A period the grid invented

The rule that places organs on this collection’s stems does not work in continuous angle. It samples the circle at a fixed number of azimuths, evaluates the repulsion from the existing organs at each one, and puts the organ at the least. Everything downstream inherits that resolution: a divergence is a difference of two grid points, so it is always a whole number of grid steps.

At the resolution the ablation work uses — 1,536 azimuths around the circumference — one step is 0.2344°. That is small against the quantities being measured, which are displacements of tens or hundreds of degrees, and it has been checked repeatedly that the answers do not move when the grid is made three times finer. This essay is about the one place where the check had not been made and the grid produced a result.

The result it produced #

Two organs removed from a settled stem at the coarsest arrangement sends it somewhere it does not come back from. Asked what the wrecked stem settles into, the routine that looks for exactly repeating motifs answered: a block of three angles — 219.84°, 220.31°, 220.78° — precessing by −59.07° per block.

That is a good-looking answer. Three is the smaller of the two parastichy numbers of the lattice that was cut, which is exactly what the two finer arrangements had given: five at 5/8 and eight at 8/13. And 360/59.07 is 6.09, so the block of three advances by one sixth of a turn, which is a two-jugate arrangement on twice the block — the same signature both finer arrangements had produced. It confirmed a prediction, in the right units, with the right internal relation between two independently computed numbers.

It is wrong, and the way it is wrong is worth more than the result would have been.

Three consecutive samples #

The three angles are 219.84°, 220.31° and 220.78°. Their differences are 0.47° and 0.47°, which is two grid steps; the three of them span 0.94°, which is four. Their mean is 220.3125°, which is a grid point.

There is no block. The stem has settled on a single divergence that does not happen to be representable on a grid of 1,536, and the sequence of placements oscillates between the two nearest representable values in a pattern that repeats with period three. The routine looking for exact repeats found an exact repeat, because there is one — it is a repeat of the rounding.

The test that settles it is to change the grid. At 4,608 azimuths the same cut on the same stem gives a block of two, with the same mean to four decimal places. A period that changes when the sampling changes and a mean that does not is a period belonging to the sampling.

What separates a real orbit from a rounded constant #

The distinction is not subtle once it is named, and naming it is the repair. A motif is an orbit if its angles are genuinely different from one another, and it is a rounded constant if they differ by about the resolution of the instrument that produced them. The quantity to look at is the span of the motif against the step of the grid.

At the coarse arrangement the wrecked stems’ motifs span three to four grid steps. At the two finer arrangements the same routine’s motifs span 574, 631, 917 and 964 steps — a hundred and thirty-four to two hundred and twenty-six degrees. Those are not near a threshold; they are three orders of magnitude apart, and any cut between twenty and five hundred separates them identically.

The test is now made wherever a period is reported, and it is made as a comparison rather than as a threshold on the period itself. A block of two is not suspicious because two is small; a block of two whose two angles differ by one grid step is suspicious because the instrument cannot tell those two angles apart.

It caught two more #

Applied to the arrangements it was not written for, the same test removes two cells from a table of twenty-two. At 5/8 a cut at offsets three and five settles at 280.43° and one at offsets four and five settles at 79.57°, each reported as a block of two with a motif spanning a single grid step.

Those two are constants, and they are a pleasing pair: 79.57° and 280.43° sum to 360°, so they are each other’s mirror. What they are not is orbits, and both had been counted as blocks of two in a table whose whole point was which numbers appear.

Two in twenty-two is not a large correction to that table. It is a large correction to a table with four rows in it, which is what the coarse arrangement had — three of the four rows were the same artefact.

The verdict should have three values, not two #

The test as stated sorts motifs into orbits and rounded constants, and that is one category short. A motif spanning three grid steps is not known to be a constant. It is known to be unresolved — the instrument cannot tell a genuine orbit of seven tenths of a degree from a constant the grid is jittering, because seven tenths of a degree is what the grid jitters by.

The distinction is not academic, because the two have different consequences. A constant is a settled divergence and belongs on the list of destinations as one entry. A small genuine orbit is a different object, and one this collection has seen: the slipped lattices precess by a whole number of turns of a surviving family, and nothing forbids a family whose turn is a fraction of a degree.

What separates them is one rerun, and the reasoning is the same as the arithmetic that exposed the block of three. An artefact’s span is a fixed number of grid steps; a real orbit’s span is a fixed number of degrees. Refine the grid threefold and an artefact’s span in degrees falls to a third while its span in steps stays at three or four; a real orbit’s span in degrees holds while its span in steps triples. The two predictions could not be further apart, and neither is a matter of interpretation.

That test has not been run on the three cells named here, so the honest description of them is unresolved, and this essay’s finding is properly stated as the block of three was not established, rather than as the block of three being a constant. The first is what the evidence supports. The second is the same overreach in the opposite direction, and writing it would repeat the mistake the essay is about.

What that leaves the tables saying #

Sorting the corrections by which claims they touch is short work and it is worth doing, because two of the three claims survive untouched.

The destination list is unaffected. Whether the coarse arrangement’s wrecked stems settle on 220.3125° exactly or orbit within a degree of it, they settle there, and the count of distinct destinations is the same either way. So the short list and the containment result stand as written.

The mirror is unaffected, and for a stronger reason. It was established by the divergence being 360° minus the original to the printed precision, together with the counted pair and the ranking of the three shortest steps — none of which is a statement about periodicity. A reflection that jitters by a grid step is still a reflection.

What is affected is any claim about which block sizes appear, which was the whole point of the table the three cells sit in. A table whose subject is which numbers turn up cannot carry cells whose numbers came from the instrument, and at the coarse arrangement that is three rows of four. The correct entry there is not a smaller number; it is a blank with a reason, which is what the routine now writes.

Why a confirming result is the dangerous kind #

The block of three was checked less carefully than it would have been if it had disagreed with anything, and that is worth saying plainly rather than leaving as an inference.

A prediction was standing: the wrecked stem’s block is the smaller parastichy number of the lattice that was cut. It had been measured twice and it had a mechanism-shaped story attached to it. The coarse arrangement had been the one place the prediction could not be tested, and the whole reason for building the two-organ intervention was to reach it. When the answer came back as three, with a precession that made it two-jugate on six rows, it did not look like a number that needed interrogating. It looked like the thing arriving.

The right protection against that is not more suspicion, which does not scale. It is to make the check that would have caught it a property of the report rather than a decision somebody has to remember to take — which is why the span is now returned beside every period, in the same object, whether or not anybody looks at it.

There is a second protection here and it is worth naming too. The claim the result confirmed has itself since failed on other lattices: swept across five rises and two branches, the block is not always the smaller parastichy number. So the artefact was confirming a prediction that was not going to survive anyway, and the two failures were found independently — one by changing the grid, one by changing the lattice. Neither would have found the other.

Why the routine was not at fault #

It is worth being clear that nothing in the code was broken. The routine’s contract is to find the shortest period p such that every value in the tail equals the value p places earlier to within half a degree, and to return nothing if no such period exists. On the coarse run that specification is satisfied by three, exactly and repeatedly, and returning it is correct behaviour.

The half-degree tolerance is the load-bearing part, and it was chosen for the right reason. Without it a genuine orbit would be missed whenever the grid rounded one of its angles differently on successive passes, and an earlier version that demanded exact equality found no period at all in runs that plainly had one. Half a degree is two grid steps: large enough to see through the rounding, and — this is the failure — large enough to see a pattern made of rounding as a pattern.

So the fault is in what was asked of the answer, not in the answer. A period is a summary statistic, and a summary statistic returned without the scale of the thing it summarises can be read as anything. The repair is to return the span alongside the period, which costs nothing and makes the ambiguity visible at the call site.

What the grid is in every other number here #

An artefact found in one place is a reason to look at the others, so the audit is worth recording rather than left implied.

Everything measured on these stems has been computed at both resolutions, and the list of what moves is short. The front — how many organs back a removal is felt — is the same integer at both. The displacement table agrees to a few tenths of a degree, which is the grid’s own step. Recovery times agree exactly. The settled divergence of an undisturbed stem agrees to four decimal places, and so does the settled divergence of a wrecked one, which is the comparison this essay turns on.

What moves is the scatter about a settled value and the period of a rounded constant, and those are the two quantities that are made of the rounding. The wrecked coarse stem scatters by 0.38° at 1,536 azimuths and by 0.06° at 4,608 — a factor of six for a factor of three in resolution, which is what a rounding error does and is not what a physical fluctuation does.

That is a usable signature. A quantity whose size falls as the resolution rises is an instrument’s quantity; a quantity that does not is the subject’s. Nothing here needed to know which of the two the scatter was until a period was read out of it.

The general shape of it #

This collection has now found the same defect three times in different clothes, and it is worth putting the three side by side.

A counter once returned the two smallest offsets rather than the two shortest hops, and the picture was no help because it drew twenty-one spirals and there were twenty-one of them. A cell-area statistic once fitted a strongly negative slope because sixty enormous boundary cells sat one ring inside the hull. And a period was once three consecutive samples of a grid.

In all three the machinery worked, the output was plausible, and the check that would have caught it was a comparison against a quantity from somewhere else — an angle recovery in the first case, a rim cut in the second, a second sampling resolution in the third. A number that comes out of one instrument and is not compared with anything cannot be wrong in any way that is visible.

The third protection, and the cheapest, is that a quantity worth reporting is almost always worth reporting twice. The mean and the period of a wrecked stem’s tail cost the same computation; one of them is robust to the sampling and one is not, and printing both would have made the artefact visible on the day it was produced rather than on the day somebody thought to change the grid.

What this does not say #

It does not say the finer arrangements’ blocks are suspect. They are not, and the test that removes the coarse one confirms them: motifs spanning hundreds of grid steps, unchanged at three times the resolution, at every offset that fails. The result they support stands and is argued elsewhere.

It does not say the grid is too coarse. For everything else measured on these stems it is not, and that has been checked: the front, the displacement table, the recovery times and the settled divergences all agree at 1,536 and 4,608 azimuths. What the grid cannot do is represent an arbitrary divergence exactly, and no grid can.

It does not say a short period is always an artefact. A block of two with a motif spanning two hundred degrees is a real two-cycle and this collection has drawn several. The test is about the span, not the length.

It does not say every constant is a mirror. Two of the three cases here turn out to be their own arrangement’s mirror and one does not, and that is a separate finding with a separate essay. The test in this one is indifferent to where the constant sits; it only says that a constant is what it is.

And it does not say the tolerance should be tightened. Tightening it would trade this failure for the one it was raised to fix, which is missing genuine orbits. The right repair is to report the span, which loses nothing.

The check that would refuse it #

Two assertions, and they point in opposite directions on purpose.

The first requires every reported period at the coarse arrangement to have a motif narrower than twenty grid steps — that is, to be a rounded constant. If a genuine orbit ever appeared there, this would fail, and it should: the essay’s claim is that the coarse arrangement does not produce one.

The second requires every period at the finer arrangement whose block is a parastichy number to have a motif wider than the same twenty steps. Without it the first check is satisfied by a threshold set above everything and says nothing about what an orbit looks like. Together they are a claim that the two populations are separable and that the separation is not close: the widest rounded constant here is four steps and the narrowest orbit is five hundred and seventy-four.