A front with no middle

The experiment is one sentence long. Grow a stem until its pattern has settled, remove a single organ from somewhere in its recent history, and let the rule place what comes next against what is left. Everything about it is a difference between two runs that share a history and differ in one organ, which is why it can say things a photograph cannot.

It has already produced two results. The first is that the removal is felt if and only if the organ removed is one of the most recent n, where n is the larger parastichy number — a spiral count obtained without counting anything. The second is that the stem does not always recover: cuts at either end of that band are undone within a few dozen organs, and cuts in the middle of it are never undone at all, in three hundred.

Both were measured at one rise, on one stem, with a front thirteen organs wide. The second one has a shape that invites an obvious reading and the obvious reading is wrong.

The thing that was measured, and the thing that was assumed #

At a rise of 0.005 the pair is 8/13 and the front is thirteen organs. The offsets that never recover are four, six, seven, eight and nine. Five of thirteen, in the middle, with the first three and the last four healing.

There are two natural readings of a band five offsets wide, and they make different predictions:

• It is a property of the rule. Five is what this rule does to a hole, and it would be five on any lattice. A front of eight would then have five unhealing offsets out of eight, and a front of five would have five out of five — the whole front.
• It is a share of the front. The middle two-fifths of the front is what cannot be repaired, whatever the front is. A front of eight would have three, a front of five would have two.

Neither is right, and the third possibility — the one that turns out to hold — is not a statement about the band at all.

Three rungs, one experiment #

The ladder gives the comparison for nothing. Lowering the rise walks the pattern up through 3/5, 5/8 and 8/13, and the front at each is the larger number of the pair: five organs, eight, thirteen.

Each rung gets the same treatment. A stem is grown to four hundred organs at a fixed rise, one organ is removed at each offset in turn, the run is continued for three hundred organs with the organ permanently missing, and a recovery is the first organ after which every later divergence stays within a degree and a half of the settled value with at least sixty of them to check. Nothing else differs between the three: same rule, same seed lattice, same azimuth grid, same recovery test.

The displacements confirm the front at each rung before any question of recovery arises. At 0.032 the organs one to five back move the next organ by 139.7°, 79.0°, 42.7°, 149.1° and 8.7°, and every older organ moves it by less than 1.4°. At 0.013 the run of large displacements ends at eight. At 0.005 it ends at thirteen. The boundary is the count, at all three.

The coarsest rung cannot be wrecked #

At the 3/5 rung every offset heals.

Not most of them: all five. The recovery times are 0, 25, 39, 38 and 8 organs for the five offsets of the front, and the organs behind the front are not disturbed enough to need repairing at all. There is no offset at this rung, and no cut point, at which a stem fails to return to 139.69° and hold it for the rest of the run.

That is worth stating plainly because of what it does to the first reading. Five unhealing offsets is not something the rule carries around with it: on a front of five, nothing at all fails. And it does the same damage to the second reading, from the other direction — a fixed share of the front would predict two unhealing offsets here, and there are none.

The middle rung splits the difference in a way neither reading predicts. At 0.013, with a front of eight, the offsets that never recover are four and five — two of eight. The recoveries either side of them are 0, 26 and 43 organs at one, two and three, and 38, 32 and 14 at six, seven and eight.

So the count of unhealing offsets across the three rungs is nought, two, five, on fronts of five, eight and thirteen. Neither constant nor a constant share.

What is constant is the edges #

Lay the three rows against each other on the same offset axis and the constant is obvious, and it is not the band.

The first three offsets always heal. At every rung, removing the organ one, two or three places back is undone: 0, 25 and 39 organs at the coarse rung, 0, 26 and 43 at the middle one, 0, 24 and 48 at the fine one. Those nine numbers are three measurements of the same thing repeated at three rises, and they barely move.

The last three or four offsets of the front always heal. At the middle rung that is six, seven and eight; at the fine rung it is ten, eleven, twelve and thirteen, which take 53, 59, 42 and 7 organs. At the coarse rung the two edges overlap and account for the whole front, which is why nothing there fails.

The band is the remainder. Thirteen organs less three at the tip and four at the far edge leaves six, and five of those six never heal — the sixth is the marginal offset the original measurement already flagged. Eight less three and three leaves two, and both never heal. Five less three and three leaves nothing, and nothing fails.

That is the third reading, and it is a statement about the edges rather than about the band. The band is not a quantity the rule produces. It is what is left of the front when two fixed-width regions are taken off either end, and it widens because the front widens while they do not.

Why the two edges are safe #

Both edges have a reason, and neither reason mentions a count.

At the tip, the hole is filled by the organ that was going there anyway. Remove the most recent organ and the next primordium appears exactly where the removed one was: a displacement of a whole divergence, and no damage at all — the arrangement afterwards is the arrangement that would have existed, with its labels shifted by one. Two and three places back are nearly the same story with a little more rearrangement, which the rule’s restoring force absorbs.

At the far edge, the hole is nearly enclosed. An organ at the outer limit of the front has neighbours on all sides that are still present, so its vacancy is a dent in a surface rather than a gap in a frontier. The next few organs are placed slightly differently and the arrangement closes over it.

In the middle, the vacancy is in the part of the surface the rule is actively placing against. The organ that fills it is then itself out of position for the organs that come after, which are placed against it. That is a cascade rather than a displacement, and the lag-one anticorrelation of about −0.6 that makes this rule self-correcting acts on displacements.

Which means the question “how wide is the unhealing band” was the wrong question, and the right one is “how wide is the front”. The front is the larger parastichy number, and the larger parastichy number is about 0.9/√h — so the band is roughly √h⁻¹ less seven organs, and it goes negative, which is exactly what a rung with no unhealing offsets looks like.

What it costs an experiment #

The intervention was offered as an experiment a botanist could run: ablate one primordium, record whether the next one came out where it was going to, and the number of organs for which the answer is no is the parastichy count. That result is unaffected — the front is the count at all three rungs.

The second result is affected, and sharply. Whether a plant can be permanently knocked off its pattern by a single ablation depends on how many contact rows it has.

A plant on a 3/5 arrangement — a young shoot, a small cone, anything at the coarse end of the ladder — will heal any single ablation, and an experimenter who worked on one would report that the rule is robust. A plant three rungs finer will fail to heal roughly a third of them. Nothing about the rule differs between the two, so a disagreement between two laboratories working on different material would be a real disagreement about nothing.

The design that follows is specific: ablate at several offsets and report the count alongside. An ablation experiment that does not say what the parastichy pair was cannot be compared with another one.

The offsets that sit on the boundary #

Two offsets do not behave cleanly, and both are where a boundary should be soft.

At the fine rung, five places back recovers — after 125 organs, the longest recovery anywhere in this work — at three cut points of five and not at the other two. It sits immediately beside the band that never recovers.

At the middle rung, five places back does not recover, and it is the outer member of a band of two. So the same offset is on opposite sides of the boundary at two rungs, which is what the edge-based reading says should happen: five is three from the tip at one rung and three from the far edge at another.

Everything else is sharp. The transition from heals to never heals is one organ wide at both rungs that have one, and it is in the same place at three cut points forty organs apart — the recovery table at the middle rung is identical at cuts made after 320, 400 and 480 organs.

What this does not say #

It does not say the coarse rung is stable against everything. One organ is one organ. A stem at 0.032 that loses two adjacent organs, or one organ and a displacement, has not been tested here, and the argument that its front has no middle does not obviously survive a larger hole — a hole of three organs in a front of five has a middle by construction.

It does not say the edges are exactly three organs. Three is what all three rungs supply and four is what the fine rung’s far edge supplies; a front of twenty-one, which is the next rung down, would be the test that separates a constant edge from a slowly growing one. That run is four times the cost of the ones here, for one number.

It does not say anything about a plant. These are stems grown by a rule that places each organ where the repulsion from the existing ones is least, and the rule is a model of form. What the measurement licenses is a prediction about what an ablation experiment would find if that rule is what a meristem does — which is the only reason to run the experiment.

The check that would refuse it #

The claim in this essay is three claims, and each of them can fail on its own. The first is that the tip edge is at least three organs deep at every rung. The second is that the far edge is at least three deep wherever there is a band to have an edge of. The third is that the band widths are all different — because a single constant would restore the first reading, and a constant share would restore the second.

All three are asserted as the stems are grown, at every rung, and a run that produced a five-offset band on a front of eight would stop the site being built. So would a coarse rung with any unhealing offset at any of three cut points, which is the strongest of the three: it is a claim that something does not happen, over a table of numbers that would be very easy to produce by accident if the recovery test were sloppy.

The recovery test is the place where sloppiness would show, and it has the hold requirement for that reason: without demanding sixty consecutive divergences inside the tolerance, an early version found the last four angles of a wrecked run and called it a recovery four organs before the horizon. Every table here is measured with the hold in place, and a wrecked stem’s tail spreads over tens of degrees, so nothing is close to the threshold.