What a forgery has to know

The previous essay left the comb reproduced by an arrangement with no rule in it, and the pair reproduced too, on every stem. That takes away the qualitative test. What is left is quantitative: the two combs a real stem shows are not equally strong, and the question is whether a forgery gets their proportions right as well as their positions.

It does not, unless it is tuned — and the tuning is in the wrong direction and by a large factor. This essay is that measurement, and it is the whole of what survives the previous phase’s claim.

The quantity #

The readout finds two combs. The main one sits at multiples of the smaller parastichy number; the second sits at the same spacing, offset by the difference of the pair. On the rule’s own stems at a rise of 0.005 the main comb averages 0.642 and the second 0.415, so the second is 0.65 of the first, and across five seeded stems the ratio runs 0.58 to 0.79.

That the main comb is the stronger looks like a definition rather than a measurement, because the readout calls the tallest comb the main one. It is not, and the reason it is not is the first surprise here: the readout finds the spacing first, and the spacing is the smaller parastichy number. Whether the comb at that spacing is stronger than the comb offset from it is then a fact about the stem.

The geometry says it should be the other way round #

At a rise of 0.005 the surface distance from an organ to the one thirteen places before it is 0.0689 of a circumference, and to the one eight places before it is 0.0745. The thirteen-neighbour is the nearer of the two.

So a process that transmits error in proportion to how hard organs push each other should transmit more along the thirteen-hop than along the eight-hop. The repulsion in this site’s placement rule falls as the cube of distance, so the ratio of the two couplings would be (0.0745/0.0689)³ = 1.26 in favour of thirteen. Any falloff at all gives the same direction; the exponent only sets the size.

Give the forgery that weighting and the prediction comes out:

The forgery’s ratio there is 1.30, against the rule’s 0.65: a factor of two, in the opposite direction, on the one quantity left.

Why the rule prefers the shorter index offset #

This is the part that was not predicted and is reported as measured.

The rule places each organ at the minimum of a sum over the previous hundred or so, weighted by the cube of distance. Its correlations are not one-step transfers along two edges; they are the response of a whole neighbourhood, and the response has a structure that the distances alone do not give.

What can be said with confidence is the negative half: the rule’s comb ratio is not the ratio of its two couplings. If it were, the rule would show the same 1.26 lean towards thirteen that the distance-weighted forgery does, and it shows 0.65 the other way.

A candidate explanation, offered as a candidate: the readout’s main comb contains the harmonics — 8, 16, 24 — and a harmonic is a step taken twice. The correlation at 16 is what survives two eight-steps, so the main comb’s score mixes one-step and two-step transfers, while the second comb at 5, 13, 21 mixes one thirteen-step with an eight-plus-thirteen. If the rule’s transfer is efficient at short chains and lossy at long ones, the comb whose members are reachable by fewer steps wins, and that is the eight-comb whatever the distances say. That would make the ratio a measure of how quickly a neighbourhood’s response decays with chain length, which is a property of the interaction range rather than of the geometry.

It is a candidate because nothing here separates it from two other stories, and naming a mechanism this site has not tested would be exactly the move the site exists to refuse.

The two other stories, so that the candidate can be compared with something. One: the rule’s response is not a transfer at all but a relaxation — a displaced organ changes the energy landscape the next few see, and the neighbourhood re-settles — in which case the comb weights are set by how the landscape’s curvature divides between the two directions, and the distances enter only through that. Two: the eight-chain and the thirteen-chain differ in how many organs lie between their endpoints in production order, so a disturbance travelling along the thirteen-chain has more chances to be overwritten by fresher noise, and the ratio measures a competition between transfer and refreshment rather than a property of either chain.

All three predict the observed sign. None of them predicts the observed size without a calculation nobody here has done, and the difference between a candidate explanation and a result is exactly that calculation.

What the tuned forgery can do #

Turn the coupling ratio the other way — make the forgery inherit more from the eight-neighbour than from the thirteen-neighbour, against what the distances say — and the ratio walks down through the rule’s value.

At 2:1 towards the eight it is 0.97. At 3:1 it is 0.75. At 5:1 it is 0.46. So somewhere around 3:1 the forgery sits on the rule, and it sits on more than the ratio: main comb 0.601 against the rule’s 0.642, second 0.453 against 0.415, recorded scatter 0.82° against 0.70°.

So the honest summary is a conditional, and the condition is about a free parameter rather than about a measurement.

If the transport is distance-weighted, the comb ratio separates it from the rule — 1.30 against 0.65, a separation of a factor of two on a quantity that five stems measure to about ±0.1.

If the coupling ratio is free, nothing does. A forger allowed to pick how much each neighbour contributes has two knobs and is being asked to match two numbers, and two knobs match two numbers.

Which of those a plant is #

This is where the essay stops being about arithmetic, and it is worth stating what would settle it rather than pretending it is settled.

A coupling three to one in favour of the further neighbour is not something a falloff produces. Every interaction that decreases with distance gives the nearer neighbour the larger share, so the tuned forgery needs its transport to be stronger where the organs are further apart — a factor of four against what distance supplies, and in the opposite direction.

There are ways that could be true. The eight-chain and the thirteen-chain are different tissue: one runs up the steeper spiral and one the shallower, and if transport happens along vascular strands rather than through contact then the strands’ geometry is not the organs’ geometry. But that is a hypothesis with its own predictions, not a free parameter, and it would be testable — which is the shape this thread keeps arriving at.

So the measurement’s value is that it converts a qualitative claim into a quantitative one with a stated escape. Before this essay: a comb shows the plant computes its pattern, which is false. After it: a comb ratio below one is inconsistent with error transport weighted by distance, and consistent with the placement rule, which is true, and which a real plant could be measured against.

Three things the ratio is not #

A quantity that survives when the qualitative test has failed attracts more weight than it can carry, so it is worth naming what this one does not do.

It is not a test of the placement rule against everything. It compares one model of error transport — inheritance along the two contact chains, with a coupling split — against the rule. A plant could be doing something that is neither, and the ratio would report whichever of the two it happened to resemble. The list of rivals this thread has tested is now three long, and the list of rivals that exist is not.

It is not independent of the rung. The ratio 0.65 was measured at a rise of 0.005, where the pair is 8/13 and the thirteen-hop is the shorter by eight per cent. At a coarser rise the same asymmetry has a different size and sometimes a different sign, so the number a plant should be compared against depends on which rung it is on — which means the comparison needs the rise, which needs the internode length and the organ size, which is two more measurements on the specimen.

And it is not a measurement anybody has made. No published work reports the autocorrelation of a divergence sequence at all, let alone the relative heights of two residue classes in it. The quantity is available from data that has been collected for two centuries — a list of divergence angles down a stem — and has not been looked at, because until the previous phase nobody had a reason to look at lags rather than at the mean.

What the previous phase should have said #

With the measurement in hand it is possible to write the sentence the previous phase was reaching for, and the difference between the two is instructive.

What it said: the comb is the first quantity here that separates a process from a form.

What it could have said: the comb separates an arrangement whose errors are independent from one whose errors are transmitted along its contact graph, and the ratio of the two combs constrains how that transmission is weighted.

The second sentence is longer, weaker, and true. It also has a property the first does not: it names the alternative it excludes, so a reader can ask whether a plant might be the alternative. The first sentence excludes “a plant that merely has a pattern”, which is not a model of anything and therefore not something a measurement can rule out.

That is the general form of the lesson, and this site has recorded it twice before under other names — a claim that survives is one whose contrary is something a plant could be.

What it would cost to measure on a plant #

The ratio is a ratio of two quantities that both need the pair to be readable, so it inherits the pair’s cost: two hundred and fifty internodes, a reading error under a quarter of a degree, a shoot slower than about two hundred and fifty nodes to the rung. It does not obviously need more than that, because the ratio is more forgiving than either comb — a common error in both numerators divides out.

What it does need, and what the previous requirements did not, is more than one specimen at the same rung. The ratio varies from 0.58 to 0.79 across five stems grown from the same rule with different disturbance seeds, which is a spread of a third, and separating 0.65 from 1.30 with a spread of a third takes a handful of plants rather than one.

The protocol, written out, is four steps and none of them is new machinery.

One. Record the divergence angle at every internode down two hundred and fifty internodes of a single shoot, to a quarter of a degree, on each of half a dozen plants of the same species at the same stage.

Two. On each plant, run the two-comb readout. Discard the plants it refuses, and record how many it refused — a refusal rate is a measurement of the sample, not a nuisance.

Three. Check that the reported pairs agree with each other and with a parastichy count made on the same shoots. Disagreement here is the periodic forgery’s signature and the reading stops if it appears.

Four. Take the ratio of the second comb’s mean to the main comb’s, on each plant, and compare the set against 0.65 and against 1.30. With a per-plant spread of about a third, six plants separate those two hypotheses at a comfortable margin and two do not.

The fourth step is the only one this essay adds, and it costs nothing extra in fieldwork — the two numbers are already computed by the readout that produced the pair. What it costs is the specimens, because the spread across stems is not small, and the reason the spread is not small is that a comb strength is a mean over three or four teeth of a correlation estimated from a few hundred numbers.