Half a turn, four at a time

The share of two-organ cuts that reverse a stem falls with the width of its front — 6.79 per cent at a front of three, 4.69 at five, none at eight. That is the ordering the reachability account predicted and it is the smaller of the two things the coarse sweep found.

The larger one is that at four of the nine coarse rises not a single wrecked cut reverses, and what those stems do instead is not settle at all.

The four-cycle #

At a rise of 0.060, thirteen of the thirty-six arrangements never repair. Every one of them settles into the same thing, and it is not a divergence. It is a sequence of four:

171.09°, 269.53°, 189.14°, 90.23°, and then 171.09° again, without end.

The four add to 719.99°, which is two whole turns over four organs. The mean is 180.0000° — half a turn — to four decimal places, and it is the same four numbers at every one of the thirteen cells.

Across the whole coarse rung, forty of the seventy-one wrecked cells end in a cycle whose mean is within half a degree of a half turn. Twenty-two reverse. Nine do neither.

Why it is not a slip #

The fine rung’s wrecked stems keep a family of the old lattice standing, and the first thing to ask of a coarse cycle is whether it does the same.

It does not, and the test is the one the fine rung’s result was established with. For each lag from one to twenty-four, take the angle from each organ to the one that many places above it, over the last hundred and twenty organs, and compare its mean with the same lag on the control. A surviving family shows up as a lag whose spread is a fraction of a degree and whose mean has not moved.

In a cycling coarse stem there is no such lag. The stem’s divergence is not even approximately constant — it takes four values in rotation, sixty-three degrees of spread — so no lag has a steady hop to begin with, and the comparison has nothing to report.

That is the sharpest way to state what the two fates are. A reversal keeps everything about the lattice except its handedness. A cycle keeps nothing: not the divergence, not a family, not the counted pair. They are the two extremes of what a removal can do, and the coarse rung is where both are available.

It is a cycle and not a rounded constant #

A period reported by a period-finder is worth checking before it is believed, because a motif narrower than the grid the azimuths are computed on is a constant the grid could not write down rather than an orbit. This collection has been caught by that once and the check has been in the machinery since.

The four-cycle’s motif spans 750 to 767 steps of the azimuth grid. The mirrored cells’ motifs span zero to four. So the two populations are not near each other and the distinction is not a matter of where a threshold is put: one kind of cell is a constant the grid rounds and the other is an orbit that crosses most of a turn on every pass.

What the four numbers are #

The motif is worth looking at closely, because its structure is not arbitrary and the arithmetic in it is checkable.

The four divergences are 171.09°, 269.53°, 189.14° and 90.23°. Two of them sit near half a turn and two sit near a quarter and three quarters — 90.23° and 269.53° are within a third of a degree of 90° and 270°. So the cycle alternates between organs placed nearly opposite the previous one and organs placed nearly at right angles to it.

The sum is 719.99°, which is two turns to within a hundredth of a degree. That means the pattern closes: organ i and organ i + 4 sit at the same azimuth, four heights up, which is exactly the statement that the four-hop is vertical and there are four files.

At the rise of 0.055 there is a second cycle as well, of period six, averaging 198.91° and counted 3/4. Its motif is 280.6°, 165.9°, 107.1°, 160.1°, 271.9° and 207.9°, summing to 1193.5° — three turns and a hundred and thirteen degrees, so it does not close, and its mean is nine degrees off half a turn. It is reported here rather than folded into the count, because a table that folded it in would be reporting one phenomenon where there are two.

Four files where there were three #

A counter shown the positions of a cycling stem returns 1/4 — or 2/4, or at one rise 3/4. Every one of those pairs has a four in it, and the lattice the stem was cut from is counted 2/3.

That is what a mean divergence of half a turn does. Two consecutive steps at 180° come back exactly to the start, so consecutive organs alternate between two sides and the pattern has files. The wobble around the mean turns two files into four: the organs on each side are not quite aligned, so what would have been two rows splits into a pair of near-rows each.

The shared factor is doing here exactly what it does at the finer rung: reporting where the divergence landed, not how many organs arrive at a time. Rotate a cycling stem by a half turn or a quarter and nothing lands on anything.

The motif read as positions rather than as steps #

The four divergences look wild — a step of 171°, then 270°, then 189°, then 90° — and the arrangement they build does not. Accumulating them turns a strange sequence into a plain picture, and the picture explains three things the essay reports separately.

Start an organ at zero and add the steps. The four organs of a period sit at 0°, 171.1°, 80.6° and 269.8°, and the fifth returns to zero. Sorted round the stem those azimuths are 0, 80.6, 171.1 and 269.8, with gaps of 80.6°, 90.5°, 98.7° and 90.2°.

That is four files at ninety degrees, distorted by about nine degrees. The “wobble around a mean of half a turn” and the “four files” are not two facts; they are the same arrangement described once through its steps and once through its positions, and the positions are the description in which nothing is surprising.

Which hops are short, computed rather than counted #

With the positions and the rise in hand, every hop length in the cycling stem can be worked out directly, and doing so predicts what a counter will report without running one.

At a rise of 0.060 an organ four places up sits 0.24 of a circumference higher and at the same azimuth, so the four-hop is 0.240 and it is the same at every organ. The others are not the same at every organ, which is the whole content of the motif being a cycle rather than a lattice. Measured in circumferences, the two-hop is 0.254 from half the organs and 0.299 from the other half; the one-hop is 0.258 from half and 0.479 from the other half; the three-hop is 0.309 and 0.508.

Three things fall out of that table and each answers something the essay reports as an observation.

The four is the shortest and is the only uniform family. 0.240 beats everything, and alone among the families it does not alternate. So a counter following chains of near neighbours finds four chains that run cleanly and nothing else that does, which is why every pair it returns has a four in it.

The second number is a near-tie among three candidates. The shortest two-hop is 0.254, the shortest one-hop is 0.258, and the shortest three-hop is 0.309 — the first two within two per cent of each other and both within seven per cent of the four. A ranking that close is decided by a small change in the motif, and the motif does change a little from rise to rise. That is exactly why the counter returns 2/4 at some rises, 1/4 at others and 3/4 at one, and it means the second number is not carrying information about the arrangement so much as reporting which of three nearly equal candidates won.

And half of every other family is long. A family whose members alternate between 0.254 and 0.299 is not a family in the sense a lattice uses the word: the chain a counter follows is short on one step and long on the next, all the way up the stem. Only the four-family is a family in the ordinary sense.

So the honest description of a cycling stem is not a lattice counted 2/4. It is a four-file arrangement with one genuine family and no second one, which a counter is obliged to report as a pair because a counter returns pairs.

That also sharpens what makes the four-cycle a state of the rule rather than damage. Every number above came out of the four divergences and the rise, with nothing fitted — and the four divergences are the same at all thirteen cells, so the whole hop structure is too. A residue of the particular damage would not reproduce a hop table to three decimal places from thirteen different cuts.

The same four numbers from different cuts #

Thirteen arrangements at one rise, and every one of them produces the same motif. That is worth reporting on its own, because it says the cycle is a state rather than a residue of the particular damage.

The cuts that reach it at a rise of 0.060 remove organs at offsets 1 and 3, 1 and 4, 2 and 3, 2 and 4, 2 and 5, 2 and 6, and so on — arrangements that differ in how much of the front they take and in where they take it. All thirteen settle on 171.09°, 269.53°, 189.14°, 90.23°, to the resolution of the azimuth grid. Two of them come out phase-shifted, starting the cycle at a different one of the four, which is what a period-four orbit reached from a different transient should do.

A residue of the damage would look different from cut to cut. A state does not, and this does not. The same thing is true of the mirrored cells at the rises that mirror: seven arrangements, one destination, agreeing to four decimal places.

No finer rung reaches it #

Wrecked stems that repeat a fixed cycle are not unusual. The 5/8 rung does it at thirty of its thirty-two wrecked cells, and the periods those cycles have are the periods of the lattice they came from — five, eight, and their multiples. That is the whole subject of the essays on what a wrecked orbit inherits.

What is unusual at the coarse rung is the cycle’s mean. Forty of seventy-one coarse cells average half a turn to within half a degree. The nearest any cell at either finer rung comes to half a turn is 175.01° at 5/8 — five degrees away — and 187.47° at 3/5, which is seven and a half.

So the coarse rung has somewhere to go that the finer ones do not, and the somewhere is the coarsest arrangement there is.

Twenty-two, forty and nine #

The three numbers do not add to seventy-one by accident, and the nine that are neither are worth naming rather than left as a remainder.

Two of them are cells whose divergence sits near half a turn and whose motif the period-finder could not resolve — the tail repeats something, but not at any period up to twenty-four with a motif wide enough to be an orbit. Six are the period-six cycle at 198.91° described above, which averages nine degrees off half a turn. One is a cell at a rise of 0.075 averaging 179.97°, which is inside the half-turn band and carries a period of eight rather than four.

None of the nine is a third destination in any interesting sense; they are two resolution failures and a neighbouring cycle. Reporting them separately rather than assigning them is the difference between a table that adds up and a table that has been made to.

The rise that does both #

At a rise of 0.075, seven arrangements never repair. Four of them reverse onto the mirror and three fall into the cycle.

The two-organ table this collection draws at finer rungs cannot be drawn here, and the reason is worth more than the picture. Its own check requires the run of felt offsets to end at the larger of the two counted numbers; at 2/3 a removal is still felt four organs back against a larger count of three. The front runs deeper than the count does, so the instrument refuses the rung rather than reporting from it.

That single row does more work than any of the others. It says the two fates are not properties of the rise — not “this rise reverses and that rise cycles” — but alternatives available to the same stem, chosen by which organs were taken out.

Without that row, a table of nine rises would have read as a rise-by-rise property, and the obvious next question would have been what changes about the rise. With it, the question is what changes about the cut, which is the same question the single-organ thread has been asking about which family survives.

The dose does not choose #

One more control, because the obvious remaining explanation is that the cycle is what a bigger cut does.

It is not. Every cell in these tables removes exactly two organs. What varies between a cell that reverses and a cell that cycles is where the two organs sat, not how many there were. At the rise of 0.075 the four cells that reverse and the three that cycle are the same size of intervention applied at different offsets.

The finer rung tells the same story from the other direction. Cuts of one to five organs there reach six destinations, the biggest cuts reach nowhere the smallest did not, and the size of the cut moves the rate of wrecking from a quarter to almost all while leaving the list of places unchanged. The dose decides whether, and it does not decide where.

So the coarse rung’s two fates join a pattern this thread has now seen three times. Which family a single removal leaves standing is decided by where it landed. Which of six places a many-organ cut reaches is decided by the arrangement and not the size. And which of two fates a coarse stem meets is decided the same way.

What a demotion is #

The word is worth choosing carefully, because “the stem falls to a coarser arrangement” is an interpretation and the measurement is narrower than that.

What is measured: a stem counted 2/3 is cut, does not repair, and ends in a four-cycle whose mean divergence is half a turn and whose positions a counter reads with a four in the pair. Half a turn is the two-ranked arrangement, which is below 1/2 on the ladder of rungs — the coarsest thing a stem can be.

What is not measured: that the stem has “gone down a rung” in the sense the rise sweep means, because the rise has not changed and the arrangement does not settle. A rung member has a divergence. This has four.

So the honest form is that a coarse stem, damaged, can lose its divergence altogether and end up oscillating about the one angle at which a stem has no spiral structure left. Whether that deserves to be called a rung is a question about vocabulary. That it is available at a front of three and not at a front of five or eight is a measurement.