The noise that arrives through the neighbours

The placement rule as written is exact. It sweeps a candidate azimuth around the apex, computes the repulsion at each position, and puts the primordium at the least of them, to the resolution of the sample grid. A real apex is not exact, and the previous phase put two kinds of inexactness into the rule to find out how much that matters.

Placement noise displaces the node after the rule has chosen: the primordium lands a little away from the minimum, by an angle. Field noise perturbs the energy profile before the minimum is taken: the rule is minimising something slightly wrong. They were chosen because they bracket the choice — one before, one after — and because an amplitude means whatever its implementation makes it mean, so a result that holds for only one kind is a result about code.

Both are idealisations. The disturbance a real shoot apical meristem is subject to is neither, and it is the obvious one.

What actually happens on an apex #

A primordium is specified at a position on the meristem surface. Then the meristem grows. Cells divide, the surface expands, and the expansion is not uniform — the whole point of a growing apex is that some of it is doing more than the rest. By the time the next primordium is being placed, several plastochrons later for the ones that matter most, the neighbours it computes against are not quite where they were put.

That is a disturbance to the rule that enters through neither of the two channels the previous phase modelled. The node was placed exactly, so there is no placement noise. The energy profile was computed correctly, so there is no field noise. But the profile was computed from displaced neighbours, which is a perturbation of the question rather than of the answer — and the phase plan’s one-line prediction was that placement noise on one node arrives as field noise for the next.

Why it is a third kind and not a re-spelling #

The obvious objection is that this is placement noise under another name. If every node is displaced, and later nodes see the displaced positions, what is the difference?

The difference is what carries the displacement into the record.

Under placement noise the node’s own recorded position is wrong. A botanist measuring divergence angles off the finished stem measures the noise directly: the angle between node i and node i+1 contains node i’s displacement and node i+1’s.

Under a jostle every node’s recorded position is exactly where the rule put it. What is displaced is the position each node’s successors saw while they were being placed. On the finished stem — and a finished stem is all anybody ever measures — the displacement is gone. The angles are the rule’s own answers.

So the two are opposite in a precise way. One is in every measurement and in no decision; the other is in every decision and in no measurement. That is not a distinction of degree, and it is why the library keeps them as separate kinds rather than as one with a sign.

There is a physical reading of the difference too, and it is not artificial. A primordium that is specified in the wrong place — because the signal that specifies it was noisy — has placement noise. A primordium specified in the right place whose neighbourhood has since deformed has a jostle. Those are different events in the development of the organ, and a model that could not tell them apart would be missing something a biologist would consider basic.

How it is implemented, and the one line that matters #

The change to the rule is four words long and it is worth quoting, because the whole distinction lives in it.

Every node carries two coordinates. x is where the rule put it — the position a botanist would measure. seen is where later nodes find it, which is x plus a displacement drawn once, at placement. The energy loop reads q.seen; the recorded divergence is computed from x.

That is all. Under the other two kinds seen and x are the same number and nothing changes; under a jostle they part company, and the rule goes on doing exactly what it did before against a set of neighbours that has drifted.

Two things about that implementation are worth defending.

The displacement is drawn once and does not accumulate. A node is displaced by a fixed amount relative to where it was placed, not by a random walk that grows with age. That is a choice, and the alternative is arguably more realistic — an older primordium has had longer to be carried away. It is not modelled because it would confound two effects: the amplitude of the disturbance and the depth of the memory, and a result that depended on both would be attributable to neither. What is here is the simplest thing that is a jostle at all, and a version with age-dependent drift is a clean extension with a second parameter.

The amplitude is in degrees of azimuth, the same unit as placement noise. That is what makes the two comparable at all, and it is the reason the phase can say that a jostle is a little gentler per degree rather than merely different. A field amplitude, by contrast, is a fraction of the barrier the rule is working against — the only scale-free quantity available, since the energy at a sample sitting on top of an existing node is unbounded — and it cannot be put on the same axis at all.

The statistic that separates them #

Since a jostle leaves no trace in the recorded angles, and the previous phase already established that the spread of those angles cannot distinguish its two kinds, the obvious question is whether anything can.

Inside the model, something can. As each node is placed, the rule also computes where it would have gone with the noise taken away and everything else — the history, the rise, the actual neighbours — left alone. That counterfactual is free: it is one extra sweep, run alongside the real one, and it answers the question that matters. Did the noise change which minimum was chosen, or merely where the node sat within the one it chose?

A node counts as having changed basin when it ends up more than half a local spacing from its counterfactual. Half a spacing rather than a fixed angle, because the spacing falls by an order of magnitude down a run and a fixed threshold would be measuring the rise.

The answers are three different kinds of number.

Placement noise: exactly zero. Not small — zero, at 0.2°, at 1°, at 2° and at 4°, the last two of which have already destroyed the lattice. This is a proof by construction rather than a measurement, and it is in the library as an assertion because it is the control every other row is read against. Any non-zero value would mean the counterfactual was being computed in the wrong place, and every jostle number in the phase would be worthless.

A jostle: one or two placements in a thousand, while the lattice is intact. Small, and not zero, and that is the whole claim.

Field noise: the same order, rising steeply once the pattern is coming apart.

So the prediction holds in the respect it was aimed at. A jostle behaves like field noise in the one way that matters: it is upstream of the choice, and it can change what is chosen.

What the prediction got wrong #

The consequence drawn from it behaves like field noise was that it would be the dangerous kind — that the most realistic disturbance would be the one most able to knock a pattern off its branch. That does not follow and is not what happens.

All three kinds destroy the lattice at about the same scatter: the largest divergence spread at which every run still has a pattern is 1.4° for placement noise, 1.6° for field noise and 1.7° for a jostle. The previous phase found this of two kinds — 2.00° and 1.68°, which it called a scale rather than a constant — and the third joins them.

The reading is that the boundary belongs to the pattern rather than to the disturbance. A lattice fails when its divergences scatter by about a degree and a half, and which of three mechanisms produced the scatter does not move where it fails.

There is a difference per degree of displacement — a jostle needs a little more amplitude to reach a given scatter, because a jostled neighbour reaches the argmin through one contribution among some thirty while placement noise goes into the recorded angle undivided. It is one step of the sweep’s grid, and it is stated here rather than called a finding.

The counterfactual, and the cost of computing it honestly #

The basin statistic depends entirely on what “with the noise taken away” means, and there are two readings of it that give completely different answers.

The wrong reading is: run the rule from the beginning with no noise, and compare the two sequences node by node. That was tried, and it fails for a reason worth recording — the noisy run and the clean one are not on the same trajectory after the first few nodes, because the noise changed a placement and every subsequent placement responds to it. Comparing them measures how far two different stems have drifted apart, which is a real quantity and is not this one.

The right reading is local: given the history this run actually had, and the neighbours actually present, where would this node have gone without the noise on this step? That isolates the current step’s disturbance from everything that came before, and it is the question the word “basin” is asking.

Computing it is nearly free for two of the three kinds. Under placement noise the counterfactual is the argmin before the displacement, which the rule already has. Under field noise it is the argmin before the profile is perturbed, which is one line taken a few lines earlier than it otherwise would be.

Under a jostle it is not free. The profile itself was computed against displaced neighbours, so the counterfactual needs a second sweep — over the same candidate positions, against x instead of seen. That doubles the cost of a jostled run, and the library does it only for that kind, and says so where it does it. A statistic that had been made cheap by approximating the counterfactual would have been a statistic about the approximation.

What this costs #

The uncomfortable consequence is that the phase has added a third thing that cannot be measured.

Matched at a recorded scatter of three quarters of a degree, the three kinds read the same. Their amplitudes differ — 0.79° of placement noise, 0.011 of field noise as a fraction of the rule’s barrier, 0.70° of jostle — but those are three quantities in different units, and none of them is what a plant hands over. What a plant hands over is a sequence of angles, and their spread, and the counted pair, and on all of those the three are indistinguishable.

That is the second phase running to end at this wall. The previous one wrote: everything a finished pattern records about its noise is shared between the two; everything that distinguishes them happened before the pattern existed. With three kinds the sentence is unchanged except for the number.

What a biologist would object to #

Three objections are worth answering, because each of them is right about something.

“A growing apex does not displace primordia randomly; it carries them outward.” True, and the model already has that: the whole geometry is a cylinder whose rise falls, which is elongation, and on a disc the elements drift outward exponentially. What a jostle adds is the irregular part — the departure from uniform growth — and irregular is what a random displacement models. A systematic outward carry is not noise and is not what this is for.

“The displacement should depend on the tissue, not on a number.” Also true, and it is the honest limit of the model. A real apex’s deformation field is structured: it is larger in some sectors, correlated between neighbouring cells, and coupled to the primordia themselves, which are stiffer than the tissue around them. A single amplitude with independent draws per node has none of that. What it does have is the property being tested — that the neighbours are not where they were put — and adding structure to the deformation would change the numbers without changing which side of the choice the disturbance arrives on.

“Primordia are not points.” They are not, and this is the deepest of the three. A primordium is a bulge with an extent comparable to the spacing between primordia, so the notion of it being displaced by a fifth of a degree is already coarse. Every model in this collection treats a primordium as a point, and every figure inherits that; it is one of the things the site’s own honest-limits notes say repeatedly. A jostle does not make it worse, and it does not fix it either.

None of the three changes the finding, which is a statement about where in the rule’s own sequence of steps a disturbance enters. That is a question about the model’s structure, and it has an answer even if every one of the model’s physical simplifications is granted.

Where the argument goes next #

Two directions come out of this, and the phase takes both.

The first is that “before the choice” and “after it” is a real distinction with a measurable consequence inside the model, and that a jostle and a field perturbation are on the same side of it while placement noise is alone on the other. That is worth pushing on, because it turns a modelling taxonomy into a claim about what kind of thing the disturbance is.

The second is that a spread throws away the order. The angles come off a stem in a sequence, and nothing so far has looked at the sequence — only at how wide it is. The previous phase’s parting prediction was that the order carries what the spread does not, and it turns out to, in a way that inverts the prediction and makes the one test in this collection a single plant could settle.