costs what surveys cost. The first is worth doing on its own, and it is the one this collection can specify completely.

Why this collection still cannot do it #

The same boundary that has been recorded for four phases, and it is not a difficulty.

Every claim on this site is checkable from this repository. The figures are generated from stated rules, the numbers are computed while the pages are built, and a reader with the source can reproduce all of it. A claim resting on a measured plant would be a different method sitting alongside that one — not a worse method, and not one this collection is equipped to hold to the same standard.

What has changed with this essay is that the specification is now cheap enough to be actionable by somebody else. The previous phase’s specification named sample sizes of four, fourteen and thirty-four specimens, and an exposure for each. This one names fifty-six internodes on one stem, an eligibility criterion, a precision requirement, and what each outcome would cost. It is the smallest and most completely specified open question the collection has produced.

What it would settle #

If it came back cleanly it would answer a question the previous two phases could only pose: whether the disturbance a real apex is subject to arrives before or after the primordium’s position is decided.

That matters because the two have different consequences for everything downstream. A disturbance arriving before the choice can, occasionally, put a primordium in a slot the arrangement did not have — one or two placements in a thousand, on the model’s numbers — and a disturbance arriving after cannot, ever. Over the length of a real stem the first is a handful of events and the second is none, and whether a plant’s pattern is subject to that class of event at all is the kind of thing a model of development ought to be able to say.

It is one plant, fifty-six internodes, and a protractor good to a few tenths of a degree.

That sentence is the smallest thing this collection has ever been able to say about an open question, and it is worth noticing what made it possible. Not a better model, and not more computation: a statistic that uses information the model was already producing and that four phases of summaries had been discarding. The order of the angles was there in every run ever made here, and every measurement taken of those runs was invariant to shuffling it.