Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit.
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.
Fig. 1 The object being summarised, seen through what it makes: the same lattice comes out over most of the sweep, which is what a top that does not move looks like from outside.
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.
Fig. 2 The distances the terms are computed from. On the surface of the stem rather than up it, which is the whole of why the answer is not what it looks like.
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.
exponent
half the profile
nine tenths
the five largest terms
1.5
33 organs
182 organs
13, 8, 5, 21, 26
2
12
120
13, 8, 5, 21, 26
3
3
30
13, 8, 5, 21, 26
4
2
8
13, 8, 5, 21, 26
6
1
3
13, 8, 5, 21, 26
Fig. 3 The columns that change, beside the column that does not. Three summaries of the neighbourhood moving over two orders of magnitude while the organs at the top of it stay put.
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.
Fig. 4 Why they are the ones: those are the lags whose step across the surface is shortest, and the term is a function of the step.Fig. 5 The same thing in the packing. An organ touches members of both contact families, and those touches are what the largest terms of the sum are.
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.
Fig. 6 And the ladder those numbers sit on: each is the sum of the two before it, which is why the top five terms are a run of consecutive members of it.
“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.
Fig. 7 How much the top carries, by exponent. At the steep end a handful of organs is everything; at the shallow end the same handful is the top of a long list.Fig. 8 And what the list is: the ladder whose members are the sums of their two predecessors, which is where the top of the profile comes from at every exponent.
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.
Fig. 9 The steps out to a lag of thirty-four, which is where the organs joining the top of the profile at shallow exponents come from.
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.
Fig. 10 The reason, drawn. On the unrolled stem, the organ placed immediately before sits most of a turn away, and the organs directly below the tip are eight and thirteen back.Fig. 11 And the same stem in the round, where the near neighbours of the newest organ are visibly not the ones placed just before it.
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.
Fig. 12 The separation stated directly: what is near in the arrangement and what is recent in the sequence are two different sets, and only one of them is in the sum.
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.
Fig. 13 The loop bound swept, and the null. If the rule’s answer were being set by how many organs the sum reaches, this figure would not be flat.
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.
Fig. 14 And what happens when a truncation is tight enough to cut into the top: the rule stops making the pattern it otherwise makes, which is the boundary of the null.
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.
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.
Fig. 15 The ranking at a second rise, where the pair is 5/8 and the top of the profile moves with it.Fig. 16 And at a third, where the pair is 3/5. The organs carrying the largest terms follow the lattice, not the index.
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.
Fig. 17 The pairs across the rises, which is what the top of the profile tracks.
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.
Fig. 18 The tail moving. Nine tenths of the profile is carried by 182 organs at one end of the sweep and 3 at the other, which is a fact about how quickly the terms die away.
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.
Fig. 19 The sweep that could not work, seen from here. It varies a tail while the argument was about a top.
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.
Fig. 20 The rule as it is written: a loop over what has already been placed, whose natural summary is a number of iterations.
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.
Fig. 21 A rule where the two units do agree: transport between adjacent cells, where the neighbour in the sequence is the neighbour in space.Fig. 22 And the surface that separates them here. On a cylinder, the organ placed a hundred and thirty-seven degrees ago is one of the furthest things from the tip.
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.
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.
Fig. 23 The number the sentence quotes, across the sweep, with the shares either side of it. All three are counts of terms and not runs of organs, and they are not the same set.
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.
Fig. 24 And the correction applied elsewhere in this collection for the same reason: comparing arrangements at matched distance rather than matched count.
“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 less than a factor of three.
Fig. 25 The narrow range, at a second rise. Whatever the exponent, the terms that dominate come from a handful of steps of very similar length.Fig. 26 The general form of the correction: two arrangements are comparable when their neighbourhoods are matched in distance rather than in count, which is a distinction this collection has had to make before.Fig. 27 And what changes when a quantity is put in the right unit: a comparison that was confounded by how crowded the arrangement is becomes a comparison of one thing.
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.
Fig. 28 The summary that was already saying so: the centre of mass of the profile moves by a factor of six where the count of organs moves by sixty.Fig. 29 And the earlier null that fits the same picture: truncating the loop the sum runs over changes nothing, because the terms that matter were never in the truncated part.
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.