The organ that guards the second slot

Near a transition the ablation experiment produces a table with a hole in it: the organs one to eight places back are felt, nine to eleven are not, and twelve is felt as strongly as anything. The organ twelve places back is also, by the hop lengths, the furthest of the eighteen offsets from where the next organ is going.

Both of those cannot be explained by the same picture of what an ablation does, and the picture that is wrong is the intuitive one.

What an ablation was thought to be #

The intervention has a natural story attached to it. The next organ goes where the inhibition from the existing ones is least; take one of them away and there is less inhibition where it was; the next organ leans that way. On that story the size of the effect follows the strength of the removed organ’s contribution, which follows its distance, which is why the response ends at the front — an organ far enough away contributes nothing, and removing nothing changes nothing.

That story survives contact with the mid-rung measurements and dies at the transition. An organ at 4.7 local spacings contributes about a thousandth of the energy at the winning slot. Removing a thousandth of the energy cannot move the minimum by a whole divergence — unless the minimum was not the only candidate.

The profile has two low points #

Draw what the rule is actually minimising, rather than the summary of it. At the height the next organ will sit at, the repulsion around the circumference is a curve with several local minima, and two of them are much lower than the rest.

The second low point is not a curiosity of this rise. It is the slot the next organ but one will occupy: a lattice’s continuation slots are one divergence apart by construction, and at the moment before organ i is placed, the profile already has a dip where organ i+1 is going.

That is the fact the whole essay turns on. The rule is not choosing between a minimum and a flat landscape. It is choosing between two slots that differ by about a seventh in energy, and a perturbation of a seventh is not a small perturbation.

Which organs hold up which slot #

Each slot is held up by the organs nearest it, and the two slots have different neighbours because they are a divergence apart.

Organ i — the one being placed — has its contacts m and n places back, at five and eight and thirteen on this rung. Organ i+1’s contacts are the organs m−1 and n−1 places back from the tip, because everything is counted from a tip that has not moved yet. So the slot after next is held up by the organs four, seven and twelve places back.

Measured rather than argued: at a rise of 0.008 the organ thirteen places back carries 14.6% of the energy at the winning slot and 0.3% at the runner-up. The organ twelve places back carries 0.2% at the winner and 16.3% at the runner-up. The two are mirror images, each one guarding the slot it is a contact of and irrelevant to the other.

Where the second slot comes from #

The second low point is not an accident of this rise or of this stem. It is what a lattice’s continuation looks like from underneath.

A cylindrical lattice at divergence δ puts organ i at i·δ around the circumference. The profile the rule minimises at the height of organ i has a dip wherever an organ would sit without colliding with the existing ones, and the two cheapest such places are i·δ — where organ i is about to go — and (i+1)·δ, where organ i+1 will go once the tip has climbed one internode. They are exactly one divergence apart, which is why the separation measured across the sweep tracks the settled divergence rather than staying at some fixed number of degrees.

The runner-up is more expensive for a reason that is also arithmetic: it is a slot for an organ that would sit one internode higher, being filled at the height of one that sits lower, so it is closer to the organs already present. How much more expensive is what the gap measures, and the gap narrows towards a transition because the lattice’s own hop lengths are rearranging.

Which means every stem on this site has had two slots the whole time. Nothing in this essay is a new feature of the model; what is new is a measurement that depends on the second one, and therefore an observation that can report on it.

The comparison that decides it #

Removing the guard lowers the runner-up’s energy by roughly the guard’s share and leaves the winner’s alone. So the runner-up wins when

the guard’s share at the runner-up slot > the gap between the two slots

and that is a comparison between two quantities that are both computed from the undisturbed stem, before any organ is removed.

Both move with the rise, and in opposite directions:

The gap narrows as the transition approaches, because the incoming lattice’s slots are becoming competitive. The guard’s share grows, because the incoming family’s hop is becoming short. They cross between 0.009 and 0.0085 — and the response develops its hole at 0.009.

That the criterion is one step out is worth being plain about. It is a first-order comparison — it holds the two slots fixed and subtracts one term — where the real experiment lets the whole profile relax and re-finds the minimum. The prediction is that the two curves cross somewhere in the middle of the sweep and that the hole opens there. Both are true, and the estimate is not exact enough to say which of two adjacent rises is the boundary.

It is worth reading the two columns as one, because the difference between them is the quantity the criterion is actually about. Subtracting gives 8.4, 3.0, 1.0, −1.3, −2.7 and −4.9 percentage points across the six rises, and the useful thing about that list is not where it changes sign but how fast.

A criterion that hovered near zero over the whole sweep would locate nothing: the sign would be at the mercy of the first-order approximation it is built on, and the one-step disagreement with the measured boundary would be evidence that the comparison is not deciding anything. This one falls by about two points per half a thousandth of rise all the way through the crossing, so it spends one step of the sweep inside two points of zero and is four or eight points clear on either side. The crossing is transversal rather than tangential.

That is what makes the one-step error tolerable and also what bounds it. A first-order comparison that is out by two points is out by one step of this sweep, and the measured boundary is one step from the predicted one — so the error in the criterion is the size the neglected term would be expected to have, rather than a residual that happens to be small. It also says what a finer sweep would buy: steps of a tenth of a thousandth would put four rises between the prediction and the measurement, and the comparison would then be wrong by four steps rather than right to one. The criterion is a statement about which side of the crossing a rise is on, and it should not be read as a statement about where the crossing is to better than its own neglected term.

What happens to the stem afterwards #

An organ placed a whole divergence from where it was going is a large disturbance, and the natural worry is that the guard cut wrecks the pattern. It does not.

Cut the guard at a rise of 0.008 and the stem is back on its lattice after twenty-six organs, holding 137.77° with a spread of 0.18° for the rest of the run. Cut the eleventh organ back — one place inside it, and silent — and the stem recovers after twenty-seven. The two are indistinguishable a few dozen organs later, which is worth knowing: the guard cut is a violent displacement that the pattern absorbs, not a way of destroying it.

The offsets that do wreck the stem at this rise are the ones the other essay names: four places back leaves the settled angle for good and settles into a repeating block of five. So a stem near a transition has both behaviours in the same table — an offset that displaces the next organ maximally and heals, and an offset that displaces it comparably and never heals — which is a useful reminder that the displacement and the fate are two different measurements.

Why the guard cut heals, and what the recovery time says it costs #

The account explains the healing as well, and then fails to explain all of it, which is the more interesting half.

A cut at the tip heals instantly because the arrangement afterwards is the one that was going to exist, one organ short and with its labels shifted. The guard cut is the same kind of event seen from the other end: the next organ goes into the slot that organ i+1 was going to occupy, and organ i+1’s slot is a legitimate slot of the lattice rather than an arbitrary position. So the arrangement that results is not a lattice with a dent in it. It is the lattice the stem was going to have, entered one organ early.

That predicts a fast recovery, and the recovery is fast — twenty-six organs, against a hundred and twenty-five for the worst healing offset in the middle of a rung. What it does not predict is twenty-six rather than nothing. A pure relabelling would settle in one organ, as the tip cut does.

The residue has a name in this essay’s own arithmetic. The runner-up is a slot for an organ that would sit one internode higher, and it is being filled at the height of an organ that sits lower. So the organ placed after a guard cut is in the right azimuth and the wrong height: correct as a member of the lattice’s angular sequence, an internode below where that member belongs. The azimuths are right immediately and the heights are wrong by one rise, and it is the height error that the following organs have to absorb.

That is a testable division rather than a story. It predicts that a guard cut’s recovery is visible in the positions and nearly invisible in the divergence sequence — one anomalous step of a whole divergence and then ordinary angles — whereas a mid-front cut’s recovery is a wander in the angles themselves. Nothing here has separated the two, and the runs that would are the runs already on disk.

Why this makes the front what it is #

The same account explains the ordinary case, which is the test of whether it is an account at all rather than a story about one strange offset.

In the middle of a rung the gap is large — 19.3% at 0.013 — and the guard’s share is small, 6.8%. Nothing past the front can flip the answer, so the response ends at the front and the front is the set of organs whose removal changes the winner’s own energy enough to move it within its own slot.

And it explains why the isolated offset is n′−1 rather than n′. The slot after next is one organ further up the stem, so its contacts are one place earlier in the history. An account that predicted n′ would be an account about the incoming lattice’s contacts of the organ being placed, and that is a different number.

The account has to survive the ordinary case as well #

A mechanism invented to explain one strange cell in a table is worth very little unless it also explains the cells that were never strange. This one does, and it is worth walking through because it changes what the front is.

Before this, the front had a geometric account: the organs within about one local spacing of the tip are the ones whose removal is felt, and there are about 1/√h of them, which matches the larger parastichy number to within a tenth. That account is about distance, and it is right about the count.

The profile account says something narrower. An organ’s removal is felt if it changes which slot wins, or moves the winner within its own slot by more than the threshold. Inside the front, the second of those is what happens: the winner’s own guards are removed, its floor shifts, and its position moves — sometimes by a couple of degrees, sometimes by a whole divergence when the hole is deep enough to swallow the organ. Outside the front, neither happens, because the organ’s share of both slots is under a per cent.

The two accounts agree everywhere except at a transition, where the geometric one predicts a front of about 1/√h = 11 organs at a rise of 0.008 — a number between the measured run of eight and the isolated twelve, and matching neither. The profile account predicts eight and twelve, separately, and says why the offsets between are silent.

That is the strongest evidence for it. A model that agrees with a simpler model everywhere the simpler one was tested, and disagrees where it was not, and is right about the disagreement, has earned its extra machinery.

Three things it predicts #

The account is worth more than the observation, because it makes predictions the observation does not.

The two slots are one divergence apart, always. Measured across the sweep, the separation is 137.3° to 138.5°, which is the settled divergence at each rise. If the second low point were an artefact of the sampling it would not track the divergence.

The guard is always the incoming number less one. Which is the prediction the sweep tests at both transitions it crosses: seven where the incoming pair is 5/8, twelve where it is 8/13.

A cut at the guard should displace by a whole divergence, not by a little. Because the mechanism is a change of slot rather than a lean. Measured: 139.0° at 0.008, against a settled divergence of 137.8°. A lean would give a few degrees.

What this does not say #

It does not say the rule has two answers. It has one: the profile has a unique minimum at every rise in the sweep, and the stems grown from it are ordinary settled lattices with a scatter of a fraction of a degree. What is being described is the runner-up, which nothing occupies until the tip has moved up.

It does not say the gap is a probability. A stem near a transition is not flipping between the two slots, however much a survivor changing hands along one rung may look like it from the outside. It takes the winner every time, and the only way to make it take the other is to remove the organ that is holding the other one up.

And it does not license reading the gap off a photograph. The gap is a property of the profile, which is a property of the model. A plant’s own version of it — if there is one — would be the difference in inhibition between two positions on a real meristem, and nothing here measures that. What an ablation experiment could be asked for is a yes or no per cut, which is a coarser thing than a gap.

The check #

Two assertions carry it, and they fail in different ways.

The first requires that the organ carrying the most energy at the runner-up slot, among the offsets past the front, is exactly the one the hop lengths name — the same reading that locates the rise at which a new family is first kept. That is checked at six rises. It would fail if the guard were merely near the predicted offset — if it were eleven or thirteen at some rise — which is what a coincidence would look like.

The second requires the two quantities to cross: that there are rises in the sweep where the guard’s share exceeds the gap and rises where it does not, and that every rise of the second kind is coarser than every rise of the first. A criterion that flipped back and forth would fail it, and so would one that never crossed at all.

Both are asserted on the same stems the response table is measured on, so the account and the observation cannot drift apart without the build stopping.