Nothing in the staircase moves

The mean number of sides a cell has in a tissue is six, and Euler’s formula leaves it no choice — not because hexagons are efficient, but because counting the edges two ways forces it. The mean squared departure from six is not forced, and it is the measurement — on heads of nine hundred organs it separates a whorled head from a random point set by a factor of eighty, where the mean separates them by nothing at all.

Swept across the divergence angle, that quantity is a staircase with dips at the rationals. And the reading that took it across head sizes established what it is made of on a spiral head: exactly the share of cells sitting on the defect rings the angle puts in closed form.

That essay ended by naming a problem with its own evidence. Every step of the staircase had been read at one head size, and a step in angle at one head size has two possible causes that a single sweep cannot tell apart.

Two things a step could be #

The first is the arrangement. Move the divergence angle and the lattice changes, different cells become defective, and $\mu_2$ moves. That is what the staircase is supposed to be reporting and it is a fact about the pattern.

The second is the rim. The defect rings sit at radii the angle determines — the same rings a closed form locates, where every disputed cell lies within two thirds of a wall spacing of one — so moving the angle moves the rings. A head of a fixed size is a disc of a fixed radius, and as the angle slides, a ring can cross that edge — in or out. The count jumps because the head suddenly contains one more ring of defects than it did, and nothing about the arrangement has changed at all. That is a fact about where somebody stopped counting.

The two are indistinguishable in one sweep and easy to separate in several. A feature that stays at the same angle as the head grows belongs to the angle. One that slides as the head grows belongs to the rim, because the rim is the only thing the head size moves.

Choosing the sizes #

At the golden angle the flip rings cross at 333, 871, 2280 and 5970 organs. Three of the four sizes here are the geometric midpoints of consecutive entries — 539, 1409 and 3690 — which is as far from an entry as a size can be placed.

The fourth is 900, and it is a control rather than a sample. It is the head size every other figure here reads $\mu_2$ at, and it sits 3.3 per cent past the entry at 871.

That is worth stating before the result rather than after it, because it is the kind of thing that is easy to present as a discovery once the answer is known. Nobody chose 900 to sit on a ring entry; it was chosen because it is a round number that gives a comfortable six hundred cells inside the rim cut, on the two lines of arithmetic every head here is built from. Had the answer to this essay’s question been yes, 900 is the one size of the four where it would have done the most damage, and every disorder reading here would have been built on it.

The window, and how finely it is read #

The sweep runs from 137.0° to 138.2° in steps of five thousandths of a degree — 241 samples, at each of four head sizes.

A step is found as a gradient rather than as a local minimum, and that choice is forced by the same thing that forced it when the dips themselves were located. At a large head a feature is narrower than the sampling grid, so no sample inside it is lower than its neighbours and a local-minimum test steps straight over it. At a small head a feature spans many samples for the same reason and again no single sample is a minimum. A gradient threshold sees both.

The threshold is a fifth of the curve’s own median, and the important thing about it is that it is the same fifth at every size. It was set by looking at the two ends: at a third the smallest head has no features at all and there is nothing to compare, and at a tenth the largest head’s ordinary sampling scatter is counted as structure. A threshold retuned per size would have manufactured exactly the nesting this essay is trying to test, which is the one way this measurement could have fooled itself.

Nothing slides #

Divide each curve by its own median — $\mu_2$ falls as one over the root of the organ count on a spiral head, and that fall is not what is in question — and the four curves line up at their features.

From 539 to 900: 4 of 4 kept, 14 added. From 900 to 1409: 18 of 18 kept, 13 added. From 1409 to 3690: 30 of 31 kept, 10 added.

52 of 53, which is 98.1 per cent.

The single exception is worth looking at rather than waving through, since it is the only candidate for a migration in the whole reading. It sits at 137.243° on the 1409-organ head, and its gradient is 0.251 — seventh weakest of that head’s thirty-one features, and a fifth above the threshold that admitted it at all. At 3690 organs the nearest features are at 137.197 and 137.273, which are 0.046 and 0.031 away, so the closer of the two misses the matching tolerance by one grid step.

Two readings of that are available and they are not the same. It could be a marginal feature that fell below threshold at the larger head, which is what the gradient suggests. It could be a feature that moved 0.031° between the two sizes, which is what a ring would do. What rules against the second is scale: the other fifty-two features sit within a few thousandths of a degree of their counterparts, so a single feature drifting by 0.031 is not a small version of what the others are doing — it is a different behaviour in exactly one case, at the weakest feature in the set. That is consistent with a threshold and unremarkable; it would take a second instance to be anything else.

So the answer to the question that reading left open is: none of them. Not one step of the staircase is a ring crossing the rim. The structure seen across the divergence angle is the angle’s, and the head size has nothing to do with where it is.

Whether there was anything to find #

A negative result is worth what the opportunity to find something was worth, and that opportunity has to be computed rather than assumed. If no ring crosses any of these rims anywhere inside the window, then “nothing slides” is not a finding — it is a sweep taken where nothing could have happened.

Rings do cross, and they cross often. Each family pair’s flip radius moves sharply with the angle — over this 1.2° window the pair that keeps eight families has its radius run from 4.09 to 5.60, and the pair that keeps thirty-four from 17.25 to 55.37 — so a fixed rim is crossed repeatedly as the angle slides. Counting them: three crossings inside the window for the head of 539 organs, three for 900, five for 1409 and four for 3690.

So there was somewhere for a ring-driven step to appear, at each of the four sizes, at angles that differ between sizes. That is the experiment working as designed.

The crossings are confounded, and two of them are not #

Reading the figure properly takes one more step, and it is the step that decides how much the result is worth.

Most predicted crossings sit within a few hundredths of a degree of a low-denominator fraction. At 900 organs the three crossings land at 137.153, 137.163 and 137.628, which are 0.010, 0.020 and 0.020 from 8/21, 8/21 and 13/34.

That is mechanism rather than coincidence. The flip radius diverges as a pair’s two families approach alignment, and alignment is what happens at a rational. So the rings pile up at exactly the angles the lattice already has features at — and at those angles the two hypotheses predict a step in the same place, and no amount of position-measuring separates them. A reading that had simply matched observed steps against predicted crossings would have found a great many matches and established nothing.

Three crossings sit clear of every fraction the attribution admits, more than five hundredths of a degree from the nearest. They are the only clean tests in the experiment:

• 539 organs, 137.208°, a ring of twenty-one entering. No step is observed there at all.
• 539 organs, 137.583°, again twenty-one. No step.
• 3690 organs, 137.518°, a ring of fifty-five. Steps are observed at 137.507 and 137.532 — but the same feature appears at 1409 organs, where no crossing is predicted anywhere near it. So it is an ordinary angle feature that happens to lie close to a crossing, not a step the crossing produced.

Two clean tests with nothing there and one explained away is thin evidence taken alone. It is not taken alone: it sits underneath the nesting result, which uses all 53 features rather than three, and the two say the same thing by different routes. The nesting test does not care whether a crossing is confounded with a rational, because a crossing is size-specific and a rational is not — which is precisely why it was the test the earlier reading asked for.

What the head size does do #

It decides how much of the staircase is visible.

Four steps at 539 organs, eighteen at 900, thirty-one at 1409, forty at 3690. Same window, same grid, same threshold — four times as many features from seven times the organs, and nothing in the curve suggests a ceiling.

That is precisely what a fixed object resolved more finely looks like. It is not what a changing object looks like, and it is not what a fixed number of features looks like either. The staircase is not a staircase with a definite number of steps that a good measurement would find; it appears to have structure at every scale the reading can reach, and the head size is the reading’s resolution.

Which reframes the factor that reading reported. The factor of eighty between a whorled head and a random one, measured at nine hundred organs, is a reading of one object taken at one resolution. The same caution applies to every other statistic read off a nine-hundred-organ tessellation here, Lewis’s law among them, where the slope on a phyllotactic head came out at 0.009 against 1.64 on a random point set — a comparison between two arrangements at one size, which is a different claim from a comparison between two arrangements. It is not wrong. But the staircase it sits on is deeper than any single head size shows, and “the staircase at nine hundred organs” is a statement about an instrument as much as about an arrangement.

Which fractions a head is large enough to show #

The added features are not added at random. They arrive at rationals of higher denominator.

A head of 539 organs shows steps at denominators 21 and 47 and nothing else. Every larger head shows 21, 34, 47, 55 and 60 — at 900 organs the steps sit at 8/21, 13/34, 18/47, 21/55 and 23/60.

The mechanism is already known here: a dip at a fraction is narrower the larger its denominator, and a small head cannot separate a narrow dip from the background. So the sequence of sizes is a sequence of resolutions, each admitting one more layer of the continued-fraction structure.

And then the attribution runs out, which is the most honest number in this essay. At 539 organs, 3 of the 4 features sit within three hundredths of a degree of a fraction of denominator sixty or less. At 900 it is 12 of 18. At 1409, 12 of 31. At 3690, 14 of 40.

The share falls because the head is resolving past the limit the attribution itself is good for. The attribution stops at denominator sixty for a stated reason: above it, every angle in this window is within a thousandth of a degree of some fraction, and saying a dip “is at a rational” stops carrying information. A head of 3690 organs resolves structure the attribution cannot name — so twenty-six of its forty steps are reported here as steps and not as anything else. They are almost certainly higher-denominator rationals. Almost certainly is not a measurement, and the sixty-limit is what stops this essay from claiming one.

What this does not establish #

That ring entries do nothing. They plainly do something, in the other direction: at a fixed angle, $\mu_2$ falls in teeth as the head grows, and that was measured exactly — the count moves in steps of twice a family number, and the fall goes as one over the radius in teeth. Nothing here revisits it. The claim is the narrow one that was open: the structure seen across the angle is not produced by rings crossing the rim.

That the staircase has structure at every scale. Four sizes over a seven-fold range show no ceiling, which is evidence and not a limit theorem. A head large enough might exhaust it, and the sizes reachable here do not say.

That 98.1 per cent is a property of the pattern rather than partly of the threshold. A feature within three hundredths of a degree is counted as the same feature, which is six grid steps; tighten that and some matches would be lost to the grid rather than to physics. What supports the conclusion is not the exact share but that no feature was found at a new angle — the failures are disappearances, not migrations, and a ring crossing the rim would have produced migrations.

And nothing here concerns a whorled or a random head. The rings are a spiral head’s rings, and the question was about a spiral head’s staircase.

What would withdraw it #

A feature present at one size and present at a different angle at the next. Across three steps in size and 53 features, there is one loss and no relocation.

A nesting share that depended on the threshold being tuned per size. It is one threshold, set from the extremes and applied unchanged; at a third the smallest head reports nothing, and the comparison says so rather than adjusting.

A feature count that fell with head size, which would mean the larger heads were resolving less rather than more. It runs 4, 18, 31, 40.

Or a step at 900 organs with no counterpart at 1409. All eighteen have one.

Still open: whether the steps stop #

The count of resolved steps rises across every size tried and the curve gives no hint of flattening, which raises a question this reading is the wrong size to answer.

If the staircase really has structure at every scale, then $\mu_2$ against the divergence angle is not a curve with features on it but something closer to a devil’s staircase — continuous, non-constant, and with a dip at every rational. That is a strong claim and the four sizes here do not come close to supporting it. What they support is that no ceiling is visible over a seven-fold range of head size.

The measurement that would decide it is a different shape from this one. Rather than widening the window, it narrows it: take one gap between two resolved steps at 3690 organs, sweep it at a hundredth of the present grid, and ask whether new steps appear inside it at the same rate the whole window produced them. A gap that stays empty at ten times the resolution puts a floor under the structure and makes the staircase finite. A gap that fills at the rate the window predicts makes it self-similar, and then the interesting quantity is not how many steps there are but the exponent relating a step’s depth to its denominator — which the dip-width measurement already defines and has only ever applied to the dips a single head size resolved.