The fragility belonged to the window

The previous phase’s prediction about a physical cut-off had three parts, and this essay is about the one that failed. It failed usefully, which is the only kind of failure worth an essay, and what it says is more interesting than what it would have said had it held.

The prediction #

The reasoning was this. An inverse first-power rule makes no lattice when it can see far enough. Cut its neighbourhood and it makes a clean one. That pattern is therefore held together by the cut rather than by the arrangement — the elements are not really finding a stable configuration, they are being prevented from noticing that no stable configuration exists — and a pattern held together that way should be brittle. Nudge it and it should fall over, where a genuinely short-ranged rule’s pattern would not.

The previous phase measured exactly that of the truncated rule and found it. Under a fifth of a degree of placement noise — a tenth of the scatter a lattice is measured as tolerating — the truncated inverse first-power rule keeps its lattice in well under half of its runs, while the inverse-cube rule at the same window keeps every one. The manufactured pattern was fragile and the real one was not, and the contrast was the strongest evidence that the first was an artefact.

So: state the neighbourhood as a falloff, run the same nudge, expect the same fragility.

What happened instead #

The truncated loop keeps 37.5% of its runs, at 26° of scatter across the ensemble — the previous phase’s number, reproduced.

The exponential cut-off at the same range keeps 100%, at 0.63° of scatter, on the Fibonacci branch in every run, settling at 137.59–137.61°.

The inverse-cube control keeps 100%, at 0.79°.

The stated cut-off’s lattice is not merely more robust than the truncated one. It is as robust as the rule this site has treated as the real one for four phases, and by a hair less scattered. Nothing in the prediction survives except the first half: the two lattices are the same lattice, and only one of them falls over.

What that means about the truncation #

The natural reading is the right one and it is worth spelling out, because it changes what the previous phase’s headline finding was about.

A recency window is not a crude version of a short-ranged rule. It is a rule whose neighbour set is chosen by an arbitrary criterion, and the pattern it produces sits in a landscape that is partly the arrangement and partly the boundary. Nudge a node and the window slides differently, a different set of neighbours is in the sum, and the landscape the next node sees has changed in a way that has nothing to do with where anything is. That is what makes it fragile: the rule is being asked to find a stable arrangement in a landscape that keeps being redrawn by an accident.

A falloff has no such property. The weight a neighbour carries is a function of where it is, so nudging a node changes the landscape by exactly the amount the nudge is worth. There is nothing to slide, nothing to redraw, and the rule spends its time doing what the rule is supposed to do.

So the previous phase’s truncation manufactures a pattern was right, and its implicit and the pattern is a fake was not. What truncation manufactured was a pattern that is real given a short-ranged rule, produced by machinery that also manufactured its own instability. Take the machinery out and the pattern stays and the instability goes.

The mechanism, in one node’s worth of detail #

It is worth following a single placement through both rules, because the difference is small enough to state exactly and it is easy to lose in the summary.

Take node i, about to be placed. Under either rule it sweeps a candidate azimuth around the cylinder and computes the total repulsion at each position. Now suppose node i−1 was nudged by a fifth of a degree.

Under the falloff, node i−1 is at a slightly different place and carries a slightly different weight, because the weight is a smooth function of where it is. Every other neighbour is untouched. The profile changes by an amount proportional to the nudge, and the argmin moves by a comparable amount. The rule then does what it always does: it puts the node between its neighbours, and since one neighbour moved slightly, the gap either side of the new node is slightly different, and the next node corrects for it. Nothing accumulates.

Under the recency window the same thing happens, plus one more. The window contains the most recent $\lceil 3/\sqrt{h}\rceil$ nodes, a count that changes as the rise falls, so at intervals a node drops out of the sum entirely — and which node drops out, and when, depends on the whole history. A nudge changes the history; a changed history changes which nodes are in the window at which step; and a node leaving the window is a finite change to the profile, not a small one. The perturbation therefore has two channels: a proportional one through position, and a lumpy one through membership.

The second channel is what the fragility is. It is not present in the falloff rule at all, because there is no membership — a node’s contribution goes smoothly to nothing as it falls behind, and the window that bounds the sum sits four widths out where the weight is under two per cent, doing nothing.

That is also why the effect could not have been anticipated from the size of the window. A larger window has fewer membership events per node and each one is smaller; a smaller window has more and larger. The quantity that matters is not how many neighbours are kept but whether the number is allowed to change, and no parameter of the model names it.

Why this makes the result harder to dismiss rather than easier #

There is a temptation to read the above as a rescue — the manufactured lattice was not an artefact after all, so nothing was wrong. That reading is not available, and the reason is worth being exact about.

The lattice is still conditional on a neighbourhood nobody has measured. At a half-weight radius of 1.5 spacings the inverse first-power rule makes a clean 8/13; past about 3.75 it makes nothing. Both of those are statements about a plant that could be false. What has changed is that the condition is now stated, and that the pattern it produces is not distinguishable from the pattern the collection’s other figures show.

That is worse for anybody hoping to tell the two rules apart, not better. Before, there was a clean discriminator: nudge the pattern and see whether it survives. Now there is not. A short-ranged inverse first-power rule and an inverse-cube rule produce lattices with the same counts, the same divergence to a few hundredths of a degree, and the same tolerance to noise. The falloff exponent — the parameter whose value came from a ferrofluid experiment rather than from a plant — has become unobservable at the level of the finished pattern, provided the neighbourhood is short.

The measurement, and what makes it one #

Three ensembles, six runs each, differing in one thing. Same rule, same rise range, same rate of decline, same number of nodes, same seed angle, same noise amplitude, same seeds. What differs between the first two rows is whether the neighbourhood is a recency window or a falloff; between the second and third, the exponent.

That matters because “this pattern is fragile” is a claim that is easy to produce accidentally. A run that fails for any reason — a counter that cannot find a pair, a transient that has not settled, a rise that fell below what the order limit can count — will register as incoherent, and an ensemble whose failures come from the machinery will report fragility that belongs to the instrument. Holding everything but one line constant is what makes the difference between two rows attributable.

The classification is also worth stating. A run counts as coherent when its divergence scatter over the last quarter is under six degrees, and that number is not a judgement: the measured spreads are bimodal with an empty decade between the modes — intact runs sit at 0.4–2° and destroyed ones at 25–45° — and the library requires that gap to exist before it classifies anything by it. Six is the middle of an empty region on a logarithmic scale, which is the only kind of threshold that does not need defending.

What a shorter range would have shown #

One might ask whether the stated cut-off is robust because it was run at a comfortable range — 1.5 half-weights, well inside the boundary — while the truncation’s 3/√h is nearer the edge of what it can support. That would make the comparison unfair.

It does not hold up. The truncated window at 3/√h nodes is, at the rises these runs end at, something like three rows of neighbours: it is the narrow end of what the loop was ever run with, and widening it to 12/√h is what destroys the pattern entirely. The stated cut-off at 1.5 half-weights is comparable in how much of the lattice it keeps and is if anything more generous. Both are inside their own pattern-forming regions and neither is at an edge.

What is true is that a stated cut-off near its own boundary would be fragile, and that is a different and unsurprising claim: any pattern near a critical point is sensitive, whatever produced it. The interesting comparison is between two rules both comfortably inside their regions, and there one is robust and the other is not.

What is left to tell the two rules apart #

If the finished pattern cannot separate a short-ranged inverse first-power rule from an inverse-cube one, it is worth asking what could.

Not the counts. Both walk the Fibonacci ladder and both land on 8/13 at this rise.

Not the divergence. 137.58° against 137.63°, which is inside the scatter of either.

Not the tolerance. Both keep every run under the nudge, at scatters a tenth of a degree apart.

Not the transitions, as far as this thread has looked: both climb the ladder as the rise falls, and the ladder is a property of the geometry rather than of the rule.

What is left is the neighbourhood itself — a measurement on the apex rather than on the pattern. How far does a forming primordium’s influence reach, and how does it decline? Those two numbers separate the rules completely, because an inverse cube is short-ranged with no cut-off at all, and an inverse first power is not; the first predicts that a wide neighbourhood changes nothing and the second predicts that it destroys the pattern.

That is a satisfying place for a model comparison to end up and an uncomfortable one for a collection that works from computation. The distinguishing experiment is not on the finished plant. It is on the apex, before the pattern exists, and it is a measurement of a chemical reach rather than of an angle.

The general form #

The transferable sentence is short and this collection has now met a version of it three times.

An arbitrary boundary does not merely add error; it adds a different dynamics. A model with an unstated cut has two mechanisms in it — the one that was written down and the one the boundary supplies — and the second can produce behaviour, and instability, that is read as a property of the first. The previous phase read the instability as evidence that the pattern was not real. It was evidence that the boundary was not real, and those are different, and no amount of care about the pattern would have separated them.

The way it was separated was to write the boundary down as a claim and run the same test again. That is the whole method: when a result depends on something unstated, state it and repeat the measurement — not to confirm the result, but because the statement usually changes it.

A note on how the prediction was made #

The prediction was a good one and it was wrong, and those two facts sit together comfortably enough to be worth a paragraph.

It was made from a mechanism — a pattern propped up by an arbitrary boundary should fall over when nudged — rather than from an analogy or a hunch, and the mechanism was correct as far as it went. Patterns propped up by arbitrary boundaries do fall over. What the prediction got wrong was the assumption that the prop and the short range were the same thing, so that removing the arbitrariness while keeping the range would keep the fragility too.

That assumption was invisible because nothing in the previous phase’s vocabulary distinguished them. Truncation, cut-off, window, neighbourhood and range were used interchangeably, which was harmless while there was only one way of cutting and became a confusion the moment there were two. The word that was missing is the one this thread had to invent: a cut can be in recency or in distance, and until those had separate names the question could not be put.

The general form is one worth carrying: a prediction that cannot be wrong in an interesting way is usually a prediction about a word. This one turned out to be about two words that had been one, and the useful part of the phase was noticing that rather than measuring what happened next.

What is left of the three predictions #

The scoreboard, since three phases of this site now hang off it:

A physical cut-off reproduces the manufactured lattice. Confirmed — 137.58° against 137.62°, same counts, scatters within a tenth of a degree.

The width becomes a hypothesis with a critical value. Confirmed, and it turned out to be two hypotheses, because the falloff shape moves the critical width by half and the contrast is what is actually invariant.

It is as fragile to a nudge. Refuted. The fragility was the window’s, and removing the window removed it entirely.

Two out of three, with the third failing towards the more interesting answer. That is a better yield than the previous phase’s directions managed — three of its four came out against the prediction — and it leaves the thread in an awkward place: the rule’s neighbourhood is now a stated claim about a plant, and the pattern it produces is indistinguishable from the pattern produced by the rule it was supposed to be an alternative to.

The thread that follows takes the same problem from the other end. Instead of asking what the rule’s neighbourhood is, it asks what happens when the neighbours themselves will not hold still — which is the ordinary condition of a growing organ, and which turns out to be a third kind of noise with a signature the other two do not have.