The survey this site cannot do

Every frequency, angle and threshold on this site is about the geometry or about a model, and every essay that quotes one says so. That sentence has been in this collection’s plan for three phases as a deferral: a real dataset, and it is the largest thing outstanding.

It is still outstanding. This essay is what replaced it, and the replacement turns out to be more than an apology, because a missing dataset can be specified — and specifying this one changed two of the four things it was going to ask for.

What was going to be asked for #

The plan, as it stood at the end of the previous phase, named four fields a survey should record: the counting radius or height, the element count, the rotational symmetry, and the handedness. Each was justified and the justifications were sound. Two of them do not survive the arithmetic.

The counting radius was first and most emphasised, on the foundation phase’s finding that the counts change with radius, so every published count is a statement about an annulus. That reasoning is right. The conclusion — that the radius is what most needs recording — is wrong by a factor of ten. Given the pair, knowing the rise narrows the divergence angle by 1.10, because a report that names the pair has already said which annulus. The radius stays on the list for two other reasons, below, and not for the one it was on it for.

Handedness — whether the spirals run clockwise or anticlockwise — was on the list and this collection has no question that needs it. Handedness is a global sign, it is equally likely either way in every model here, and no claim on this site would be affected by any distribution of it. It comes off the list. It was there because it is a thing one notices about a plant, which is not a reason.

The element count stays, promoted. It is what says which rung a count was taken on, and the previous essay measures that what a count is worth scales as $221°/mn$ — so the rung is not context, it is most of the information. It also fixes the null for the census question, which depends on the rise a survey pooled its counts at.

The rotational symmetry stays, unchanged. The previous phase established that the counts genuinely cannot decide jugacy: an ordinary lattice at 180.5° counts 2 and 4 with symmetry of order 1, and a genuine bijugate stem counts the same pair with order 2. Only the symmetry separates them, it is a glance at the specimen, and nobody records it.

And one that was never asked for #

The individual gaps between rings, on any organ that shows more than one transition.

Nothing in the previous plan mentions them, because until this phase there was no question they answered. The cylinder field’s three new essays supply one: an organ’s shape exponent varies along its axis, a single fitted exponent returns the harmonic mean of what its steps report, and the variance the fit averaged over is exactly what the gaps carry and the fitted exponent throws away.

The requirement that comes with it is the most demanding on the list — five rings per specimen, each to within about three per cent, from a dozen specimens of one species — and the smallest useful version of it needs no exponent at all: four rings on a dozen cones, and the question is whether the gaps increase from tip to base.

The specification #

With the sampling stated: specimens chosen without reference to their pattern, from a named population, with the uncountable ones recorded as uncountable rather than dropped.

Why a field costs nothing at the bench and cannot be recovered later #

The phrase free at the moment of counting runs through this specification and it is worth unpacking, because it is the whole reason a small survey has never happened.

Counting a sunflower head is the expensive part. Somebody has to obtain the head, hold it flat, find a family of spirals, trace it round, and not lose count — several minutes of concentration for one number, and the number that comes out is the pair.

Every field on the list above is a note taken while already doing that. Counted at the rim is four words. Primordia arrive singly is a glance at the youngest ones. A ring position is a ruler laid against a cone that is already in the hand. None of them requires going back to the plant, obtaining a second specimen, or doing anything twice.

And none of them can be added afterwards. A published count of “34 and 55” cannot be annotated with where it was taken; the head has gone. That asymmetry — trivial to record, impossible to reconstruct — is what makes a specification worth writing before rather than after, and it is the same asymmetry the cylinder field found for ring gaps: the gaps are what was measured, before anything was done to them, and a fitted exponent is a summary that cannot be un-summarised.

What it would settle, and what it would cost this collection #

A specification is only worth writing if it could come out against the people writing it, so it is worth saying what each result would do to this site.

If the Fibonacci share came out near the geometry’s 14.7 per cent, the interpretation this collection has held since its expansion phase — that continuity from a coarse start is what makes plants Fibonacci — would be in serious trouble, and the grown-history measurement that supports it would have to be re-examined as a property of the rule rather than of plants. Four specimens would start that conversation and fourteen would settle it.

If multijugate patterns came out at two per cent rather than fifteen, the previous phase’s k-jugate machinery would remain correct and would be about a curiosity. The essays that treat bijugate teasel as a real and separate family would need their emphasis changed, though not their arithmetic.

If a conifer cone’s ring gaps came out constant, the ogive model of its shape is wrong, or the ladder does not apply to it, and the cylinder field’s last three essays would need rewriting. One specimen at three rings starts that, and a dozen at four settles the direction.

And if the gaps came out decreasing, something is wrong that this collection has no account of at all, since no surface of revolution that widens outward produces it.

Each of those is a real exposure. The collection would be poorer in a specific and stateable way, which is the property a set of claims should have.

What it would not settle #

Three things this collection would like to know and a survey of finished plants cannot supply, listed because a specification that promises everything is not one.

Where the noise in an apex enters. The emergence field measures that a disturbance to the choice can change a pattern’s branch and a disturbance to the position cannot, and that the two are indistinguishable in the finished pattern’s divergence scatter. Only watching primordia as they are specified separates them, which is live imaging rather than a survey.

Whether the rate threshold is real in plants. The branch threshold at about ninety nodes per rung is a statement about a growing shoot, and a finished stem carries no record of how fast it grew. A survey could correlate branch outcome with something like internode length as a proxy, and a proxy is what it would be.

And the divergence angle itself, at the precision this site works at. The angle between two visible organs on a mature stem includes whatever the internode has done since — torsion, differential growth, a season of carrying a leaf — and the tolerance measured in the emergence field is about two degrees. A field measurement is not going to resolve that, and the honest form of every angle question here is the one this collection already uses: ask about the pair.

Why this is the right substitute for the data #

The temptation, when a promised deliverable does not arrive, is to substitute something adjacent and not say so. This collection’s own record on that is mixed and worth citing: the previous phase called for a continuous formulation of the dynamical model, abandoned it for a good reason, and delivered a closed form for the branch points instead — which reached further than the plan and was not the plan.

The difference here is that the substitute is about the missing thing rather than beside it. The specification is what a dataset would have to be, computed from the same machinery every other claim on this site rests on, and it makes the absence concrete in a way that four phases of noting it did not.

It also produced two corrections that only doing the arithmetic could produce. The counting radius was the headline request for three phases and is worth ten per cent for the question it was justified by. Handedness was on the list and answers nothing. Neither of those was going to be discovered by asking more loudly.

Which species, and why the question has no general answer #

A specification that says “specimens” without saying of what is not finished, and the reason this one stops short is worth stating rather than hiding behind a placeholder.

The census question needs a population in which the pattern is countable at a stated rung, and different organs put that rung in different places. A sunflower head counted at the rim reaches 34/55 and is worth a tenth of a degree; the same species counted near the middle reaches 13/21 and is worth eight tenths. A pine cone reaches 8/13 at best over its countable length. A rosette may not reach past 3/5.

So “fourteen plants” is fourteen plants of a species whose rung is stated, and mixing species is mixing nulls — the geometry’s Fibonacci share depends on the rise, from 67.5 per cent at a coarse one to 14.6 per cent at a fine one, so a pooled survey is being compared against a null it cannot compute.

The recommendation that follows is narrower than the one this collection has been making. One species, one rung, fourteen specimens answers the census question cleanly. Fourteen specimens of fourteen species answers nothing, however carefully each is counted, and it is closer to what the existing literature consists of.

That is a fifth correction to the original plan, and the least expected: the request was for a survey, and what is actually needed is a survey of one thing.

The one-line version #

Fourteen plants, counted at a stated rung, with a note of whether the primordia arrive singly or in pairs. That is the census question and the jugacy question most of the way, and it is an afternoon.

A dozen fir cones, four ring positions each, and the question is whether the gaps go up. That is the shape question at its smallest, and it is a second afternoon.

Neither has been done here, and neither needs anything this collection has. What they need is a plant, which is the one thing a set of computed figures cannot supply, and saying that precisely — with the sizes, the fields and the exposures attached — is what this phase could do about it.

What happens to this deferral now #

A deferral that has been carried for three phases and is being carried into a fourth deserves an account of why, and of what would change it.

The honest reason is a boundary rather than a difficulty. This collection is a set of computed figures with a stated rule behind each one; its whole method is that every claim is checkable from the repository, and a claim resting on somebody’s field notes would not be. Adding measured data would not be an extension of the method, it would be a different method sitting alongside it, and the questions of provenance, error and selection that come with it are precisely the ones this specification spends its length on.

So the deferral is not being carried because the work is hard. It is being carried because the work is of a different kind, and this phase has done the part that is of this kind.

What would change it is somebody else’s data. A published survey with the rung recorded would settle the census question against the arithmetic here without anybody needing to grow anything, and the specification exists partly so that such a survey, if it exists somewhere unnoticed, can be recognised as sufficient. The fields are stated, the sizes are stated, and the comparisons are computed. What is missing is fourteen plants and a note of where each one was counted.

The general form, for a collection that has a gap like this #

Most bodies of work have one — a measurement everything else is waiting on, which never arrives and gets mentioned in every summary. There is a pattern in how they are handled that this phase would recommend against, and a substitute.

The pattern to avoid is repeating the request. The data is still missing is true, gets truer, and does nothing; and it has a cost that this phase found by accident, which is that a request repeated is a request that stops being examined. Three phases of leading with the counting radius, and the two minutes of arithmetic that would have shown it worth a tenth were never spent, because the reasoning behind it was sound and nobody re-derives a sound argument.

The substitute is to compute the specification. What fields, how many specimens, what each would decide, and what would have to be withdrawn if it came out the other way. That is finite work, it can be done entirely from inside, and it converts an absence into a statement — one which is itself checkable, which is more than the absence ever was.

It also has the property that it can fail usefully. This one did, twice: a field that was worth a tenth of what its prominence implied, and another that answers nothing at all. Neither error was visible while the request was a request.

Where this leaves the wrong field #

This field of the collection exists to take the confident statements the subject arrives with and give each one a test and a number. The nautilus is out by a factor of 2.14. Sunflower spirals are Fibonacci far less often than the geometry is generous — 14.6 per cent of divergence angles at a fine rise — and what makes plants Fibonacci is continuity rather than optimality. The golden angle’s claim to be special survives, in a sharper form than the version usually told, and it is arithmetic rather than packing.

Every one of those is a statement about geometry with a plant-shaped hole in it, and this essay is the shape of the hole drawn accurately for the first time.

That is a real advance and it is not the same as filling it. The field’s essays will continue to say this is about the geometry wherever they quote a frequency, because they still are. What changes is that the sentence now has a footnote with a number in it: fourteen plants, one species, one rung, and here is what each outcome would do to the argument.

A claim with a stated price for refuting it is in a different condition from a claim with an open-ended appeal for data, even when neither has been refuted. This phase’s contribution to the wrong field is to have put the price on.