The front that reads one short

The front result says that removing one organ is felt out to the larger of a stem’s two spiral counts and no further. Put to a design of fourteen cells — seven rises, two branches — it holds at eleven of them exactly. At the other three the run of felt offsets stops one place early: a 4/7 stem whose front reads six, a 7/11 stem whose front reads ten, an 8/13 stem whose front reads twelve.

There is a cheap way to make that go away. The boundary is decided by a threshold — the next organ counts as moved if it moves by more than two and a half degrees — and the offsets in question move it by 1.88°, 2.34° and 1.41°. Lower the threshold to one degree and all fourteen cells agree.

That would be a mistake, and not a subtle one. A threshold chosen so that a claim comes out true is not a measurement of anything, and the resulting statement — the front is always the larger parastichy number — would be unfalsifiable by construction. So the three cells are reported as they are, and this essay is about what they turn out to be.

What the boundary looks like when it is sharp #

Start with a cell where nothing is in doubt, because the contrast is the whole argument.

At a rise of 0.010 the golden stem carries 5/8. Removing the organ one place back moves the next one by a large angle; so does two, three, and on out to eight. The displacement at offset eight is 8.91°. At offset nine it is 0.47°, at ten less still, and it stays there out to twelve.

That is a step with a factor of twenty across it, and any threshold between one and eight degrees returns the same answer. The measurement does not depend on the number chosen, which is what a good boundary looks like.

Now the same measurement at a rise of 0.016, on the same branch and the same lattice. The displacement at offset eight is 2.58° — barely over the threshold — and past it, 0.70°. The step is still there and it is much shorter.

The three short cells are the three newest rungs #

The pattern in that is not obvious from the rises and is obvious from the ladder.

A stem’s parastichy pair changes as its rise falls, and the rises at which it changes are computable: they are the rises at which two lattice steps have equal length, solved from the stem’s own settled divergence rather than fitted. The golden branch enters 5/8 at a rise of 0.0182 and leaves it at 0.0069. The Lucas branch enters 4/7 at 0.0249 and leaves it at 0.0095.

Line the design’s cells up against those boundaries and the three short cells are the three that have most recently entered their rung.

The golden 5/8 cells sit at 1.13, 1.39, 1.82 and 2.27 times below their entry rise, and the displacement at offset eight runs 2.58°, 4.92°, 8.91°, 11.95°. The Lucas 4/7 cells sit at 1.19, 1.53, 1.91 and 2.50, and the displacement at offset seven runs 1.88°, 6.33°, 11.25°, 15.47°. The Lucas 7/11 pair gives 2.34° and 5.16°; the golden 8/13 cell, alone in its rung and only 1.14 below its boundary, gives 1.41°.

Every one of those sequences increases, at every step, without exception. The three cells that read short are the three lowest values, and each is the topmost cell of its own rung.

The three cells, one at a time #

Averages and trends hide the cases they are made of, so here are the three, with what each one actually shows.

The Lucas stem at a rise of 0.020. Its settled divergence is 101.76° and a blind counter returns 4 and 7. The run of felt offsets is one through six; offset seven displaces the next organ by 1.88° and offsets eight through eleven by less. Its rung entry is at a rise of 0.0249, so it sits 1.19 times below the boundary — the second-highest position in the whole design.

The Lucas stem at 0.008. Divergence 99.08°, counted 7 and 11. Offsets one through ten are felt; offset eleven gives 2.34°, which is within a fifteenth of the threshold, and past it the largest anywhere is under half a degree. It entered 7/11 at a rise of 0.0095 and sits 1.19 below it.

The golden stem at 0.006. Divergence 137.93°, counted 8 and 13. Offsets one through twelve are felt, offset thirteen gives 1.41°, and it entered 8/13 at 0.0069, sitting 1.14 below. It is the newest rung in the design and it gives the smallest displacement in the design.

The ordering by position in the rung is exact: 1.14, 1.19, 1.19 for the three short cells, against 1.13 for the one cell that clears the threshold by a fourteenth of a degree and 1.39 and upwards for everything that clears it comfortably. There is one apparent inversion — the golden 5/8 cell at 1.13 sits lower in its rung than any short cell and reads correctly — and it is the useful part of the data rather than an embarrassment: at 2.58° it is on the threshold, and it says the effect is a continuous shortening of the last stair rather than a switch that flips at a particular depth.

Why the newest offset is the weakest #

The reason is what a rung boundary is.

At a boundary, two candidate second families have equal step length. Just below it, the new family’s steps are barely shorter than the old family’s, and the new count has only just become the larger of the pair. So an organ exactly that many places back is, at that rise, only marginally a member of the neighbourhood the rule is sensitive to. Further down the rung its steps shorten relative to everything else and it becomes solidly one.

So the front does not have a hard edge that sometimes slips. It has an edge whose last stair varies in height with position in the rung, and near the top of a rung that stair is under two degrees.

This also predicts something the design did not set out to test and confirms it in passing: the second-newest offset should be stronger, and it is. At the golden 8/13 cell — the shortest-lived rung in the design — offset thirteen gives 1.41° and offset twelve gives 24.6°, which is why the front reads twelve rather than eleven.

What the threshold is, and why it is not moved #

It is worth stating what the threshold is set from, since the argument above depends on not adjusting it.

Two and a half degrees is a tenth of the local spacing between organs at the rise the ablation work is centred on, and four steps of the azimuth grid the runs are computed on. It was fixed before any of this was measured, on the grounds that inside the front displacements run from a few degrees to a hundred and seventy and outside it they run under half a degree — so anything between half a degree and two degrees gives the same table at the cells where the boundary is sharp.

At the three short cells that empty band is narrow, and the honest description is that the measurement has become threshold-dependent there. Reporting eleven of fourteen with the other three named, their displacements printed and their position in the rung explained is what that description looks like. Reporting fourteen of fourteen at a threshold of one degree would be reporting the threshold.

The same shape, elsewhere in this collection #

A boundary that is sharp in the middle of a rung and soft at its top is not a new kind of object here, and the earlier cases are worth putting beside it.

The response of a stem to a removal is an interval of offsets only in the middle of a rung. Near a transition it acquires a hole: a stretch of quiet, and then one isolated offset well outside the front that is felt strongly. That was measured on the golden branch and its position was pinned to the count coming in at the next rung, less one.

The blind counter has the same character. Given an ideal lattice in the middle of a rung it returns a pair with a wide margin — the third-shortest step is much longer than the second. Near a boundary the second and third are nearly tied, and the counter’s answer, while still correct, is one a tenth of a degree of noise could change.

So this essay’s finding is the third instance of one thing. Every property of a lattice that depends on which family is second degrades near the rise where two families are tied, and the front is such a property, because the front’s depth is the larger of the two counts. Reading that as a defect of the front measurement would be reading a property of lattices as a property of an instrument.

A consequence for anybody doing this on a plant #

The result has a use, and this essay changes what it is worth in a specific way.

The attractive thing about the front is that it turns a spiral count into a sequence of yes-or-no answers, which is a far cruder measurement than counting spirals on a photograph. This says the crudeness has a limit that depends on where the specimen sits on its own ladder. On a stem well inside a rung, the last offset of the front displaces the next organ by ten degrees or more and any competent observation catches it. On a stem that has just changed rung, the last offset displaces it by one or two degrees, and an observer will systematically read the front one short.

That is a bias rather than noise: it always errs in the same direction, and it errs on exactly the specimens that have most recently changed their counts. A census of ablation experiments that did not record where each specimen was on its ladder would under-report the front, by one, on a subset it could not identify afterwards.

The repair is cheap and it is the same one this collection keeps arriving at: record the counts and the geometry. A specimen whose pair has just changed is identifiable — its two families have nearly equal step lengths, which is measurable on the same photograph the counts come from.

What a soft edge does to the yes-or-no reading #

The front’s appeal is that it converts a count into a run of answers to a much easier question. It is worth working out what a soft last stair costs that conversion, because the answer is not “a bit of noise”.

Suppose an observer removes an organ at each offset in turn on a set of clones and records whether the next organ moved. Inside the front the displacements are tens of degrees and the answer is unambiguous however the observation is made. At the last offset of a stem near the top of its rung the displacement is one to two degrees, which is comparable to the divergence scatter this collection measures in its own noisy runs and comparable to what a protractor on a real apex would resolve.

So the failure mode is not a wrong count with a wide error bar. It is a count that comes out exactly one less, confidently, with every other offset behaving perfectly. An experimenter would have no internal signal that anything had gone wrong: the run would be consecutive, the region past it quiet, the boundary clean.

That is the reason to state the three cells rather than tune them away. A known bias of exactly one, on an identifiable subset of specimens, is something a census can correct for. An unstated one is a census that reports the wrong integer.

What this does not say #

It does not say the three cells refute the front result. At each of them the run of felt offsets ends one place early and everything past it is quiet, so the shape of the result is intact; what varies is the height of the last stair. A cell whose front read four short, or whose felt offsets were scattered rather than consecutive, would be a different matter and none does.

It does not say the strengthening is linear, or that it has been characterised. Four points per lattice on two lattices, plus two shorter series, is enough to say the ordering is always the same and not enough to fit anything. The claim is monotonicity and nothing more.

It does not say the threshold is arbitrary. It is set from the spacing and from the grid, and at eleven of fourteen cells it could be moved by a factor of five with no effect. The three cells are where that stops being true.

And it does not explain the two cells that are not in the pattern. The Lucas stem at a rise of 0.024 carries 3/4, which the ideal ladder says it should already have left; it is holding a pair past its own boundary, so it has no position in its rung to measure and it is excluded from the comparison rather than fitted into it. That hysteresis is real and is another thread’s subject.

The check that would refuse it #

Two assertions run with the figure, and the second is the interesting one.

The first is that within each lattice appearing at two or more rises, the displacement at the offset equal to the larger parastichy number grows at every step down the rung. That is four sequences with no exceptions allowed, and a single inversion anywhere stops the collection being built. It is the assertion that would fail if the pattern were a coincidence of two cells.

The second is that every cell whose front reads short is one whose newest offset falls under the threshold — which sounds circular and is not. The front reading short means the run of consecutive felt offsets ends early, and it would also happen if some offset in the middle of the run were quiet, or if the offsets past the boundary were noisy enough to make the run ambiguous. The assertion says the short reading is always attributable to that one offset, and it would fail if a short cell were short for any other reason.

There is a third guard on the cells that are excluded. A stem carrying a pair the ideal ladder says it has already left has no measurable position in its rung, and rather than fitting one it is dropped from the comparison and reported as lagging. That is one cell of fourteen, and saying so is cheaper than the alternative, which is a design that quietly contains a specimen it cannot place.