Which minimum was chosen

There is a sentence in the previous phase’s notes that reads like a summary and is actually a claim: only noise arriving before the choice can change what is chosen. It was written as a way of describing what two kinds of noise had done, and it had no measurement behind it — no quantity was called “what is chosen”, and nothing counted how often it changed.

This essay is that quantity.

What an argmin is, and why it has two failure modes #

The rule does not compute a number and report it. It sweeps a candidate position around the apex, computes a profile of the repulsion at each position, and returns the location of the least value. That is an argmin, and an argmin is a different kind of object from a value.

A value degrades smoothly. Perturb the inputs a little and the output moves a little, in proportion. That is what most measurements do and it is what most intuitions about noise assume.

An argmin does not. Perturb the profile a little and, almost always, the reported location moves a little — the minimum shifts within its own well. But sometimes the perturbation makes a different well the lowest, and then the reported location jumps by a whole spacing or more. The output is a step function of the input almost everywhere, with a set of measure zero where it jumps.

So a rule of this shape has two distinguishable failure modes under noise, and they are not degrees of the same thing:

• The answer moves within its well. The primordium is a little off where it should be. The pattern is blurred.
• A different well wins. The primordium is somewhere else entirely. The arrangement has been changed, not blurred.

The second is what a change of branch is made of. A stem that walks from the Lucas ladder to the Fibonacci one does it by placing nodes in wells the old arrangement did not have, one at a time. If noise can do that, it can restructure a pattern; if it can only do the first, it can only make one untidy.

Measuring it #

Run the rule twice at each step: once as it is, and once with the noise removed and everything else — the history, the rise, the actual neighbours — left exactly as it is. The second answer is what this node would have done had this one step been quiet. Compare the two.

A node has 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 over a run, and a fixed threshold would be measuring the rise rather than the disturbance.

Why half a spacing, and not something else #

The threshold deserves an argument rather than an assertion, because a share statistic is only as good as the line it counts across.

Too small a threshold and every placement counts: the counterfactual and the real answer differ by something at every step under any noise at all, so a threshold near zero would report a share of one for all three kinds and measure nothing.

Too large and nothing counts: a threshold of two spacings would ignore genuine moves into the adjacent well, which is precisely the event being looked for.

Half a local spacing is the natural line because it is where the wells are. The profile a lattice hands the rule has minima roughly a spacing apart — that is what having a spacing means — so a displacement of less than half a spacing leaves the node nearer its own minimum than any other, and a displacement of more than half puts it nearer a different one. The threshold is a statement about the geometry of the landscape rather than a tuning parameter.

The reason it is stated in local spacings rather than in degrees is the one this site keeps running into. A run starts at a rise of 0.4 and ends at 0.004, so the spacing falls by a factor of ten from top to bottom; a fixed angular threshold would count nothing near the top and everything near the bottom, and the resulting share would be a measurement of where in the run the nodes are. The site’s earlier essays record the same lesson about counting bands, transition ladders and noise amplitudes: on a stem whose rise declines, a threshold in absolute units is a threshold that means something different at each end.

The structural zero #

Placement noise changes no basin. Not “changes few” — changes none, at 0.2°, at 1°, at 2° and at 4°, and the last two of those have already destroyed the pattern.

This is a proof by construction rather than a measurement, and that is exactly why it is in the library as an assertion. The displacement is applied after the argmin has been taken, so the node is by definition in the well the rule selected, however far it has since been pushed. There is no configuration of the model in which the answer could be anything else.

An assertion that cannot fail is usually worthless. This one is worth having for the opposite reason: it is the check that the counterfactual is being computed in the right place. If a run ever reported a placement-noise basin change, it would mean the comparison was against something other than the pre-displacement argmin — and every jostle number in the phase would be an artefact of the same bug. The zero is a self-test disguised as a result.

It also has a physical reading, which is why it is not merely a tautology. It says that a disturbance to a primordium after it is specified cannot change the pattern’s arrangement. It can move a primordium, blur the angles, and — at large enough amplitude — destroy the lattice by making the angles meaningless. What it cannot do is put a primordium into a slot the arrangement did not have. That is a statement about development with content: whatever jostling happens to a primordium after its fate is decided is cosmetic as far as the pattern’s structure goes.

The small positive numbers #

The other two kinds change one or two placements in a thousand while the lattice is intact. Three things about that number are worth saying.

It is small on purpose. The amplitudes at which it is measured are amplitudes that leave a working pattern — under a degree of scatter, well inside the boundary where a lattice gives way. Push harder and the share rises steeply, but then the pattern is coming apart and a basin change in a destroyed run means nothing: a sequence with forty degrees of scatter has no wells to be in.

It is not zero, and that is the claim. The library’s threshold separates a small positive number from a structural zero rather than separating two small numbers, and the assertion is written that way: it requires an amplitude at which every run is coherent and at which some placements have nonetheless gone elsewhere.

One in a thousand is not nothing over the length of a stem. A run here places some three hundred nodes; a real shoot places thousands over a season. At one in a thousand, a plant makes a few of these a year — and each one is a node in a slot the arrangement did not have, which the pattern then has to accommodate or correct.

What the pattern does about it #

That last point invites the obvious question, and the answer connects this thread to the third one.

The rule is self-correcting. A node placed in the wrong well leaves a gap that the next node falls into, and the arrangement pulls itself straight over the following few placements. Measured directly: injecting a twentieth of a degree of noise into a run and comparing against the same run without it gives a residual of about half a degree, and injecting four times as much gives less than twice as much. Nothing runs away. Two unrelated angle sequences would differ by 104°.

So a basin change is not a permanent restructuring. It is a local error that the rule repairs, and the repair is why a share of one in a thousand does not accumulate into a different pattern over three hundred nodes. This is the same fact as the positive correlation between consecutive divergences that the sequence thread turns on, seen from a different side.

It is also why the previous phase found what it found about branches: noise does not substitute for a slow rate. A branch change requires the arrangement to be walked from one ladder to another, which is many basin changes in a consistent direction, and a self-correcting rule undoes them faster than a random disturbance makes them.

Field noise and a jostle, which the statistic does not separate #

The basin count draws one line and it is worth being clear which. It puts a jostle and a field perturbation on the same side and placement noise on the other, and it says nothing at all about the difference between the first two.

That is not a defect of the measurement; it is the measurement being exactly as informative as its construction allows. Both kinds perturb the profile before the argmin, so both can move the winner from one well to another, and the counterfactual sees only that the winner moved. Where the disturbance lives — in the field the rule is reading, or in the neighbours that generate the field — leaves no mark in the comparison.

Whether those two are really different is a fair question. They are, in a way a biologist would care about: a field perturbation is a disturbance to the signalling chemistry, and a jostle is a disturbance to the tissue. One is a claim about a morphogen and the other about mechanics. They also differ in the model — a field perturbation is redrawn at every step, and a jostle is drawn once per node and then persists, so a jostled neighbour goes on being displaced for as long as it is in the neighbourhood.

That persistence is the thing that ought to leave a signature somewhere. It does not leave one here, and the sequence thread finds that it does not leave one there either: the two kinds are within the sampling band of each other on the autocorrelation as well. Two independent instruments draw the same line and neither draws a second one, which is worth recording as a limit rather than hoping a third instrument would do better.

Why the distinction is not visible from outside #

Everything above happens inside the model. On the finished stem the three kinds are matched: at the same recorded scatter their spreads agree within a fifth, their counted pairs are the same, and their lattices give way at the same place.

The counterfactual is not something a plant can be asked for. It requires running the same apex twice, identical in every respect but one step, and no experiment does that. So the cleanest result in this thread is a result about a model, and the question of whether a real disturbance arrives before or after the choice is untouched by it.

That is the honest position and it is not a comfortable one. A model that can distinguish two hypotheses internally, and whose distinguishing quantity has no observable counterpart, has produced a taxonomy rather than a test.

What the argmin structure says about the whole model #

There is a wider point here that this collection has been circling for four phases, and the basin statistic makes it sayable.

This site’s central model is an optimisation. Every essay about it is, underneath, an essay about where a function is least, and almost every surprise the model has produced has come from properties of argmins rather than from properties of plants. The packing essays found that three criteria pick three different winners, because which function is being minimised decides everything and no criterion is privileged. The exponent essays found that what matters is contrast rather than convergence, because a term that is the same at every candidate position moves no minimum however large it is. The truncation essay found that an arbitrary boundary manufactures a pattern, because the boundary enters the landscape rather than the answer. Now the noise essays find that where a disturbance enters relative to the argmin decides what it can do.

Four findings, one shape. A model built on “put it where the function is least” inherits the pathologies of argmins, and those pathologies are invisible to anybody thinking of the model as being about a function. The value of the energy at the chosen position is not the subject; nothing on this site plots it. What is the subject is which position won, and the answer to that is unstable in exactly the places where a well-behaved value would be stable.

It is a good reason to be suspicious of the model in general, and it is also the reason the model is worth having: a rule that merely reported a number would have none of this structure, and would also produce no lattice.

What would make it a test #

Two routes, and the phase takes the second.

The first is to find an observable consequence of basin changes that accumulates. There is not obviously one, precisely because the rule self-corrects: the errors are repaired, and what is repaired leaves no trace in the finished arrangement.

The second is to stop looking at the arrangement and look at the sequence. The angles come off a stem in an order, and every measurement in this collection so far has thrown the order away — a scatter is a summary that is invariant to shuffling. If a disturbance that changes the question leaves a different temporal signature from one that changes the answer, the order would carry what the spread cannot.

It does, and by a margin that is not close. The essay after next is that measurement.

The refusal #

The library refuses a noise whose kind it does not recognise, by name, and the reason is the ordinary one on this site: a misspelling should produce an error rather than a figure.

Ask for thermal noise and the rule stops and says which three kinds it knows. Without that, the string falls through every branch of the implementation, no noise is applied at all, and the run comes back as a clean noiseless lattice under a caption describing an experiment about noise. That is the exact shape of failure this site’s gates exist for — a plausible figure with a wrong caption, passing everything — and it costs one line to make impossible.