The last of three quantities

A rung is a range of rises over which a counter returns one pair. Sweep the rise inside it and three things move: the pair does not, by definition; the settled divergence slides; and which of the two contact steps is the shorter changes hands once. Sweep across a transition and the pair moves too.

For a long time every result about which family survives a removal was scored on a census that varied all three at once. This essay is the bookkeeping: what two designs have now removed, what is left, and why what is left cannot be tested the same way.

The problem, stated as an experiment #

Every experiment here is a removal. Take a stem, take out one organ, watch what the rule does next, and record which lag’s hop is still rigid three hundred organs later. Repeat at a range of rises and a range of offsets and the result is a census.

A census’s rows differ in the rise, and the rise is not one variable. It is three variables tied together by the rule, so a rule that predicts the survivor from the counted pair and a rule that predicts it from the ordering will agree on most rows for no reason at all. That is not a subtlety about statistics; it is a statement that the census cannot tell them apart.

How badly they agreed #

The measure of the problem is a number an earlier essay produced. Positioned inside its own rung, every census row acquires a fraction nobody had computed, and eight of the ten lattices that wreck sit past their rung’s handover — twenty-five of thirty cuts.

So the census was not sampling the ordering. It was very nearly holding it. Two readings scored twelve and eleven of thirty and looked like rivals, and on most rows they were the same reading with two names.

That is what a design has to fix, and a design fixes it by holding something on purpose rather than by accident.

The first design #

A band. Near the rise where the two contact steps change places, the settled divergence has a shallow floor, so a run of rises either side of that crossing share an angle to a twentieth of a degree while the counted pair holds and the ordering reverses.

Two quantities held, one moved. Fifty-five wrecked cuts on two branches, and the family left standing never changes — including at the rises where the survivor is the longer step, which is where the shortest-hop reading would have had to be right.

The second design #

A matched pair. Down the golden branch the divergence curve turns, so a value it takes on one rung it takes again on another. Five such pairs exist; four of them wreck at both rises.

One quantity held, one moved, and the third moved with it. At every one of the four the two stems keep different families, so the divergence is not sufficient to decide which one survives.

What each design cannot say #

Neither is an account. A band says the ordering is not the actor and says nothing about what is; a matched pair says the divergence is not sufficient and says nothing about what is.

Both are negatives, and negatives of the weakest useful form: this quantity, on its own, does not determine the answer. Neither rules a quantity out of an account with several terms in it. What they rule out is the shape of explanation in which one of them is the whole story — and that is the shape most of the candidate readings here have taken.

The third quantity has no design #

The counted pair is what is left, and it is the one quantity here that has never been held still while the other two moved.

The reason is structural. Holding the pair is what a rung is, and a rung moves the divergence and the ordering together. There is no rise at which the pair is held and the divergence is varied independently of the ordering, because both are functions of the rise and the rise is one number.

So the pair is the last candidate standing and is also the least directly tested of the three. That is an uncomfortable position for a result to be in and it is better said than left implicit.

What a third design would need #

A stem with the same counted pair, the same divergence, the same ordering, and something else different — or a way of changing the pair without changing the rise. Neither exists in the rule as written.

There is one candidate that does not need the rise at all. A jugate stem — organs arriving several at a time — is an ordinary lattice seen k times over, and its counted pair is k times an underlying one. That changes the pair by a factor while leaving the underlying arrangement in place, which is closer to a pair-only variation than anything the rise can do.

Whether a jugate cut is comparable to a single-jugate one is a separate question with its own measurement, and it is a real route rather than a hopeful one.

The pair fixes the menu and the offset chooses from it #

The survivor depends on the pair and the offset together sounds like a surrender, and one result already in hand makes it much narrower than that.

A surviving lag has to divide one of the counted numbers, because a lag survives only if its hop lands on a sublattice the removal left intact. So on a 5/8 stem the candidates are the divisors of five and of eight — one, two, four, five and eight — and nothing else, out of the twenty-four lags a census records. On an 8/13 stem they are one, two, four, eight and thirteen, which is the list that made a period of four on an 8/13 lattice look like an exception and turn out not to be one.

Read that way the division of labour is clean rather than vague. The pair fixes the menu; the offset picks from it. A menu of five items out of twenty-four is most of the work, and it is done by the quantity this essay has just finished saying is the least directly tested — which is a reason to take the pair seriously rather than to hold it under suspicion.

What a jugate design would decide, stated in advance #

That also turns the jugate route from a promising idea into a design with a prediction, which is what it needs before anybody grows one.

A two-jugate stem whose underlying arrangement is 5/8 is counted 10 and 16. Its menu of divisors is therefore one, two, five and ten from the first, and one, two, four, eight and sixteen from the second — a list that contains ten and sixteen, which the underlying arrangement’s list does not.

So the two readings are separated by a single observation. If the survivor is a function of the counted pair, a wrecked jugate stem should sometimes keep a lag of ten or sixteen, and those numbers exist nowhere in the arrangement’s own geometry. If the survivor is a function of the underlying lattice — the pair being merely how a counter describes it — then the survivors should be five and eight, exactly as on a single-jugate stem, and ten and sixteen should never appear.

Neither outcome is the safe one. A ten would say the counted pair is doing real work rather than standing in for the geometry, which is a strong claim about a quantity that is blind to almost everything else. A run of fives and eights would say the pair has been a proxy all along, and that all that scoring of survivors against it was scoring them against the underlying lattice under another name.

That the prediction is stated before the stems are grown is the whole point of writing it here rather than after.

Two designs and one census, compared #

The census has thirty wrecked cuts and varies everything. The band has fifty-five and varies one thing. The matched pairs have four comparisons and vary one thing.

Weight of evidence and cleanliness of design run in opposite directions here, which is the ordinary situation. The census is where the readings were found and is the worst place to test them; the two designs are where they can be tested and carry far fewer rows. A result that survives both is a result that survived a large dirty sample and a small clean one, and neither on its own would be worth much.

The arithmetic of what is left #

Three quantities, two ruled out as sufficient, one standing. That is not the same as one quantity being the mechanism, and the difference matters enough to write out.

If the survivor were a function of the counted pair alone, then every stem with one pair would keep one family whatever the offset. It does not: a 5/8 stem keeps the 5 at some offsets and the 8 at others. So the pair is not sufficient either, and the honest statement is that the survivor depends on the pair and the offset together, with the divergence and the ordering ruled out as sufficient on their own.

What the two designs agree about #

They agree on something neither was built to test. A band’s cuts keep one family at each offset across every rise of the band; a matched pair’s two stems keep exactly the counted numbers their pairs share.

Read together, both say the answer is a property of the contact structure — which chains exist and which one the removed organ belonged to — rather than of any continuously varying quantity. A band varies two continuous quantities and changes nothing; a matched pair holds one continuous quantity and changes everything, by changing which chains there are.

The cost of the two designs #

Worth recording because it is the argument for doing this kind of thing again. The band cost a sweep of the rise at two parts in a thousand and a cut at every offset of every rise it found: several hundred grown stems. The matched pairs cost an overlap test that is free, a refinement of twenty-five stems a pair, and sixteen cuts at each of ten rises.

Against that, the census cost thirty cuts and settled nothing. The designs are ten times the compute and they are the only part of this thread that has closed anything.

What a plant sees #

A specimen has a counted pair and does not have a measured divergence. So of the three quantities, the one the field can report is the one still standing, and the two that have been ruled out are the two the field cannot see.

That is a convenient result and it should be said carefully, because convenience is where a reading gets over-read. The convenience is real and it is about the specification rather than the mechanism: the ablation this collection has specified records counts, and these two designs say the counts are the right thing to record.

Where the ordering result now stands #

It was carried by two bands and it is now carried by more. Four of the ladder’s six handovers had never had a band built on them, and building them puts the same claim on four counted pairs rather than two.

That is worth separating from this essay’s arithmetic. The three-quantity bookkeeping is about which designs exist; the number of bands is about how much weight one of them carries. Both changed here and they changed independently.

What is still open #

The offset. It is the second variable in every one of these experiments, no design here holds it, and the survivor plainly depends on it. A reading that predicts the survivor from the offset scores twenty-five of thirty and fails at five, and the five are not a fine-end artefact.

And the mechanism. Nothing in these two designs says why the contact structure should decide what a removal leaves standing. The displacement profile of a wrecked stem is the nearest thing to an answer this collection has, and it is a description of the damage rather than of the rule that produced it.

What a fourth quantity would look like #

Three is the number of things a rung moves, and it is worth asking whether it is the number of things there are.

A rung is defined by what a counter returns, so “the counted pair” is a label on a range of rises rather than a physical quantity. Inside that range the two contact steps have lengths, the neighbourhood has a size, the front has a depth, and each of those is a continuous function of the rise. Any of them could be the actor and none of them is separated by either design here.

So the three-quantity bookkeeping is bookkeeping over the quantities this thread has named, not over the quantities that exist. A band’s own widest member is the case where that bites: the survivor at a fixed offset changes somewhere inside it, with the pair and the divergence both held, so something the two designs do not name is varying.

The offset, which no design holds #

Every experiment here has two coordinates and the designs hold one of them.

A cut is named by a rise and by an offset — how many places back from the growing tip the removed organ sat — and the survivor depends on both of them. The offset rule scores twenty-five of thirty on the census, which is far better than anything stated over the rise, and it fails at five rows that are not a fine-end artefact.

Neither a band nor a matched pair varies the offset in a controlled way. A band holds the offset and varies the rise; a matched pair varies both, because the two stems of a pair wreck at mostly different offsets. So the offset’s own account is scored on the census, with all the mixing that implies.

Why the counted pair is a strange thing to be left with #

It is not a continuous quantity. A stem is 5/8 or it is 8/13; there is no rise at which it is halfway between, because the transition is where a counter’s answer changes.

That makes it an unusual candidate for a mechanism. The placement rule is a minimisation over distances and every quantity in it is continuous; the counted pair is a reading taken off the result, by machinery that is never shown the divergence. A discrete label being what decides a continuous process is either a sign that the label is standing in for something continuous, or a sign that the process has a genuinely combinatorial step in it.

The shape of the damage points at the second: two adjacent chains exchanged is a combinatorial event, not a small displacement, and the chains are what the counted pair counts.

What this essay did to the two designs #

Both got wider and both got a limitation.

The band went from two to six, and the sixth one holds all three quantities — a control nobody had. The claim it carries narrowed from “constant across a band” to “constant across a handover”, because a wide band turned out not to be a narrow experiment.

The matched pair is new and its limitation is that it produces four comparisons. There is no fifth on this ladder: the coarse end wrecks at nothing and the fine end has no further rung to pair with.

Neither of those is a defect to be repaired. They are what the ladder holds, and a design tested on everything a ladder holds has no sample left over.

The one line #

The rise moves the counted pair, the settled divergence and the step ordering together. A band holds the first two and shows the third does not decide the survivor; a matched pair holds the second and shows it is not sufficient either. The counted pair is the one left, it is not sufficient on its own, and it is the one quantity here that no design can isolate — because holding it is what a rung already does.