One way round, seventeen times

Two chains change places above a hole: one displaced forwards, one displaced backwards, by equal and opposite amounts, on neighbouring chains, on seventeen of the thirty wrecked cuts in the census.

“Neighbouring” is a symmetric word. It says the two chains are one apart and it says nothing about which of them went which way. That is a question with two possible answers on every row, and it turns out to have one answer on all of them.

The measurement #

Label each chain by the lag of its members counted from the removed organ, so that chain 0 is the chain the organ was on. Take the chain displaced forwards and the chain displaced backwards, and subtract their labels round the period.

The answer is +1 on every one of the seventeen rows. The backward-displaced chain is always the one immediately above the forward-displaced one. Not one row has it the other way round.

Why that is a result and adjacency was not #

Adjacency is one bit per row and it was already known. Direction is a second bit, and it is the bit that separates two accounts of what happened.

A swap — two chains trading positions — is symmetric. It has no preferred direction and would give plus one about half the time and minus one about half the time. A shift — one chain moving up a place and displacing its neighbour down — is not symmetric, and gives the same sign every time.

Against what a coin would do #

Seventeen rows agreeing on a two-sided question is one in a hundred and thirty thousand under a fair coin. That is the whole of the statistics, and it is worth being clear that the coin is the right null here.

It is the right null because nothing in the design picks a direction. The chains are labelled from the hole, the displacement is a signed difference between two runs, and the sign convention is fixed before any row is read. There is no step at which an analyst chooses which chain to call forward.

The rows are not one lattice seen seventeen times #

They come from eight lattices across both branches, at rises from 0.005 to 0.026, with offsets from three to nine organs back and surviving lags of 4, 5, 7 and 8. Four of the eight lattices contribute more than one row and four contribute one.

That matters because seventeen rows from one stem would be seventeen readings of one event. These are seventeen cuts, each with its own control, each grown separately.

The four lattices contributing several rows are worth a second look for the same reason. On the 0.010 stem the three cuts share a control and differ only in which organ was removed, so their three plus-ones are less independent than three rows from three lattices. Counting the lattices rather than the rows gives eight agreements, which is one in two hundred and fifty-six — weaker than one in a hundred and thirty thousand and still not a coin.

Two accounts, and what each predicts elsewhere #

The swap account and the shift account are not only different stories about one measurement. They predict different things about a quantity this thread already has: the slip, which is how far the wrecked stem’s own divergence sits from its control’s.

A shift by one place moves every organ above the hole one position earlier in the sequence, which changes the divergence by a whole turn divided by the surviving lag. That is exactly what the slip has been measured at — 72.0° at a lag of 5, 45.0° at a lag of 8, 51.4° at a lag of 7 — on every wrecked cut in the census.

A swap of two chains moves nothing else and predicts a slip of zero. So the slip was already deciding between these two accounts, measured for a different reason, and it decides for the shift.

What the labelling does not do #

A reader who has followed the chain numbering will notice that the convention is a choice: numbering by the cut run’s own ordinals rather than by the control’s shifts every chain label by one.

It does not touch this result. The orientation is a difference between two labels, and adding a constant to both leaves the difference alone. So the one result in this thread with no exceptions is also the one that survives the convention being wrong.

What a shift would look like #

The chain that lost an organ is one organ short. The organs above the gap are each placed one position earlier in the sequence than they would have been, so the chain they belong to has shifted along by one.

If that is what happens, the chain immediately after the shifted one takes an organ that used to belong elsewhere, and the two are displaced in opposite directions by one step. Which is what the measurement says: equal, opposite, adjacent, and one way round.

But the direction is the wrong way for the simplest version #

The simplest shift account predicts that the chain losing an organ is the one displaced, and the chain losing an organ is the hole’s own, chain 0. On ten rows chain 0 is the backward one, which fits; on seven it is not involved at all.

And on the ten where it is, the backward chain is the hole’s and the forward chain is the one below it — the chain of organs one place earlier in the sequence, which is below the hole rather than above it. The simplest picture has the disturbance propagating up from the gap, and the measurement puts one half of it underneath.

Which is impossible, and therefore a labelling fact #

Nothing below the hole moves. The two runs share their history to the last digit below the removed organ, so a chain “below the hole” in the labelling is a chain whose members are above the hole and whose lag is congruent to a smaller residue.

That is the moment to be careful with words. The chains are residue classes, not places, and “one below” means one lower in a cyclic label. Reading it as a physical direction is exactly the error the labelling was introduced to avoid.

So what the result is, stated carefully #

Order the chains by their lag from the removed organ, modulo the period. Then in that cyclic order the forward-displaced chain immediately precedes the backward-displaced one, on every row.

That is a statement about a cyclic ordering, and its content is that the exchange has a handedness. An arrangement and its mirror image would give opposite signs, and every stem in this census turns the same way.

Which is a prediction #

Grow the stems the other way round — start them at the mirror of their seed angle, so every parastichy runs the other way — and cut them identically. The prediction is that the orientation reverses on every row.

If it does, the result is about handedness and nothing else. If it does not, the orientation is a property of the labelling and not of the arrangement, and this essay is about an artefact. Thirty runs, and it has not been done.

The prediction is cheap and it is the whole test #

That is worth saying plainly. Every other check in this essay is a way of making seventeen rows into better evidence for a claim about seventeen rows. The mirror run is the only one that could show the claim to be about the wrong thing.

It costs thirty stems and thirty controls, which is the same as the census already grown. Nothing about it is difficult and it is not done, and the honest place for that is here rather than in a footnote.

The sizes, which are the same result again #

The exchange’s size is 88.0° to 147.2° across the seventeen rows, against settled divergences of 99.1° to 138.0°. One organ’s step, to within twelve per cent, which was the earlier reading.

A shift by one place in the sequence displaces an organ by one divergence step, which is what a size of one divergence means. So the size and the direction are two readings of the same account, and both fit it.

And the size has a residual the direction does not #

The size misses one divergence step by up to twelve per cent, and that miss has a pattern in it. The direction has no residual at all: it is plus one on every row, with nothing left over.

Which is the ordinary difference between a discrete reading and a continuous one. A sign has no error bar and an angle has, and it is why a result about a sign carried by seventeen rows is stronger than a result about an angle carried by the same seventeen.

What the thirteen other rows say #

Nothing, and they cannot. A row whose exceptions are three chains rather than two has no forward-and-backward pair to orient, so the question does not arise on it.

Those thirteen are not counter-examples and they are not evidence either. They are rows where the measurement is undefined, which is a third category and one this thread keeps having to name — an offset that recovers rather than wrecks is the same category one level up, and the census reports it as its own state rather than as a missing value.

The refusal that goes with it #

Asking for the orientation of a row with three exceptions is refused rather than answered by taking the two largest. That is a small piece of machinery and it is the one that keeps seventeen from becoming thirty.

A rule that would return a direction for every row would have returned seventeen plus-ones and thirteen numbers computed from whichever two exceptions happened to be biggest, and the second group would have looked like scatter around a signal.

Why this was not found earlier #

Because the earlier reading asked whether the two exceptional chains were adjacent, and adjacency is symmetric. The measurement that gives the direction is the same subtraction with the sign kept, and keeping a sign is a decision somebody has to make.

That is the shape of most of the leavings in this collection: not an experiment nobody could afford, but a quantity computed and then reduced before it was looked at. The profile itself was nine thousand numbers read for two for five rounds.

What it would take to break it #

One row the other way round. Seventeen is enough to make a coin implausible and it is not enough to make an exception surprising: an eighteenth row at minus one would put the result at seventeen of eighteen and change it from a law to a tendency.

Which is what the position of the exchange already is, on the same seventeen rows. Two readings of one pair, and one of them has no exceptions and the other has seven.

The asymmetry between them is instructive. A tendency invites a search for the column that explains the exceptions, and four columns have been tried. A law invites a search for the assumption that manufactured it, and one candidate has been named and not tested.

What the seventeen rows have in common #

Nothing that would produce this by construction. They are seventeen separate pairs of runs, each grown from its own seed, each cut at its own offset, each compared against its own control. The only thing shared is the placement rule.

That is worth checking rather than asserting, because a shared implementation is exactly the thing that can manufacture a shared sign. The check is the one this thread already runs on every continuation: grown to a length outright must equal grown shorter and continued, to the last digit, and it does.

Where this leaves the account #

A wrecked stem, on the seventeen rows that carry a clean exchange, is its control with one chain shifted along by a place and its neighbour pushed the other way. The size says one divergence step, the direction says the shift runs one way, and the slip says the whole arrangement turns by a turn over the surviving lag.

Three quantities measured for three different reasons, all consistent with one picture. What is missing is the mirror test, which is the only one that could show the picture to be about the labelling rather than about the stem.

The one line #

On all seventeen wrecked cuts that carry a clean exchange, the chain displaced forwards immediately precedes the chain displaced backwards in the cyclic order of lags from the hole — one in a hundred and thirty thousand under a coin, and unaffected by the labelling convention it is stated in.

The account it fits is a shift by one place rather than a swap, and the test that would show it to be a property of handedness rather than of labelling is thirty mirrored runs that have not been made.