The comb was never the rule

Five rounds of this site have been building one instrument. It reads a list of divergence angles — no coordinates, no rise, no model — and returns first a number, then a pair, then a claim about mechanism. The last of those has just been withdrawn, and this essay is the audit that goes with a withdrawal: what each round of work claimed, what still stands, where the mistake was general rather than local, and what would have to happen for the claim to be made again.

The ledger #

earlier in this collection: the sequence has a memory. The correlation between one divergence and the next is not zero — around −0.6 at a fixed rise — and it comes from the rule’s self-correction: an organ placed to one side of its minimum leaves a gap that pulls the next one back. Stands. It is a statement about a sequence and it was never offered as evidence about mechanism.

Then: the spectrum names the family. Autocorrelate the sequence out to thirty lags and the peaks are periodic at the smaller parastichy number. Sixty internodes are enough. Stands, and it is the workhorse: a number that a botanist’s list of angles gives up without any coordinate being measured.

Then: the angles give the pair. The significant lags are two arithmetic progressions, the second offset from the first by the difference of the pair, so the reading is two integers rather than one. Two hundred and fifty internodes. Stands, with a caveat the work here adds: a disturbance that repeats can forge the first comb, and only reading several stems catches it.

Also then: the instrument has a ceiling one rung above where it works. The window needs three teeth and has to stop short of the larger number’s comb, and at 13/21 there was said to be no window that satisfies both. Withdrawn, and for a reason with nothing to do with this thread: the band is [39, 63) and never empty, and what was actually stopping the reading was the number of azimuths the rule samples. Two withdrawals in one round of work from one round’s open questions is worth noting as a pattern — both were claims made in a plan file at the end of a round of work, and neither was measured before it was written.

Then: a comb is evidence of a rule. Build the same lattice kinematically with independent errors and there is no comb, so the comb belongs to the process rather than the form. Withdrawn. Errors transmitted between contact neighbours reproduce both combs and the pair on an arrangement with no rule in it.

And now: the comb ratio constrains the transport. The second comb is 0.65 of the main one on the rule’s stems and 1.30 on a distance-weighted transport, so the ratio distinguishes them — unless the transport’s coupling split is left free. Stands, conditionally, and the condition is stated in the claim.

The shape of the mistake #

The control that failed was not badly executed. It was exactly what it said it was: the same lattice, the same divergence, the same rise, the same parastichy pair, the same scatter, built without a rule. Every one of those was checked.

What it varied was whether there was a rule. What it held fixed, without saying so, was how the errors relate to each other — and it held it at the one value that makes the comb vanish.

A control is a claim of the form this observable requires that ingredient, and it is only as strong as the space of alternatives it searches. Holding one thing fixed while varying another tests whether the varied thing matters at that value of the fixed thing. That earlier work’s control showed that a rule is sufficient for a comb and that independent errors without a rule are not. It never showed that a rule is necessary, because it never varied the ingredient that turned out to supply the comb.

The distinction between sufficient and necessary is the whole of it, and it is worth saying that the earlier claim would have been unimpeachable if it had been written with the first word. A placement rule produces a comb is measured, it stands, and it is useful. A comb requires a placement rule is the sentence that was written, and it is a claim about everything that could produce a comb rather than about the one thing that was built.

A control’s strength is the fraction of the space it visits #

There is an arithmetic form of that, and it turns the lesson from a caution into a thing to do before building a control.

The alternatives here have two dimensions. Is there a rule? — two values. What structure do the errors have? — at least four: independent, remembered, periodic, transmitted between contacts. That is eight cells, and a control’s job is to establish that an observable requires the ingredient it varies.

That earlier control visited two of the eight: rule with independent errors, and no rule with independent errors. It varied the first dimension at one value of the second, which is exactly what it claimed to do — and the cell that eventually broke it, no rule with transmitted errors, was never built.

So the control was a quarter of one dimension of an eight-cell space, and its conclusion was stated over the whole space. Nothing about that is careless; the second dimension had no name at the time, and a dimension nobody has named is a dimension nobody enumerates.

Which is the procedure the audit produces. Before building a control, write down the dimensions of the alternative space and mark which cells the control visits. A control that visits two of eight cells is a real measurement and it supports a claim about two cells. Writing the fraction down does not require knowing what the unnamed dimension is; it requires only asking whether there might be one, and recording the answer as a bound rather than as a conclusion.

The two withdrawals have a shape #

Six items in the ledger and two are withdrawn, and the two are not a random pair.

Read the six again asking a different question: was each item, when it was written, a measurement or a conclusion? The four that stand are measurements — a correlation computed, a spectrum read, two combs found, a ratio scored. The two withdrawn are conclusions: there is no window that satisfies both, and the comb belongs to the process. Neither was a number. Both were arguments written down at the end of a round of work and carried forward as though they had been established.

That gives the audit a rule it can apply to itself before the next withdrawal. An item written as a conclusion rather than as a measurement is the one to re-check first, and marking each item as one or the other at the time it is written costs a word.

This is a general enough trap to be worth a name, and the site has walked into its relatives twice. The interaction-range thread recorded a recency cut-off that manufactured a lattice — a parameter of the program producing a result read as a property of the model. The rising-stem work records a sweep of a neighbourhood cap that returned the same number at every setting and was read as robustness, when the parameter was not binding. Both are cases of a variable being held fixed invisibly.

The version here is the sharpest of the three, because the fixed variable was not a parameter at all. It was an assumption in the null model, and a null model’s assumptions are the least visible thing in an experiment: they are what the comparison is against, so they never appear in the result.

What a control has to do, written down #

The failure is general enough to be worth turning into a rule this site can be held to, since the alternative is discovering it again.

Name the ingredient the observable is claimed to require. Here: a placement rule. That part was done.

Then name what the alternative is allowed to have. This is the step that was skipped. The kinematic lattice was allowed the divergence, the rise, the pair and a scatter — all quantities the positions have — and was not allowed any structure in its errors, because nobody wrote down that errors have structure as a thing an alternative could have.

Then ask what else could supply the observable. The observable was correlation at multiples of the parastichy numbers. The question what else produces correlation at those lags has one obvious answer — anything that travels between the organs at those offsets — and it was available before any code was written. It was not asked because the control had already come back clean.

And test the alternative that is closest to the thing being ruled out, not the one that is easiest to build. The independent-error lattice was the easiest alternative to build, which is why it was built. The transported-error lattice is the nearest one, which is why it is the one that mattered.

The last of those is the operative one and it has a tell: an alternative much simpler than the thing being ruled out is probably the wrong alternative. A model of a plant that has no interaction between organs at all is not a rival theory of phyllotaxis. Nobody holds it. Ruling it out was never the job.

Why no longer stem fixes this #

The natural response to a failed discrimination is more data, and it does not work here. The two arrangements agree on every quantity this site can compute from a list of angles or a list of positions: divergence, rise, parastichy pair, transitions, contact families, side-count distribution, hop lengths, both combs. Doubling the stem doubles the precision on each of them and does not add a quantity.

The one quantity that does differ is the ratio, and the previous essay measured how far a free parameter takes that.

What would settle it #

Every discrimination this site has managed has the same shape, and it is worth naming because it says where to look next. Two instruments that share no code path, applied to the same object, agreeing. The angle readout and the position counter. The recovered angle and the angle the head was built at. The counted pair and the ladder’s prediction.

Every one of those is a statement about form, and form is what a finished plant has. The claim that has just failed was an attempt to reach past form using only a finished plant, and the failure is not an accident of this particular control: a finished arrangement carries the order its organs were made in only through its errors, and errors can be structured by anything that travels along the contact graph.

So the next evidence has to come from somewhere else, and the obvious somewhere is an intervention. The placement rule makes a prediction that a transport of errors does not:

Remove a primordium, and the next organ moves.

Under the rule, the next organ goes where the repulsion from the existing ones is least, so deleting one of them changes the landscape and moves the minimum by a computable amount that depends on which neighbour was removed. Under any transport model, deleting a primordium removes a source of error and changes nothing about where the next organ goes, because nothing in that model decides where organs go.

That is a real experiment — primordium ablation is done, with a laser, and has been for decades — and it is a real prediction, because the displacement is computable from the rule with no free parameter once the interaction range is fixed. Neither of those is a small claim, and neither is made here: computing the ablation response is the principal thing the work here leaves undone, and it is left undone deliberately rather than attempted at the end.

The experiment, priced #

The ablation prediction is worth a paragraph of specifics, because “an intervention would settle it” is the kind of sentence that sits in a plan file for a long time without becoming anything.

What is removed. One primordium, at a stated position in the sequence, before the next one is initiated. The rule’s neighbourhood at a rise of 0.005 is about a hundred organs deep, but the weight is concentrated: the two contact neighbours carry most of the repulsion the next organ feels, so ablating one of those is the case with the largest predicted effect and ablating an organ forty places back is the control that should show nothing.

What is measured. The divergence angle of the next organ, and of the two or three after it. Not the position of the ablated organ’s replacement, which is a different and much-studied question.

What the two models predict. The rule predicts a displacement in a stated direction — away from the surviving neighbour, since the removed one was pushing the other way — of a size set by how much of the local repulsion the removed organ was contributing. A transport model predicts nothing at all: it has no opinion about where organs go, only about how their errors are correlated, so its prediction for the mean divergence after an ablation is the same as before.

And what would make it a real test rather than a demonstration. A displacement that is merely nonzero is consistent with a dozen mechanisms. The test is the size against the falloff: the rule’s prediction depends on the interaction range, which this site has already measured against two other observables, so the ablation displacement is over-determined. A rule fitted to the ladder and to the transitions predicts the ablation response with nothing left free, and a number that comes out right under those circumstances is worth more than any amount of correlation in a finished stem.

That is the experiment, and the reason it is not here is that computing the prediction properly means handling what happens to the sequence after the ablation as well as to the next angle — the pattern re-settles, and re-settling is where the interesting part is. Doing that badly at the end of a round of work is how a number nobody has checked gets into a plan file and is quoted for three rounds of work.

What a reader should take from the withdrawal #

Three things, and the third is the one this site is actually about.

The measurement was right and the interpretation was wrong. Every number the earlier work published is reproduced here — the comb heights, the pair, the lengths, the refusals. What changed is what the numbers are evidence for, and that changed because a new alternative was built and measured rather than because anything was recomputed.

The withdrawal was produced by the site’s own machinery. The assumption was written into the note left with it as untested, whatever comes next tested it, and the test was designed to be able to fail. A claim that has never been given a chance to fail is not a claim this collection is willing to make, and that is the whole of why that earlier work’s control was written down as a control rather than assumed.

And a weaker true claim is worth more than a strong false one. A comb rules out independent errors and constrains how transmission is weighted is a sentence a botanist can act on. A comb shows the plant computes its pattern was a sentence nobody could act on, because its contrary — a plant that merely has a pattern — is not a model of anything.