The boundary belongs to the pattern

The previous phase measured how much divergence scatter a lattice survives, in two independent ways, and got 2.00° and 1.68°. It recorded the agreement carefully and declined to make much of it — two numbers agreeing to twenty per cent is a scale rather than a constant, and two is a small sample of ways to break something.

There are three ways now. The third is a different kind of disturbance in a different unit, and it lands in the same place.

The three, and why their amplitudes cannot be compared #

Each kind of noise has an amplitude, and the three amplitudes are three different quantities.

Placement noise is a displacement of the node after the rule has chosen, measured in degrees of azimuth. That is what a ruler would report, which is why it was the first kind implemented.

Jostle is a displacement of the neighbours, also in degrees of azimuth, so it is at least in the same unit as placement noise even though it acts through a completely different channel.

Field noise is a perturbation of the energy profile, and its amplitude is a fraction of the barrier the rule is working against — the gap between the median energy around the circle and the least. It has to be scale-free like that, because the energy at a candidate position sitting on top of an existing node is unbounded, so a fraction of the range would be a fraction of an accident of the sample grid.

Three units: degrees, degrees through a different channel, and a dimensionless fraction. Asking which is larger is meaningless. What can be asked is what each one produces, and that is where they become comparable.

The measurement #

For each kind, sweep the amplitude, and at each amplitude run an ensemble and ask what share of the runs still have a lattice at the top of the stem. Take the largest amplitude at which the share is one — every run intact — and record the divergence scatter there.

Placement noise is intact to 0.8° of displacement, at 1.42° of scatter. Field noise is intact to a barrier fraction of 0.015, at 1.64°. A jostle is intact to 1.0° of displacement, at 1.72°. The three scatters agree to nineteen per cent, which is the width of a single step of the amplitude grid.

Five measurements now, counting the previous phase’s two: 2.00, 1.68, 1.42, 1.64, 1.72. They are not all measurements of the same thing under the same conditions — the earlier pair used a finer amplitude grid and a different classification cut — and the sensible summary is that the boundary sits somewhere between one and a half and two degrees, and does not depend on which of three mechanisms is doing the breaking.

Why a share and not a mean #

The statistic is the share of runs that keep a lattice, and the scatter is averaged over the intact runs only. Both of those are deliberate, and the second is a repair of a mistake this site made and recorded.

Comparing the two noise kinds by ensemble mean scatter put them five degrees apart, which looked like a real difference between them. It was not. At the amplitude where half the runs are destroyed, the mean is an average of 1.9° and 30° and describes neither population — it is a measurement of where the threshold is, dressed as a measurement of a quantity. Restricting to intact runs put the two kinds a third of a degree apart.

The general form is worth having: a mean over a bimodal population is a measurement of the mixing proportion. If the two modes are “intact” and “destroyed”, the mean moves as the share moves, and reporting it as a property of the disturbance attributes a threshold’s position to the wrong quantity.

The classification itself is what makes that possible. A run counts as coherent when its divergence scatter is under six degrees, and six is not a judgement: the measured spreads are bimodal with an empty decade between the modes — intact runs at 0.4–2° and destroyed ones at 25–45° — and the library asserts that the gap is there before classifying anything by it. A cut in the middle of an empty region on a log scale is the only kind that does not need defending.

The conversion rates, since they are the interesting part of what differs #

The amplitudes cannot be compared directly, but the rate at which each kind converts amplitude into scatter can be, and it is where the three kinds’ differences actually live.

Placement noise is the most efficient converter. It goes into the recorded angle undivided: a displacement of node i by an angle ε shows up in the divergence before it and the divergence after it, at full size. Nothing intervenes.

A jostle is less efficient. The displacement of a neighbour reaches the recorded angle only through its effect on where the next node is placed, and that node is responding to some thirty neighbours at once. One of them moving by ε moves the argmin by rather less than ε — the neighbourhood averages it down — so a jostle needs a larger amplitude to produce the same recorded scatter. Measured, it needs about a quarter more.

Field noise is in its own units and the conversion is not a ratio of like quantities at all, but the same structure applies: a perturbation of the profile moves the argmin by an amount set by how steep the profile is near its minimum, which is a property of the arrangement rather than of the perturbation.

That quarter is the whole of the difference between the two displacement kinds as far as tolerance goes, and it is one step of the amplitude grid, which is why this essay does not make a finding of it. It is worth stating because it is the correct reading of a fact that could easily be read the other way: a jostle is not gentler in any interesting sense, it is merely diluted on the way in. Per unit of scatter produced — which is the only comparison that means anything — it is no gentler at all.

There is a small prediction hidden in that dilution, and it is worth recording as untested. If the averaging over neighbours is what dilutes a jostle, then the dilution should depend on how many neighbours the rule sees — so a short-ranged rule, with fewer effective neighbours, should convert jostle into scatter more efficiently than a long-ranged one. Nothing here tests it, because everything here is at one exponent and one neighbourhood. It is one line of sweep away for a phase that wants it.

What “the boundary belongs to the pattern” means #

The three kinds differ in everything except where they break the lattice. They enter the rule at different steps; two can change which minimum is chosen and one cannot; their amplitudes are in different units and the conversion between them is not a constant. If the tolerance were a property of the disturbance, three such different disturbances would have no reason to agree.

The reading is that the pattern has a failure threshold of its own, and each kind of noise reaches it at whatever amplitude gets it there.

The mechanism is the rule’s self-correction. A lattice is an arrangement in which each element sits between its neighbours, and the rule restores that: a node placed a little to one side leaves a gap the next node falls into. Measured, a twentieth of a degree injected comes back as about half a degree of residual and stays there — the gain is bounded, and four times the amplitude does not give four times the residual.

A pattern maintained by a restoring process fails when the disturbance per step exceeds what the following steps can pull back. That is a statement about the restoring process — how strongly the arrangement pulls, which is set by the rule and the geometry — and not about what is doing the disturbing. Hence one threshold, three mechanisms.

What it is worth to somebody measuring a plant #

Modestly good news and a larger piece of bad news.

The good news: a scatter measurement has a stopping rule. A stem whose divergence angles scatter by much more than two degrees is not a lattice in the sense this collection’s machinery uses, and every count taken from it, every recovered angle, every inferred rise, is a statement about something that does not satisfy the assumptions. That is a useful thing for a survey to be able to say before it starts analysing, and it does not depend on knowing anything about where the noise came from.

The bad news is the one this thread keeps arriving at: the boundary being universal means the scatter carries no information about the mechanism. A stem at one and a half degrees is a stem on the edge of losing its pattern, and that is all it is. A stem at three quarters of a degree is comfortable, and that is all it is. Nothing in the number says whether its primordia were displaced after being specified, or placed in slightly wrong positions, or specified correctly against neighbours that had drifted.

What a real stem’s scatter looks like, as far as anybody can say #

It is fair to ask where a plant sits on this axis, and the honest answer is that this collection does not know and has said so for four phases.

Published divergence measurements exist, and they are not usually reported as scatter. A paper gives a mean angle — 137.5°, or a number near it — and often a standard deviation over specimens rather than along a stem, which is a different quantity: variation between plants includes whatever differs between plants, and the number this essay is about is variation between consecutive internodes of one plant. The two are conflated often enough that a reader has to check.

What can be said from the model’s side is what each regime would look like. Under half a degree of scatter, a stem’s angles cluster tightly and its counted pair is unambiguous at every band. Between one and two degrees, the pattern is intact and approaching its limit; a blind counter still returns a clean pair, and the divergence recovered from that pair is still narrow. Past two, the counting machinery starts refusing — not returning a wrong answer, refusing, which is the behaviour this site’s recovery routines were built to have — because no lattice is consistent with the offsets it finds.

That refusal is worth more than it sounds. A survey that measured a stem too noisy to be a lattice would, with this machinery, be told so rather than handed a plausible number. The wrong field’s essays make the same point about counts: the value of a measurement is entirely conditional on the thing measured satisfying the assumptions, and machinery that refuses is what turns a conditional into a check.

What is genuinely a constant here and what is not #

Worth separating, because a threshold that turns up repeatedly invites being read as a law.

Not a constant of nature. It is a property of this rule, at this exponent, on this geometry, at these rises. Nothing has been measured on a plant. The number would move if the falloff moved, if the rate moved, or if the counting machinery classified differently.

Not obviously a constant across the ladder either. Every measurement here is at the same rise range, ending at 0.004. The previous phase’s tolerance-by-depth work found the tolerance tightens as the pattern gets finer, which is what one would expect — a pattern with more, closer neighbours has less room — so the single number quoted is the tolerance at one place on the ladder rather than everywhere.

What does look robust is the independence from the kind of disturbance, which is the claim this essay makes and the only one it makes. Three mechanisms, one threshold, at a fixed place on the ladder. That is a statement about the pattern being maintained rather than about the pattern being a particular size.

The ensemble, and why five runs is not laziness #

Every number above is a share over an ensemble, and the ensembles are small — three to five runs a point. It is worth saying what they are for, because a reader used to statistics will reach for a sample-size objection that does not apply.

The runs are not samples from a population. They are the same deterministic rule with different noise seeds, which is to say they are the same experiment repeated with a different draw of the thing being studied. What varies between them is only the noise, and the quantity being estimated — the share that keep a lattice — is bimodal and sharply thresholded, so the share goes from one to zero over about one step of the amplitude grid. Estimating the position of a step function to within one grid step does not need many runs; it needs the grid to bracket the step, which is asserted.

What would need many runs is estimating a rare event, and the phase has one of those: the basin-change share is one or two in a thousand, and it is measured over three hundred placements per run rather than over runs. The unit of replication is the placement, not the stem, and there are a thousand of them.

The one thing a small ensemble genuinely cannot do is separate two thresholds that sit within a grid step of each other. That is exactly why this essay declines to make a finding of the jostle’s quarter-of-a-step advantage, and it is the honest limit of the design rather than an oversight in it.

The separation the previous phase learned the hard way is also in force here: members differ by their noise seed and not by their starting angle. Drawing a different starting divergence per member does not separate anything — at a coarse rise the only arrangement is 1/2, every history collapses onto it within four nodes, and five “independent” runs return the same divergence to three decimals. A share of five identical runs is a share of one, and it is reported as five.

The shape of the finding #

There is a family of results this belongs to, and naming it is useful.

A system with a restoring force has a tolerance set by the restoring force. That is almost a tautology, and the reason it is worth measuring anyway is that “almost a tautology” is where the interesting failures hide: it would be perfectly possible for a rule to have a restoring behaviour that worked against one channel of disturbance and not another. A rule whose self-correction acted on positions but not on the field it reads would tolerate placement noise well and field noise badly, and the two boundaries would be nowhere near each other.

They are near each other, which says the correction acts on the arrangement rather than on any particular route into it. That is a real property of the model, it is the kind of property that could have come out otherwise, and measuring it is the difference between a tautology and a result.

It also sets up the thread’s last question. If the scatter is blind to the mechanism, and the tolerance is blind to it too, then everything a finished stem offers as a summary is blind. What is left is not a summary at all — the order in which the angles arrived — and that turns out to see the difference immediately.