A disturbance that is not passed on

One disturbance on this site does all the damage. Give a stem’s organs errors that each one inherits from the two organs it touches — the ones m and n places back, which is what a parastichy pair is — and an arrangement with no placement rule in it reproduces the comb in the divergence sequence, the second comb, and the pair itself on eight seeds of eight. That result retired the comb as evidence of a rule, and a later measurement retired the ratio of the two combs as well.

Everything in that thread was measured with one implementation, and the implementation does two separable things.

Two properties, one stream #

The transported disturbance is a recursion. Organ i takes a share of what organs i−m and i−n were displaced by, and those displacements were themselves inherited from organs further back. So the stream has:

Structure. Its errors are correlated at the two contact offsets, which is the hypothesis about plants — an organ’s disturbance is transmitted by the organs it touches.

History. An error that enters the stream is still in it hundreds of organs later, spread across every sum and difference of the two offsets, and the stream wanders slowly because those contributions accumulate.

Nothing measured so far can say which of the two is responsible for what. The hypothesis about plants is the first; the second arrived with it because a recursion is the easiest way to write the first down.

The control, and what makes it one #

The control is a moving average over the same offsets. Organ i takes a share of the fresh deviates drawn for organs i−m and i−n — the innovations, not the displacements those organs ended up with.

That keeps the structure and removes the history. It is one line of arithmetic different from the transport, and the difference is which array is read from.

Three properties make it a control rather than a different experiment.

It carries the same lags. At a coupling of a half the transport has correlations of 0.418 and 0.414 at the two contact offsets and the average has 0.308 and 0.303. Same lags, same sign, same order of magnitude.

It carries their difference too. Both are correlated at lag five, which is thirteen minus eight: 0.176 for the transport and 0.126 for the average. That matters because the second comb — the observable that carried this site’s last discriminator — lives at that offset. Organ i and organ i−5 both contain the innovation from organ i−13, in both streams, for the same reason.

And it carries nothing else. Past the two offsets the average’s correlation is 0.000, 0.002, 0.008, 0.002, 0.004, 0.006 at lags 16, 21, 24, 26, 29 and 34 — zero to three decimal places at every one — where the transport has 0.175, 0.298, 0.081, 0.176, 0.170 and 0.172.

Why a recursion was the natural way to write it, and what that cost #

It is worth asking how the two properties came to be welded together in the first place, because the answer is not carelessness.

The hypothesis is about a plant: a primordium that is displaced pushes the organs it touches, so its error appears in theirs. Writing that down, the obvious sentence is organ i’s error is a share of the errors of the organs it touches, plus its own — and the errors of the organs it touches are the errors they actually have, which include what they inherited. That is a recursion, and it is the honest transcription of the hypothesis.

The moving average is a different sentence: organ i’s error is a share of the disturbances that arrived at the organs it touches, plus its own. As a model of a plant it is worse, because it supposes an organ can feel what happened to its neighbour without feeling where its neighbour ended up.

But the results this thread produced were not about plants. They were about what a class of disturbance can manufacture, and the class was defined by a property — correlation at the contact offsets — that both sentences have. Every conclusion was stated in terms of that property, and only one of the two objects was ever run.

So the cost was not a wrong model. It was a conclusion phrased in terms of a property when it depended on a property the object also happened to have.

How much history each one has #

“History” can be measured rather than described. Take the mean of a hundred consecutive draws, and then the variance of those block means across the stream. For independent draws that variance is the stream’s own variance divided by a hundred; anything above that is power at frequencies below one per hundred organs — which is what a slow wander is.

At a coupling of a half the transport’s block means carry 5.5 times a white stream’s variance and the average’s carry 2.2. At a coupling of 0.7 the transport’s carry fifty-five and the average’s 2.6.

The average’s 2.2 is not one, and it should not be: correlating at lags eight and thirteen does put some power at low frequencies. What it does not do is grow with the coupling, because there is nothing to accumulate.

The arithmetic of why lag five is shared #

The difference of the two offsets is where the second comb lives, and both streams carrying it is not an accident of the implementation. It falls out of the construction in one line, and it is worth doing because the second comb is the observable that survived longest.

In the moving average, organ i’s error contains the innovation of organ i−8 and that of organ i−13. Organ i−5’s error contains the innovations of organs i−13 and i−18. The two share the innovation at i−13, so their errors are correlated — at lag five, with a coefficient that follows from the couplings alone.

The same holds in the transport, with more terms, because the inherited quantities contain everything the innovations do and more.

What that means for the comparison is that neither stream has been given an advantage at the observable being tested. The second comb has a mechanism in both, and if the average manufactures no second comb it will not be for want of the correlation that produces one.

What the two streams look like beside each other #

Correlations at lags are an austere way to describe a disturbance, and the two objects are easier to hold in mind as behaviour.

The shared stream is jumpy. Each organ’s displacement is mostly its own fresh deviate with a fraction of two older ones added, and those older ones are themselves independent draws, so nothing persists. Watch a hundred organs and the mean of them is close to zero; watch the next hundred and it is close to zero again. It has texture at two spacings and no tendency.

The inherited stream drifts. An organ that is displaced to the left passes part of that to the two organs it touches; they pass part of what they now have to the organs they touch, and so on up the stem, so a single early disturbance leaves a bias that decays over hundreds of organs rather than tens. Watch a hundred organs and their mean is somewhere; watch the next hundred and it is somewhere else, five times further from zero than chance would put it.

That is the entire difference, and it is why the two can be told apart by a statistic as crude as the variance of a block mean. It is also why the inherited stream is the more dangerous one for an arrangement: a lattice can absorb an organ being displaced, and what it cannot absorb is every organ in a stretch being displaced the same way.

And it stays the same size #

A control that is bigger than the thing it controls for is not a control. The transport has a problem here that had to be corrected once already: its recursion inherits from two lagged terms and divides by the normalisation of one, so at a coupling of 0.7 its deviates have a standard deviation of 2.01 rather than one. Comparing at equal nominal amplitude compares a disturbance with one twice its size.

The average has unit variance by construction, at every coupling: 1.006 measured at 0.5, 0.7 and 0.9. Its two inherited terms are independent of its own fresh term and of each other, so the variances add to exactly what they were built to add to.

Every comparison in this thread is made at equal measured spread, with each stream divided by its own standard deviation, so this is bookkeeping rather than a result — but it is bookkeeping that has gone wrong here before.

The couplings each one can be run at #

A practical difference, which turns out to matter for the comparisons the control is for.

The inherited stream’s recursion is stable only while its coupling is small. At 0.5 and 0.7 it produces a stream with a finite variance; at 0.9 the two inherited terms feed back hard enough that the variance grows without bound, and the stream has no standard deviation to normalise by at all.

The shared stream has no such limit. Its variance is the sum of the squares of its three coefficients, whatever the coupling, because none of its terms depends on its own past. It can be run at 0.9 or at 0.99 and it is the same size as ever.

That asymmetry is why the essays that follow can answer the obvious objection to their results. When a control fails to reproduce an effect, the first question is whether it was simply driven too weakly — and here the control can be driven harder than the thing it is a control for.

The history is not a feature, it is a pole #

The two block-mean numbers deserve to be read together rather than as two measurements, because the pair of them says what kind of quantity the history is.

The transport’s block means carry 5.5 times a white stream’s variance at a coupling of 0.5 and fifty-five at 0.7. That is a factor of ten for a factor of 1.4 in the coupling, and it does not stop there: at 0.9 the recursion no longer has a finite variance at all. So the history is not a property the transport has a certain amount of. It is a quantity with a singularity in the coupling, and the couplings the comparisons are made at sit on the approach to it.

The average’s block means carry 2.2 at a coupling of 0.5 and 2.6 at 0.7 — a change of a fifth for the same change in the coupling, and no divergence anywhere, because there is nothing to accumulate. Its structure-derived low-frequency power is bounded and nearly flat.

Two consequences follow, and the second is a caution about every number in this thread.

The two properties are separable in a stronger sense than “one can be removed”. One of them is bounded in the coupling and the other is not, so no choice of coupling makes the transport into the average or the average into the transport. They are different objects rather than two settings of one.

And any conclusion drawn about the transport is a conclusion at its coupling. A quantity that runs 5.5, 55 and unbounded across three settings is not a quantity a single measurement characterises. When the transport is found to do something the average does not, the size of the difference is a fact about the coupling chosen, and the only way to know whether it is a fact about the mechanism is to read it at more than one — which is why the couplings are named beside every number here rather than left to a caption.

What this is not #

It is not a hypothesis about a plant. No process transmits a displacement between two organs without transmitting the displacement they themselves received; an organ that pushes its neighbour pushes it from wherever it actually is. The average is a deliberately unphysical object built to hold one property fixed while removing another, which is what a control is for.

It is not a claim that the transport is wrong. The transported account is the physically natural one and it remains the strongest rival to the placement rule. What the control makes possible is to say which of its two properties each of its successes depends on.

And it is not a new observable. Nothing here proposes a way of reading a stem that did not exist before, and nothing here is a measurement of a plant. The readouts are the ones the site already has: the comb, the second comb, their ratio, the recorded scatter, and whether a counter shown the positions agrees. Nothing about the instruments changes; the only new thing is a second disturbance to point them at.

What it is for #

Two questions, each with its own essay.

Does the forgery need the history? The transported disturbance manufactures a comb on a lattice with no rule in it, where a plain memory from organ to organ manufactures nothing. If the average does too, then correlation at the contacts is enough and the forgery is about the neighbour graph. If it does not, the forgery needs re-transmission, which is a stronger claim about a plant than merely having neighbours that share errors.

Does the damage need the history? A transported disturbance costs a lattice more than an independent one of the same size — a stem survives half as much displacement. If the average is equally harmful, the damage is done by the correlation at the contacts; if it is harmless, the damage was the accumulation.

The two answers turn out to be different, which is the useful outcome: one of the transport’s two properties is responsible for one of its two effects, and the other for the other.

That is also the outcome a control is least likely to produce and most worth having. A control that reproduces everything says the isolated property was never doing anything; one that reproduces nothing says the two properties cannot be told apart by these instruments. A control that reproduces exactly one of two effects has assigned both, and it does so on a single line of arithmetic difference between two streams that are otherwise the same object.

The check #

Three assertions run on the streams themselves, before anything is driven through anything.

The first requires the two to agree at the contact offsets to within 0.15 in correlation, and to agree at their difference — so a control that had quietly lost the structure would fail rather than produce a clean and meaningless negative.

The second requires the average to have no power past the two offsets and the transport to have some, which is the property being isolated.

The third requires the transport’s block means to wander at least twice as much as the average’s. That is the one number the whole comparison rests on, and it is asserted at the coupling the comparisons are made at rather than at the coupling where the difference is most flattering.

Where that measurement can and cannot be made #

The block-mean comparison in this essay is made on the disturbance streams themselves, and that is the right place for the question it answers, which is about their structure. It is not the place a plant supplies.

A divergence is a difference of two organs’ errors, and differencing removes exactly the power a block mean is sensitive to. So the ×5.5 and ×55 measured here become 1.06 and 0.83 when the same statistic is computed on the divergences those streams produce — indistinguishable from the 0.96 independent errors give.