A rule that cannot heal a hole

The intervention does not end when the next organ has been placed. The organ is still missing, the stem keeps growing, and what it does over the next few hundred organs is a second experiment that costs nothing extra to run.

It has an obvious expected answer, and the expected answer is wrong for a third of the offsets.

The rule is self-correcting, and this site has the number #

The expectation comes from a measurement made several essays back. The divergence sequence of a stem grown by the rule has a lag-one autocorrelation of about −0.6: a divergence larger than the settled angle is followed by one smaller, and the deficit is made up almost at once.

That is a restoring force, and a restoring force is what heals a disturbance. A displaced organ is pulled back. Noise of any of the three kinds this site distinguishes leaves a pattern that is still the same pattern.

So the natural prediction is that a removal is a large disturbance which the rule undoes: a transient of some tens of organs, then the lattice again.

What actually happens, offset by offset #

Each ablated stem is continued for three hundred organs and watched for a recovery — the first organ after which every later divergence stays within 1.5° of the settled 137.84°, with at least sixty of them to check, so that the last few angles of a wrecked run cannot be mistaken for a return.

The pattern of it is worth setting out plainly, because it is not what a restoring force predicts.

Cut at the tip and there is nothing to heal. Removing the most recent organ costs nothing at all: the next primordium appears where the removed one was and the sequence continues, so the recovery time is zero. Two places back takes 24 organs and three places back 48.

Cut at the far edge of the front and the stem recovers. Ten, eleven, twelve and thirteen places back take 53, 59, 42 and 7 organs. Five places back takes 125, which is the longest recovery measured and still a recovery — on three cut points of five, as the section on reproducibility below has to report.

Cut in the middle and it never recovers. Four, six, seven, eight and nine places back are still off their lattice after three hundred organs, with mean divergences of 186°, 228°, 184°, 186° and 182° and spreads of 54° to 83°.

It is worth dwelling on how different those two behaviours look in the same axes, because the difference is not one of degree. A healed cut wanders for a few dozen organs and then produces divergence after divergence within a tenth of a degree of 137.84°, indefinitely. An unhealed one produces a sequence whose mean is nowhere near the settled angle and whose spread is comparable to the whole circle — and which is nonetheless not disordered at all, as the next essay takes up. Nothing in between was observed at any offset: no stem drifted slowly back, and no stem recovered and then lost it again.

Where the boundary is, and where it is not #

The five offsets that never heal are 4, 6, 7, 8 and 9 out of a front thirteen organs wide. They are neither the contact offsets nor the non-contact ones: 5, 8 and 13 are the three shortest hops at this rise, and the set of unhealing offsets contains 8, excludes 5 and 13, and includes four offsets that are not contacts at all.

There is a reading of that list which the numbers support better than the list itself does, and it is worth putting alongside. The offsets that fail are 4 and 6 through 9. Written out, that is a contiguous stretch of the front from four places back to nine with a single organ of recovery sitting inside it — and the organ sitting inside it is the one offset in the whole table whose verdict is not reproducible. Five places back recovers at three cut points and never recovers at the other two.

So the honest description is not five scattered offsets. It is a band six organs wide, from 4 to 9, whose interior contains one marginal offset — or a band of four with an isolated failure two places above it, and the measurement made here cannot tell those apart. What settles it is more cut points at five, which is the cheapest experiment this essay leaves behind: the same runs, the same threshold, one offset, twenty starting heights instead of five. If five fails at, say, eight of twenty, it belongs inside the band and the band is contiguous; if it recovers at every one of the extra fifteen, then four is an island and the description needs to say why.

The difference is not cosmetic, because the two readings predict different things about the coarse arrangements. A contiguous middle band scales with the front: on a front of five the middle is one or two organs and the band should be one or two organs wide. An isolated failure at a particular offset does not obviously scale with anything.

What the unhealing set does line up with, on either reading, is age within the front. Both ends of the front are safe and the middle is not. The reading that fits is geometrical rather than arithmetical: a hole at the very top is filled by the organ that was going there anyway; a hole at the bottom of the front is nearly enclosed by organs that remain, so almost nothing is missing; a hole in between is a real vacancy in the part of the surface the rule is actively placing against, and the organ that fills it is then itself out of position for the organs that come after.

That is a cascade rather than a displacement, and a restoring force that acts on displacements does not act on it. The rule is self-correcting against being pushed and is not self-correcting against being cut.

The cascade can be watched happening. Take the cut six places back. The next organ goes 102° from where it was going, into the vacancy. The organ after that is now placed against a neighbourhood whose most recent member is 102° out of position, so it too is displaced, and by a comparable amount; and the one after that against a neighbourhood containing two displaced organs. Within four organs the front has been rebuilt out of organs none of which is where the undisturbed stem would have put it, and there is no arrangement of the ones that remain that the rule can walk back to. The restoring force is still there — the lag-one correlation is still negative on the wrecked stem — and it is restoring towards a different lattice.

That is the sense in which a deletion is not a large displacement. A displacement leaves the neighbourhood intact and moves one member of it. A deletion changes what the neighbourhood is, and every organ placed afterwards inherits the new one.

Noise does not rescue it, which is what makes it an attractor #

A deterministic model that gets stuck in a bad state is a suspicious object. Real apices are not exact, and a rule that only fails when it is run without noise would be reporting a property of the arithmetic.

So the same cut was made with a jostle in the run — the third kind of noise, the realistic one, where each organ is placed exactly and then the organ it sits on grows, so what the next organ computes against has moved.

At a jostle of 0.1° the wrecked stem’s tail is 186° ± 83° — indistinguishable from the noiseless run. At 0.4°, which is nearly the amplitude at which the lattice fails on its own, it is 149° ± 92°: still nowhere near 137.8°, and now with the disturbance doing most of the spreading.

Noise does not put the stem back. Whatever state a mid-front cut leaves it in, that state is stable — an attractor with a basin, rather than a knife edge the arithmetic happened to balance on.

It is worth being precise about what “stable” means here, because the site has a statistic for it. basinChanges asks how often a disturbance sends an organ to a different minimum than the same neighbours would have handed back with the disturbance taken away — the rule’s own counterfactual, run alongside it. A jostle changes the choice once or twice in a thousand placements while the lattice is intact; placement noise, which arrives after the choice, changes it exactly zero times at every amplitude tried, including amplitudes that have already destroyed the pattern.

A deletion changes the choice once, at the moment of the cut, and then never has to change it again — because after the cascade the rule is choosing correctly among a set of organs that is no longer the one it started from. The comparison puts the intervention in a different class from all three kinds of noise: it is the only disturbance in this collection that moves the pattern from one attractor to another and leaves it there.

The recovery times are not a rate #

The eight offsets that heal do so in 0, 24, 48, 125, 53, 59, 42 and 7 organs, reading outwards from the tip. Two things about that list are worth stating and one is worth refusing to state.

The two: it is not monotone in the offset, and its largest member sits directly beside the unhealing band. Five places back is the last offset before the band begins and it takes 125 organs — two and a half times any other recovery, and on two cut points of five it does not recover at all — which is what a boundary looks like from the safe side. Thirteen places back takes seven, which is what a boundary looks like from the far side of the front, where the cut was barely a cut.

The refusal: these numbers are not a decay constant and should not be fitted to one. A recovery time here is the first organ after which the sequence stays inside 1.5° of the settled angle for at least sixty organs, which is a threshold-crossing on a trace that wanders rather than a time constant of an exponential. Changing the threshold to a degree or to two degrees moves the individual times by tens of organs while leaving the verdicts — healed, not healed — untouched. The verdicts are the measurement; the times are a description of the traces.

That distinction matters more here than it usually would, because a recovery time is exactly the kind of quantity a reader would want to compare against a plastochron. Two and a half plastochrons versus a hundred and twenty-five is a real difference and it is the one to carry away; the difference between 42 and 53 is not.

What the sharpness costs, and what it buys #

The boundary between healing and not is one organ wide — five recovers, four does not — and a boundary that sharp deserves a check that it is not an accident of where along the stem the cut was made.

It is not, with one exception that is worth more than the rule it breaks. The whole table was repeated at five cut points forty organs apart, and fifteen of the sixteen offsets give the same verdict at every one of them.

The sixteenth is five places back, which recovers at 125 organs at three cut points and does not recover within three hundred at the other two. Five is the offset immediately beside the band that never heals. So the one place the verdict moves with where the cut was made is the boundary — which is where a boundary should be soft, and it is the only evidence here that the band has an edge rather than a wall.

The table was also repeated at 4,608 azimuths, where the front’s width and every displacement inside it agree with the coarse grid to within a quarter of a degree.

What it buys is a second observable an experiment could put to a plant, and one that is easier than the first. The displacement of the next organ needs a measurement made within one plastochron of the ablation. The failure to heal needs no timing at all: it is visible in the finished shoot, weeks later, as a stretch of stem whose divergences never return to what they were below the wound.

There is a third use, and it is the one that makes this worth a set of essays rather than a paragraph. The first essay’s experiment measures a number — the front’s width, which is the larger parastichy number. This one measures a map: which removals a pattern can undo. Two accounts that agree about the first can disagree about the second, and the transported-error account has nothing to say about either, since in it no organ’s position is a function of which organs exist. So the two experiments are not one experiment run twice; they are a count and a susceptibility, and a plant that gave the first and not the second would be saying something specific about how its apex differs from the rule.

Three things this does not say #

It does not say a plant cannot heal an ablation. Surgical experiments on apices are a century old and the usual report is that the pattern reorganises and carries on. This says something narrower and checkable: in a model whose only response to a vacancy is to re-take the minimum, the vacancy’s position in the front decides whether the pattern comes back, and the boundary is one organ wide. A plant that heals every ablation is evidence about what a real apex does that the model does not — regrowth of the wound, a plastochron that is not fixed, a primordium whose radial position is free — and that is a more useful finding than agreement would have been.

It does not follow from the first essay’s result. The front is thirteen organs wide and the unhealing band is five of them, and nothing in the front’s width predicts which five. Both facts came out of the same runs and only the first was expected.

And it does not explain which five offsets fail. Age within the front is the description that fits, and it is a description rather than a mechanism: it says where the unhealing band is and not why it ends where it does. A cut four places back fails and a cut five places back recovers, and nothing measured here distinguishes those two positions except that one of them is a contact offset — which cannot be the reason, since eight is a contact offset and fails.

Work that wanted the mechanism would need to watch the cascade organ by organ at many more cut points than the seven used here, and would probably want a smaller quantity than a verdict: how far out of position the k-th organ after the cut is, as a function of both k and which organ was removed. That is a surface rather than a table, it is affordable, and it is the obvious next thing to compute.

One arrangement heals a hole and not two #

The stems here that cannot heal are all at fine arrangements. At the coarsest one every single-organ cut repairs, which was reported as an immunity and is really a statement about how large a disturbance is needed.

Two organs is enough. Four pairs of offsets in sixty-four never repair at a rise of 0.032, and three of those four settle on the mirror image of the lattice they were cut from — same counts, same hop lengths, opposite hand.