The nearest organ is not the nearest neighbour

Four different summaries of the rule’s neighbourhood disagree about its size by a factor of ten and agree exactly about the order: every one of them falls as the falloff exponent rises. A quantity that changes by a factor of sixty by one measure and six by another, with the ordering identical under both, is a quantity whose ordering is being set by something the measures have in common.

They do. It is the top of the profile, and — unlike the window that was mistaken for it — it does not move.

The measurement #

At the tip of a settled stem, take every organ already placed, compute its term — one over the distance to the power of the exponent — and sort the terms by size. Then, instead of counting how many terms are needed to reach some share, simply read off which organs the biggest ones belong to.

At every exponent from 1.5 to 6 the five largest terms belong to the organs 13, 8, 5, 21 and 26 places back. The same five, in the same order, at every one.

Those five numbers are the lattice #

The stems are grown at a rise of 0.005, where the counted pair is 8/13 and the next number up the ladder is 21. So the five organs carrying the largest terms are the two contact families, the number below them, and the two above — 5, 8, 13, 21 and 26, which is twice 13.

That is not a coincidence and it is not a new fact; it is the definition of a contact family restated in the rule’s own arithmetic. What is new is the consequence for the sweep.

How much of the profile the top five carry #

“The five largest terms” is a claim about an ordering and it is worth attaching a size to it, because five terms out of three hundred could be a rounding or could be almost everything.

At an exponent of 6 the five largest carry very nearly all of it: half the profile is one organ and nine tenths is three, so the five are already past the point where anything else contributes measurably. At an exponent of 1.5 they carry much less — half the profile takes thirty-three organs — but they are still the five largest, and the organs joining them as the share is raised are 34, 3, 39 and 47, which are the next entries on the same ladder and the near misses beside them.

So the strength of the claim varies across the sweep and its content does not. At every exponent the terms are ordered the same way, by the same lags, for the same reason — the steps of those lags are the shortest steps on the lattice. What the exponent changes is how much of the total the leaders take.

The organ placed just before is not a neighbour #

The lag that is conspicuously absent from the list is 1.

Organ i − 1 is the most recent thing the stem did. It is one plastochron away in time and it is 137.5° away round the stem — nearly two fifths of a turn — which makes it one of the furthest organs from the tip among the last few dozen. Its term is small at every exponent, and steeply smaller as the exponent rises.

So “the rule looks at the last thirty organs” is a sentence that sounds like a statement about depth and is really a statement about an index. What the rule looks at is a set of positions, and the positions that matter are the contact families — which are a property of the lattice rather than of how far back the counting goes.

The same fact from the other end: a truncation test #

There is an independent measurement in this collection that says the same thing and was made for a different reason, which is the best kind of corroboration available.

The loop bound — how far back the sum is actually taken in the implementation — was swept from one local spacing to six, and it changes nothing: the wander of the divergence moves by a factor of 1.30 across the sweep, against 3.28 between random seeds at one setting.

That null is exactly what the top of the profile predicts. If five organs at lags 13, 8, 5, 21 and 26 carry the ordering, then any loop bound reaching past twenty-six includes all of them and any bound reaching past a couple of local spacings does too, because a local spacing at this rise covers several of those lags at once.

So two measurements made for different purposes agree. Sweeping how far the sum reaches changes nothing because the terms that matter are near the front of it; sweeping how fast the terms fall away changes the tail and not the front. The neighbourhood, in the only sense that affects an outcome, is a fixed set of five organs.

The measurement does not depend on the rise #

Five lags at one rise is one lattice, so the same reading is made at the two other rises this thread uses.

At a rise of 0.013 the counted pair is 5/8 and the five largest terms belong to lags 8, 5, 3, 13 and 16. At 0.032 the pair is 3/5 and they belong to 5, 3, 2, 8 and 10. In each case the list is the pair, the number below it on the ladder, the number above it, and a double of one of them — the same shape as the 8/13 case, moved.

The three lists have an exact form rather than a family resemblance, and writing it out turns a shape into a prediction. Take the counted pair (m, n) with m the smaller. Then the five organs carrying the largest terms are, in order of their step length:

the larger number n, the smaller number m, the ladder term below m, the ladder term above n, and twice n.

Checked against the table: at 8/13 that reads 13, 8, 5, 21, 26 — and 26 is twice 13. At 5/8 it reads 8, 5, 3, 13, 16, and 16 is twice 8. At 3/5 it reads 5, 3, 2, 8, 10, and 10 is twice 5. Three rises, three pairs, one rule with no free choice in it.

So the top of the profile is four consecutive terms of the ladder plus one number that is not on it at all. That last member is the one worth noticing, because it says the top five is not simply the five shortest ladder steps — a double is not a ladder term, and it beats the next ladder term up for a reason that is geometric rather than arithmetic.

The rule also predicts, which is what makes it worth stating. On a stem at 13/21 the five should be 21, 13, 8, 34 and 42, and on one at 2/3 they should be 3, 2, 1 and — where the ladder below runs out — whatever the geometry supplies. Neither rung has been read this way, and either would refuse the rule in one line.

That is the right kind of dependence for the claim to have. The five organs are not a universal constant of the rule; they are the contact families of whatever lattice the stem is on, and they move when the lattice does. What does not move them is the exponent.

It is worth saying what would have falsified this. If the five largest terms had been lags 1, 2, 3, 4 and 5 at a steep exponent and lags 13, 8, 5, 21 and 26 at a shallow one, then the exponent really would be sweeping which organs the rule holds, and the neighbourhood counted in organs would have been the right quantity all along. The measurement was made in the expectation of seeing exactly that.

Why the ordering could never have told anybody anything #

Now the negative result from two essays back looks different.

The exponent changes how fast the profile’s tail falls away. At 1.5 the hundredth organ back still contributes something; at 6 it contributes nothing measurable. What the exponent does not change is which organs are at the top of the profile, because that is set by which lags have the shortest steps, and that is set by the lattice.

A disturbance is corrected by whatever the rule is holding on to. If that is the same five organs at every exponent, then sweeping the exponent is not sweeping the thing a filter argument was about — which is why every definition of the depth gave the same ordering, and why the ordering did not produce the crossover the argument predicted.

Where the wrong unit came from #

It is worth asking why “organs” was ever the unit, because the answer is not carelessness and the same mistake is available in several other places.

The rule is written as a loop over the organs already placed. The implementation has a bound on that loop. The natural summary of “how much of the loop matters” is therefore a count of iterations, and a count of iterations is a count of organs. Every step of that is reasonable and the result is a quantity indexed by sequence position being used to answer a question about distance.

The mistake is easy to make because the two units agree for almost every other kind of rule. In a reaction-diffusion model on a line, the n-th cell back really is n cells away. On a stem it is not: the arrangement wraps, and the wrapping is the entire subject.

The same confusion is what makes a plastochron and a spacing feel interchangeable when they are not, and it is worth carrying out of this thread: on a stem, “recent” and “near” are different words.

What this does to “the rule looks at N organs” #

The sentence appears in this collection more than once, and it should be read differently from here on.

It is not false. Thirty organs really do carry nine tenths of the profile at an exponent of 3, and that is a fact about the sum. What it does not license is the inference anybody would draw from it — that a disturbance shorter than thirty organs is averaged away and one longer than thirty organs is passed. That inference treats the thirty as a window in the sequence, and the thirty is a count of terms selected by size from all over it.

The distinction has a measurable consequence. The organs carrying nine tenths of the profile at an exponent of 3 include lags well past thirty — the deepest of them is further back than the count itself, because the set is assembled by size rather than by position. So “the last thirty organs” and “the thirty organs that matter” are different sets, and only the second has anything to do with the rule.

The unit was wrong, and there is a right one #

“Organs” is a unit of index. The rule works in a unit of distance, and the distances that matter are the contact steps.

Measured that way the neighbourhood is small and nearly constant: the five largest terms belong to five lags at every exponent, and their steps span a narrow range — the two contact steps differ by nine per cent at this rise, and the five together span a factor of exactly two.

The word exactly is doing work and the reason is worth a paragraph, because it is a piece of arithmetic rather than a measurement. On a cylindrical lattice the displacement from an organ to the one k places above it is k times the divergence round the circumference and k times the rise up the stem. For the lag that gives the shortest step, doubling k doubles both components without the azimuth wrapping past a half turn — that is what makes the shortest step short in the first place — so the step of lag 2n is twice the step of lag n, to the last digit.

At a rise of 0.005 the five steps are 0.070, 0.073, 0.091, 0.111 and 0.140 circumferences, for lags 13, 8, 5, 21 and 26. The last is twice the first because 26 is twice 13, and the whole neighbourhood the rule is working in therefore spans one octave in distance and nothing more.

That is a much stronger statement than “a narrow range”, and it explains why no summary counted in organs could capture it. A count of organs spans two orders of magnitude across the exponent sweep; the distances span a factor of two and do not move at all. The exponent decides how steeply the terms fall away within that octave and outside it, and it cannot decide which lags are in it, because that is fixed by where the lattice puts them.

That reframing has a consequence for the outstanding question. If the operative scale is the contact scale, and the contact scale barely moves with the exponent, then a corner in a filter argument should not move with the exponent either — and the earlier sweep’s failure to find one may have been a failure to look in the right quantity rather than evidence that there is none.

Where the corner has to be, if there is one #

The reframing gives a number, and it is worth extracting before the next measurement rather than after it.

A filter argument needs a scale in organs, because a disturbance’s correlation length is counted in organs. The neighbourhood’s scale in distance is one octave and says nothing directly. But the five lags carrying it are 5, 8, 13, 21 and 26, and those are index numbers — so the index scale the rule actually has is the spread of that set, from five organs to twenty-six, and not the thirty or the hundred and eighty-two a share-based count reports.

That is a fivefold spread rather than a sixtyfold one, and it has a shape. A disturbance correlated over three organs is shared by none of the five, since the closest pair of them is three apart and the rest are further. One correlated over ten organs is shared by the 5, the 8 and the 13 and not by the 21 or the 26 — so part of the top moves together and part does not, which is the mixed case a filter’s corner sits in. One correlated over forty organs is shared by all five, and the whole neighbourhood translates.

So the corner, if there is one, should sit somewhere between ten and thirty organs, and it should sit there at every exponent, because the five lags do not move. That is the prediction, and it is also the explanation of the earlier failure: the sweep varied the exponent from 1.5 to 6 and asked where each rule’s corner was, and the answer was that they all have the same corner, so no ordering of corners could appear. A sweep that varies a parameter the answer does not depend on returns a flat table whether or not the effect is real.

The measurement that follows from it is different in kind. It does not sweep the exponent at all. It sweeps the correlation length finely across the ten-to-thirty band at a single exponent, and asks whether the drift through has a knee in it. The earlier sweep visited correlation lengths of 2.8, 9.5 and 32.8 organs, which straddles the band with one point and resolves nothing inside it — three points cannot show a knee they are wider apart than.

There is a corner. It was looked for in the wrong quantity, it is in the right one, and it sits where this essay says it should.