The response with a hole in it

The intervention returns a count. Remove the organ k places back, place the next one against what is left, and record whether it moved: the answer is yes out to the larger parastichy number and no beyond it, so counting the yeses counts the spirals without measuring an angle or a position.

That was established at three rises, and the three were chosen in the middle of their rungs. This essay is what happens between them, and the answer is that the sentence above stops being true in a specific and informative way.

Why the middle of a rung was chosen, and by whom #

It was not a matter of taste. The figure that first showed the response offset by offset carries a slider over the rise, and the slider is snapped to the three measured rungs rather than sweeping continuously. The reason is written into it: three even steps from 0.032 to 0.005 pass 0.0185, which is a hundredth of a rung from a transition, and the figure’s own assertion refuses that rise because the counted pair and the front disagree there.

A refusal is a result that has not been looked at. The figure was right to refuse and the refusal was recorded as the obvious next experiment. This is it.

Thirteen rises, one experiment #

The sweep runs from 0.020 to 0.005: the coarse end sits on the 3/5 rung, the fine end on 8/13, and 5/8 occupies the middle. At each rise a stem is grown to four hundred organs, an organ is removed at each of fourteen offsets in turn, and the first organ placed afterwards is compared with the control — the same history continued with nothing removed.

Two things are held fixed and both matter. The azimuth grid is 1,536 samples throughout, four times the site’s usual, because the quantity is a displacement in degrees. And the cut is always made after the same number of organs, so the history each stem is cut from is as settled at one rise as at another.

On a rung it is an interval #

At 0.016, 0.013, 0.011 and 0.0095 the answer is exactly what the earlier work says. The felt offsets are one through eight, the pair is 5/8, and every offset past eight moves the next organ by less than 0.94° — a tenth of what the smallest felt displacement is, and a fifth of what a plant’s own scatter would be.

The same at the fine end: at 0.005 the felt offsets are one through thirteen against a pair of 8/13, and at 0.006 they are one through twelve. At the coarse end, 0.020 gives a pair of 3/5 and felt offsets one through five.

So on nine of the thirteen rises the response is a run, and it ends where the count says.

Approaching a transition it is not #

Between 0.009 and 0.0065 the run still ends at eight — and something else happens past it.

An organ twelve places back, whose removal at 0.0095 does not move the next organ at all — 0.00°, below the azimuth grid — moves it at 0.008 by 139°, a whole divergence. And the organs nine, ten and eleven places back, which sit between the run and it, move the next organ by 0.0°, 0.5° and 0.2°.

That is not a tail. A response that decayed with distance would show the displacement falling from eight outwards, and it does not: it falls to nothing and then comes back to its largest possible value. The set of felt offsets has a hole in it, three organs wide, and past the hole one more offset is as strongly felt as anything inside the front.

The lone offset is not any offset #

Twelve is not a contact. At this rise the hops in order of length are eight, five, thirteen, three, sixteen, eighteen, ten — and twelve is the longest of the eighteen offsets measured, at 4.7 local spacings. The organ whose removal moves the next one by 139° is the one furthest away from it.

What twelve is is one less than thirteen, and thirteen is the larger number of the pair this stem is climbing towards. The same shape appears at the other transition in the sweep, which is the check that it is not a coincidence of one number: at a rise of 0.020, on the 3/5 rung with 5/8 coming, the run ends at five and the lone responder is at seven — one less than eight.

So the isolated offset is predicted rather than fitted: read the three shortest hops off the settled lattice, take the shortest hop longer than the pair — that is the incoming number — and subtract one. The prediction is made from the geometry of the undisturbed stem and tested against an experiment on a disturbed one.

The incoming front arrives outermost first, and its own edge arrives last #

The order in which the hole fills is worth reading off the table rather than summarising, because it is not the order the two-front picture would suggest at first sight.

The offsets that join the response as the rise falls are twelve, then eleven, then ten — outermost first, working inwards towards the run that is already there. And the offset that is never in the list until the pattern has actually changed rung is thirteen itself. At 0.006 the response is a run of twelve; thirteen only becomes felt at 0.005, on the far side of the transition, where the count is 8/13 and the front is thirteen deep — which is the same growth of the front across a rung read from the other end, in displacements rather than in which offsets wreck.

So the incoming arrangement announces itself with everything except its own boundary. Reading the table as the new front growing behind the old one would predict the opposite: a front is the band of the most recent n organs, and a thirteen-wide band coming into existence should include its thirteenth member as soon as it includes any of them.

What arrives instead is one organ short of the incoming count, and then two, and then three — which is the same n − 1 that names the lone responder at both transitions the sweep crosses, extended one place at a time. That makes the lone cell and the filling of the hole one phenomenon rather than two, and it says the quantity being tracked is not a band at all but a set of offsets counted down from the incoming number.

Which is a transition detector #

Turn it around and it is an instrument. An ablation experiment near a transition returns two numbers rather than one: the run, which is the count the plant has, and an isolated responder, which is one inside the count it is about to have.

That is a different kind of measurement from any other on this site. The counting instruments here read an arrangement: positions, or angles, or a spectrum of angles, all of them measured on a pattern that is already there. This one reads what the pattern is about to do, and it does so on a stem whose visible count has not changed — at 0.008 a counter shown the positions says 5/8, the angles say 5/8, and the ablation says 5/8, and 13 is on its way.

The cost is that the identity between the boundary and the count — the clean result the intervention was built for — is a mid-rung property. Within about a fifth of a rung of a transition the largest felt offset is twelve while the count is eight, so an experimenter who took the largest felt offset as the count would report thirteen minus one on a plant with eight rows.

There is a second use for it, and it is the one an experimenter would reach for first. A plant photographed once gives a count and nothing else; a plant photographed twice, weeks apart, gives a count and possibly a change. The ablation gives the change from a single occasion, because what it reads is not the arrangement but the shape of the profile the next organ is being placed against — and that profile knows about the incoming pair before any organ has been placed according to it. Whether a real apex would cooperate is a separate question, but the design is cheap: the offsets to test are the ones just past the front, which is where nobody would otherwise look.

The economical experiment is the one that cannot see this #

There is a practical consequence, and it is the kind that only shows up once the answer is known.

An experimenter running this on apices spends one apex per offset. The count is the length of the run, and a run is over as soon as an offset comes back quiet. So the obvious economy — cut outwards from the tip, stop at the first offset whose removal does nothing, report the count — is exactly right about the count, costs the fewest apices, and cannot see a transition at all. It stops at nine and the informative cell is at twelve, with three silent offsets in between whose only job is to look like the end of the experiment.

That is worth stating as a design rule rather than as an observation, because the economy is not a careless one. Stopping at the first silence is what a careful experimenter short of material would do, and it returns the right number: the run ends at eight at every rise in the transition window, so the count is never damaged by the shortcut. What is lost is a second reading that the same apices were four cuts away from supplying.

So the specification is to cut past the first silence by at least half the front again. At a count of eight that is four more apices, twelve rather than eight, and the four are the ones that separate a plant sitting comfortably on its rung from one about to leave it. Whether they come back empty is itself the measurement — a full run of silences past the front is a stem in the middle of a rung, which is a statement about the plant and not a failed experiment.

It is not the grid and it is not the cut point #

Two ordinary explanations are available for a lone cell in a table, and both are excluded.

It is not the azimuth grid. At 1,536 samples the displacement at twelve places back is 139.0°; at 4,608 it is 139.2°. The quiet offsets are quiet at both.

It is not where the cut was made. The stem was cut after 300, 360, 400, 440 and 500 organs, and the row is identical every time: 0.0°, 0.5°, 0.2°, 139.0°, 0.9°, 0.0° at offsets nine to fourteen. That is five independent histories, each settled to a different length, giving the same table.

And it is not a threshold artefact. The displacement at the lone offset is 139°, against a threshold of 2.5° and a largest quiet displacement of 1.2°. No choice of threshold between one degree and a hundred changes the table.

What fills the hole #

Follow the sweep past the transition and the hole closes from the far side. At 0.007 the offsets eleven and twelve are both felt; at 0.0065 it is ten, eleven and twelve; at 0.006 the response is a run of twelve; and at 0.005 it is a run of thirteen and the pair is 8/13.

So the picture over the whole sweep is a front of eight with a second front growing behind it: first its outer edge alone, then its outer two offsets, then three, then the two fronts merge and the pattern is on the new rung. What the lone responder is, is the leading edge of the pattern’s next arrangement, felt before anything about the visible pattern has changed.

The displacements themselves, which are not small either #

It is worth putting the numbers beside each other, because a table of filled and empty cells hides how violent the isolated response is.

At a rise of 0.008 the felt offsets one to eight move the next organ by 137.8°, 83.2°, 54.4°, 164.1°, 25.8°, 97.7°, 124.9° and 12.0°. The quiet offsets nine, ten and eleven move it by 0.0°, 0.5° and 0.2°. And offset twelve moves it by 139.0°, which is larger than five of the eight offsets inside the front and within a degree of the largest displacement any offset produces anywhere in the sweep.

So the isolated cell is not a marginal detection at the edge of a threshold. It is the strongest kind of response the experiment can return: the next organ does not lean towards the vacancy, it goes into a slot a whole divergence away, which is what this rule does with a hole rather than with a nudge.

That also settles a question the shape of the table invites. One might imagine the front has simply become ragged near a transition — that the response is weakening unevenly out to some larger radius. It is not. Inside the run every offset is felt strongly, outside it three consecutive offsets are silent to a fifth of a degree, and past them one offset is felt as strongly as anything. A ragged front would not produce three consecutive silences with a maximal response behind them.

What this does not say #

It does not say the response is unstable near a transition. Every number in the table is reproducible at two grids and five cut points. The response is not noisy there; it is a different shape there.

It does not say a real plant would show this. A single ablation on a real apex has a surgical footprint, a wound response and a healing time, none of which are in this rule. What the measurement licenses is the prediction that a plant whose organs are placed where the inhibition is least would show a lone sensitive offset at one inside the incoming count — which is worth stating because it is a strange enough prediction that finding it would be evidence and not finding it would be evidence too.

And it does not yet say why. The reason twelve is the offset that matters, and why it stops mattering above 0.009, is a question about the profile the rule minimises rather than about the arrangement — the same place the survivor’s own change of hands along a rung turned out to live — and it is the subject of the next essay. The short version is that the profile has two low points a divergence apart and the twelfth organ back is holding one of them up.

The check #

Three assertions run as the stems are grown, and each one refuses a different mistake.

The first requires the response to be an interval at every rung rise — so a rung that developed a hole would stop the build, which is the way this claim could turn out to be an artefact of something shared by all thirteen runs.

The second requires the response not to be an interval at the four transition rises, and requires the offsets in the gap to be quiet to under a degree. A tail would fail it: a response that merely decayed slowly would have the intervening offsets somewhere between the front’s displacements and nothing, and they are not.

The third requires the isolated offset to be exactly the incoming number less one, where the incoming number is read from the hop lengths of the undisturbed stem rather than from where the rise at which a new family is first kept has since been located. That is the one that could most easily be wrong, because it is a prediction from one measurement to another, and it is checked at both transitions the sweep crosses.

The same sweep with two organs removed #

The hole in this essay is a property of the single-organ response near a transition. The two-organ version of the experiment has a boundary in the same place — the run of felt offsets ends at the larger parastichy number at every gap — which is what says the second removal has not changed what is being measured.