Errors that pass between organs

That earlier work’s control was a lattice with independent errors, and the two essays before this one have taken the two obvious ways of making them not independent. A memory manufactures nothing. A periodicity manufactures a comb and then names a partner that changes from stem to stem, which a second specimen catches.

This essay builds the third, and it is the one that was always going to be awkward, because unlike the other two it is not a contrivance at all: nobody has to choose a number for it, and a plant that transmits anything transmits it here. It is simply what happens if an organ’s error is passed on by the organs that touch it.

Why this is not a contrivance #

The choice of eight and thirteen looks like cheating, and the argument that it is not is the whole essay, so it is worth putting first.

A parastichy pair is not a fact about a plant’s arithmetic. It is a fact about its geometry: m and n are the index offsets whose two organs come out closest together on the surface. The organ eight places before this one, and the organ thirteen places before it, are the organs it is touching. That is what the spiral rows a botanist counts are — chains of organs in contact, each one m or n steps back in the order of production.

So a disturbance transmitted by contact is correlated at m and at n automatically. It does not have to be told the pair; the pair is where the contact is. Any mechanism at all — a physical push between swelling primordia, a shared vascular strand, a locally depleted substrate, a mechanical stress field in the tunica — has this property. It is hard to think of a disturbance a plant could have that is transmitted at all and not transmitted at these offsets, and the difficulty is the point rather than a rhetorical flourish.

What it does to the readout #

The disturbance is built the plainest way that has this property. The error at organ i is a fraction of the error at organ i − 8 plus the same fraction of the error at i − 13, plus a fresh draw, with the whole thing normalised so the recorded scatter is the same half degree as every other arrangement in this thread.

Nothing else changes. The nodes sit at exact multiples of the divergence before the displacement. No minimum is taken anywhere. No organ has any influence on where any other organ goes; the influence is only on how wrong it is.

The comparison that matters is against the control it replaces.

The claim that has to be withdrawn #

The earlier work wrote:

The comb is the first quantity here that separates a process from a form, and it does so because it lives in the order the angles arrived in, which a finished pattern does not carry.

The first half of that is right and the second half is doing work it cannot do. A finished pattern does not carry the order the angles arrived in — but a disturbance on a finished pattern does, if it was transmitted between organs, because transmission happens in order. The arrangement in the figure at the top of this essay is a finished pattern with a history in its errors and no rule in its positions, and that possibility was not in that work’s list.

So the claim as written is withdrawn. A comb in a real plant’s divergence sequence is not evidence that the plant computes its pattern.

The withdrawal is worth sizing against what it leaves, because “withdrawn” reads as total and is not. This thread names five arrangements a comb could come from: independent errors, a memory, a periodicity, a contact transport, and a placement rule. Before this essay the comb was said to pick out the last of the five. After it, the comb picks out the last two — and it still excludes the first three, which is three of five accounts refused by one reading.

So the instrument has gone from separating one account from four to separating two from three. That is a real loss and it is not the loss of the instrument: a measurement that halves a five-way space is worth taking, and what it has stopped being is a measurement that settles the question on its own.

How much transmission the forgery needs #

The coupling here is 0.7, which means an organ inherits seven tenths of each of two neighbours’ errors before its own fresh draw. That is a great deal of transmission — before normalisation the organ is mostly its neighbours — and the essay does not ask whether a plant would have that much.

It is the obvious question and it is the one that would put a number on the withdrawal. If the comb needs a coupling of 0.7 to forge and a real primordium passes a tenth of its displacement to the organs it touches, then the comb is evidence again, with a threshold attached: a comb means transmission of at least so much.

What is known is a range rather than a threshold. The forgery is reported at 0.5 and at 0.7, and both work; the recursion stops having a finite variance somewhere above 0.7, so the upper end is a limit of the construction rather than of the phenomenon. Nobody has run it at 0.3, 0.2 or 0.1.

That sweep is the cheapest outstanding measurement in this thread and it decides how much the withdrawal costs. A forgery that survives to 0.1 takes the comb away entirely; one that fails below 0.4 hands it back with a stated condition, and the condition is a quantity a botanist could in principle argue about from tissue mechanics rather than from angles.

Until it is run, the honest form of the withdrawal is conditional: a comb is not evidence of a rule on a plant whose errors are strongly transmitted, and how strong is unmeasured.

And the control that caught the last forgery does not catch this one #

The periodic forgery was convicted by reading more than one stem: its partner is an accident of the pattern drawn for that stem, so two specimens disagree. That requirement went into the specification and it is a good requirement.

It does nothing here. This forgery returns 8/13 on eight seeds of eight, because the offsets it couples at are the arrangement’s own and every stem of the species has the same ones. So the control added to catch one forgery is silent about the next, and the two failures are independent.

That is worth stating as a general caution about specifications built by accretion. Each requirement in this one was added to close a hole somebody found, and a requirement closes the hole it was written for and no other. A specification grown that way is a list of past failures rather than a proof against future ones, and its length is not a measure of its strength.

It is worth being exact about what is left, because it is not nothing:

A comb is evidence that something was transmitted between contact neighbours. That is a real and non-trivial statement about a plant — it rules out a pattern whose organ positions are independent draws around an ideal lattice, which is the null model a botanist would otherwise be arguing against.

The placement rule is one process with that property, and not the only one. The rule transmits because each organ is placed against the ones near it. A mechanical push transmits because organs touch. The angle sequence does not distinguish them, and the next essay measures how nearly it comes to.

And that earlier work’s argument for why the comb exists never used the rule. Read it again: correlation travels between neighbours, so the lags carrying it are a·m + b·n for small a and b, and a one-step chain has b = 0 or ±1. Every word of that is about the neighbour graph. The placement rule appears nowhere in it. The explanation was right, and it was an explanation of something weaker than what it was attached to.

What “form against process” turns out to mean #

The distinction this thread has been trying to measure is usually put as form against mechanism: a photograph of a finished plant shows the form, and the question is whether anything in it constrains the process that made it. The that earlier work’s answer was that the order the organs arrived in is not in the photograph and is in the angle sequence, so the angle sequence carries process where the photograph carries only form.

That answer needs a correction rather than a replacement, and the correction is narrow enough to state in a sentence. The angle sequence carries whatever was transmitted along the order of production, and a placement rule is not the only thing that can be.

The arrangement at the top of this essay is a strange object worth looking at directly. Its positions are a form: node i is at exactly i times the divergence, decided before any organ existed, with no reference to any other organ. Its errors are a process: each one was computed from two earlier ones, in order, and could not have been computed in any other order. So it is a form with a history bolted onto it, and the history is only in the mistakes.

That is not as artificial as it sounds. A plant whose organ positions were determined by something other than local interaction — a pre-patterned field, say, or a genetically specified sequence — would still have organs that touch, and whatever passes between touching organs would still pass in the order they were made. The angle sequence would still carry a comb. What the comb reports is the existence of transmission along the contact graph, and that is a much weaker statement than the existence of a rule that chooses positions.

It is also, on reflection, the statement the measurement was always making. The readout finds correlation at the contact offsets. Correlation at the contact offsets is what transmission along contacts produces. The step from there to the plant computes its pattern was an inference about which process does the transmitting, and nothing in the data picked one.

How much coupling a plant would need #

The forgery works over a range and the range is worth pricing, because it is the range a survey would have to argue a plant is outside of.

The coupling is the fraction of a neighbour’s error an organ inherits. At 0.3 — each organ taking three tenths of each of its two contact neighbours’ errors — the readout refuses on every stem: the comb sits at 0.090 against a band of 0.073 and does not clear. At 0.4 it reports on five stems of eight, at 0.5 on all eight. So the threshold is between the two, and what it means physically is that an organ has to inherit something like four tenths of each contact neighbour’s displacement before the signature appears at all.

The transition is sharp in the way a threshold on a mean usually is. The comb grows roughly as the square of the coupling — 0.09, 0.14, 0.21, 0.32, 0.55 at couplings of 0.3 to 0.7 — while the band it has to clear does not move, so a factor of two in the coupling is a factor of four in the comb and the difference between refusing everything and reporting everything is a coupling change of one tenth.

Is that a lot? It is more than nothing and less than everything, which is an unsatisfying answer, and the honest form of it is that nobody knows. The quantity has never been measured on a plant, because until this essay there was no reason to want it. What can be said is that it is not an extreme assumption: organs in a meristem are packed at the density where they are touching, they grow while in contact, and a displacement of one is not obviously attenuated by more than half before it reaches the next.

So the forgery is not ruled out by implausibility, and a survey that found a comb could not appeal to implausibility either.

Where the forgery breaks down #

It is not free, and the boundaries are worth having because they are what a survey would have to rule out.

Below a coupling of about 0.4 there is nothing. The inherited fraction is too small, the comb does not clear the band, and the readout refuses on every stem.

Above about 0.8 the arrangement stops being a lattice. Inherited errors accumulate: each organ takes most of two previous errors and adds its own, so the variance grows along the stem until the recorded scatter is 102° and the points are spread over the whole circumference. That is the same incoherence the placement rule reaches at a disturbance of 0.4, and it is refused for the same reason — there is no pattern left to read.

So the window is a coupling between roughly 0.4 and 0.8, which is not a narrow target. Which of the two things a coupling does — correlating the errors, and handing them on — the forgery actually needs is a question a control settles later. A process that passes half of an organ’s error to its two contact neighbours is not an extreme assumption about a plant; it is close to what “organs push each other” would mean.

The upper edge is worth one more sentence, because it is a real difference between the forgery and the rule and it points at where a discriminator might eventually be found. The forgery’s errors accumulate: nothing pulls a displaced organ back, so the variance of the displacement grows along the stem and the arrangement drifts off its own lattice. The placement rule’s errors do not, because the rule is self-correcting — an organ placed to one side of its minimum leaves a gap that pulls the next one the other way, which is what this site measured as a lag-one correlation of −0.6 earlier in this collection.

That difference is visible in the long-run behaviour of the scatter and invisible in a window of a few hundred organs at a coupling in the working range, which is why it does not appear as a discriminator here. On a stem long enough for the drift to show, it would — and no plant is that long.

What this does not touch #

Two things, and stating them keeps the retraction from becoming a larger claim than it is.

It does not say the placement rule is wrong. Nothing here is evidence against the Douady–Couder rule, which this site has tested from a dozen other directions and which reproduces the ladder, the transitions, the bifurcation and the round trip. It says one observable that was offered as evidence for the rule does not distinguish it from a rival — and a rival that reproduces one observable is not a rival that reproduces the model. This essay’s arrangement has no ladder in it, cannot change parastichy pair as the rise falls, and has to be handed the divergence angle it is built at. The placement rule computes all three.

And it does not say the comb is worthless. A quantity that rules out independent errors is worth measuring, and it was the site’s own standard that made the retraction necessary: the control was stated, the assumption behind the control was written into the note left with it as untested, and whatever comes next tested it. A result that survives a test it was designed to fail is worth more than one that was never given the chance, and this one did not survive.

What survives as a discriminator is therefore a ratio rather than a presence, and the ratio has since been shown to carry an instrument setting of its own — a fifth of its value came from the grid the azimuths are placed on rather than from anything the stems were doing. Both corrections point the same way: the qualitative test is gone and the quantitative one has to be quoted with the settings it was measured at.