What a count cannot decide

Count the spirals on a head and two numbers come back. It is the oldest measurement in the subject, it is what a botanist can make in a field without instruments, and this collection makes it on positions rather than on an assumed angle so that it means the same thing on a run as on a plant.

It is also, by construction, blind to almost everything. That sentence is meant literally rather than rhetorically, and the rest of this essay is an inventory of what it is blind to, an inventory of what it is not, and an argument that the two lists are the same fact.

A counted pair is a topological reading. It follows chains of nearest neighbours and reports how many there are, and it returns the same answer for every arrangement in which those chains connect the same way. A whole rung — a range of rises — is by definition the set of geometries about which it says one thing.

That blindness is the measurement’s virtue. It is why a count is robust to how a plant grew, to how it was photographed, and to where up the stem it is read. It is also why four separate results this collection now holds are results about what the count cannot decide.

The virtue and the limitation are not two properties that happen to coexist; they are the same property described twice. A measurement is robust to a variation exactly when it is blind to it. Asking a counter to distinguish two stems inside a rung is asking it to stop being robust to the thing that makes it useful in the field, and no better counter would fix it — a counter that could tell them apart would be reporting something other than a spiral count.

Four things it does not fix #

Which contact step is shorter. Rank the lags by the distance the placement rule actually uses and the two contact families come first and second. Which of them is first reverses inside a single rung without the pair changing.

How deep the front is. The run of organs at which a removal is felt grows as the rise falls, so the number of places a cut can land and fail to heal grows with it — across one rung, from one offset to five. The larger counted number is often quoted as the front’s depth, and on the rises where both have been measured the two agree; what the pair cannot do is track the depth as it changes across a rung, because the pair is the thing being held while the depth moves.

Which family a wrecked stem keeps. At one offset on one lattice the answer changes between the coarse end of a rung and the fine one, which is the result that started this round.

And whether a comparison of two rules has a corner in it. Sweeping the rise moves the contact numbers, and the shape of the comparison changes with them — a corner, then none, then a corner again. That fourth one is different in kind from the first three and is worth separating. The first three are quantities a count holds constant; this one is a comparison between two rules whose outcome turns out to depend on the lattice they are run on. A count cannot decide it not because the count is coarse but because the thing being asked about is not a property of a single arrangement at all.

What it does fix, and why that is the strong part #

The temptation after four such results is to conclude that counting is a weak measurement. That would be the wrong lesson and the collection’s own strongest result argues against it.

The pair fixes which lags exist. A wrecked stem keeps one hop rigid, and at twenty-nine of thirty offsets that hop is one of the two the counter names. Not one of the twenty-four lags a stem could in principle keep — one of two, and the two are exactly what the count reports.

That is a restriction from twenty-four possibilities to two, made by a measurement anybody can take in a field, and none of the four results above touches it. The count decides the menu and not the choice.

The menu result is also the one with a mechanism behind it, which is why it is robust in a way the others are not. A rule minimising a sum of inverse powers of distance is holding its nearest neighbours; the nearest neighbours are what a contact family is; and a count is a way of finding the contact families from positions. So the restriction is not an empirical regularity that a wider census might overturn. It is close to a definition, which is exactly the kind of result a blind measurement can deliver.

Why the two get confused #

Because the menu is usually short enough to look like a choice.

On most lattices the two contact numbers are the only plausible answers to anything, so a rule stated over the pair and a rule stated over the geometry agree on nearly every row. They come apart only where the geometry moves and the pair does not — which is inside a rung, and inside a rung is exactly where no census had ever looked.

The agreement is not a coincidence and it is worth understanding rather than noting. The counted numbers are the lags of the two shortest steps, so any quantity that depends on those steps depends on the pair too — through it, but not only through it. Two stems with the same pair have the same two lags and different step lengths, and a rule stated over lags will agree with one stated over lengths until the lengths reorder. Which they do, once per rung, at a rise no count reports.

There is a second reason and it is linguistic. A rung is called a plateau, and plateau suggests a set of equivalent stems. It is a plateau in one reported number laid over a geometry that slides continuously beneath it, and the finest sweep this collection has run finds no flat stretch anywhere inside one.

Three things it can decide that look like it cannot #

The list of blindnesses is easy to over-extend, and three near-misses are worth recording so that the boundary is a line rather than a mood.

Whether an arrangement is jugate. A pair whose numbers share a factor is the signature of a pattern arriving several organs at a time, and a count reports the factor directly. What a count cannot do is confirm it — a wrecked stem can be counted at a pair with a shared factor and have no rotational symmetry at all — but the question of whether the factor is there is decided by the pair alone.

Which family is the larger. Trivially decided, and it matters because several rules in this collection are stated over “the smaller counted number” and are therefore safe from everything in this essay.

And whether two arrangements are on the same rung. Also decided by the pair, by definition — which is what makes a rung sweep possible at all. The sweeps that found the blindnesses depend on the count being reliable about exactly the thing it is reliable about.

What a counter is checked against #

None of this is an argument for trusting the count less. It is checked here in ways a field measurement cannot be.

It is made on positions rather than on an assumed divergence, so it cannot inherit the answer it is being used to test. It is checked against arrangements written down from formulas where the right pair is known in advance. And it is checked for band dependence, because a count taken at the wrong radius is a different number.

What the checks establish is that the count is right about what it reports. They say nothing about whether what it reports is what a given question needs, and that second thing is what four results have now had to discover separately.

That is a distinction worth carrying beyond this subject. A measurement can be validated exhaustively — against ground truth, against alternative implementations, against its own edge cases — and every one of those checks is a check on fidelity. None of them is a check on sufficiency, which is a relation between the measurement and a question rather than a property of the measurement at all. A collection that validates well and never asks the second question will accumulate correct numbers that do not settle the arguments they are quoted in.

The practical form #

Two sentences, and they are the useful residue of the whole round.

If a claim can be evaluated from a counted pair alone, a count is sufficient for it. Which lags are available, whether an arrangement is jugate, which family is larger — all decided by the pair.

If a claim mentions a length, a depth, a divergence or an angle, a count is not sufficient, and the rise has to be reported beside the pair or the claim is underdetermined. This is the rule the sampling essay arrives at from the other direction, and the two together are the round’s method result.

The second sentence needs one refinement, and it is the difference between a rule that works and a rule that looks as though it does.

Reporting the rise beside the pair is not quite enough, because the readings that depend on the rise do not depend on its raw value. They depend on where that rise sits between the ends of its own rung. The two contact steps change places part way along a rung; the front deepens along it; the settled divergence slides across the whole of it. Each is a function of position within a rung, and the same raw rise is a different position on a different rung and on the other branch. A row carrying 0.013 carries a number that cannot be interpreted without knowing which rung it fell in and where that rung ends.

So the sufficient description of a stem, for every claim in the second category, is the pair and the fraction of its rung the rise sits at. The pair says which rung; the fraction says where inside it. Between them they fix the divergence, the step ordering and the depth of the front — which is precisely the list a count alone leaves undetermined.

That is more demanding than it sounds, because the fraction is not observable without sweeping the rung. A rise is a number anybody can write down; the ends of the rung it falls in have to be found by growing stems either side of it until the pair changes. Every row published here carries the first and not the second, which is why the position within a rung can be recovered only for rungs that have been swept, of which there are two.

The consolation is that the requirement is finite. A rung’s ends are a property of the rung rather than of any experiment, so they are found once and then reused by every row that ever falls inside it.

What a plant would have to be measured with #

The practical consequence points outward, and it is uncomfortable.

If a claim about ablation depends on the rise, then testing it on a real plant requires the rise to be measured on that plant — and the rise is not a quantity field botany reports. A count is; a divergence sometimes is; the vertical spacing per organ relative to the circumference is not, and it is the quantity that has decided every result in this round.

It is derivable in principle from the same photographs a count comes from, because the contact geometry fixes it: two contact steps of known lags and known lengths determine the rise. That is arithmetic on quantities a careful photograph carries, and it is not currently part of anybody’s protocol — including this collection’s own specification for a survey it cannot do.

Where this leaves the subject #

The count survives all of it, in a smaller and better-defined role than the one it has usually been given.

A plant shows a pair, and a pair is a robust, transferable, checkable fact about it. What a pair does not carry is the rise, and the rise is what decides several of the things this collection has spent a round finding out it decides. Anyone comparing a run to a plant needs both, and until now this site has been recording one of them.

The honest summary is that the subject’s founding measurement is sufficient for the subject’s founding question — what pattern is this — and insufficient for most of the questions that follow it. That is not a criticism of the measurement and it is not news to anybody who has looked closely at one. It is simply a thing worth writing down in a collection whose whole method is to state what each instrument can and cannot settle, and which had gone eleven rounds without stating it about the one instrument every essay uses.