Which chains changed places

Remove one organ from a settled stem, continue against a control that shares its history, and every organ placed afterwards sits some angle from where the same organ sits in the control. Folded on the lag the stem kept, those angles are not three hundred numbers but a handful of levels — one per chain, held to a few degrees, repeating for as long as the run goes on.

Most chains sit together and two do not: equal and opposite to within a twentieth, on neighbouring chains, on seventeen of the thirty wrecked cuts in the census. Two neighbouring chains displaced by equal and opposite amounts is what a pair of chains that have changed places looks like.

What was left #

The chains were labelled by residue, and a residue is a label. It said the two exceptional chains were adjacent and it said nothing about where they sit relative to the hole — whether the exchange happens next to the removed organ, or somewhere else, or in a different place each time.

That question is arithmetic on indices already computed. The profile is indexed by lag from the removed organ, so a chain’s label can be read as a lag rather than as a residue, and the chain the removed organ was on is the one whose lag is a multiple of the period.

The convention, stated once #

Chains are numbered from the removed organ: chain 0 is the chain it was on, chain 1 is the chain of the organ one place above it, and so on round the period. That is the control’s indexing, and it is the indexing the displacement is defined in — every row of the profile is a difference between the two runs at one control index.

There is a rival convention. The cut run has one fewer organ below the window, so numbering by its own ordinals shifts every label by one. It changes neither of the two results here, because both are differences: the orientation is a difference between two labels and the tally is over a fixed offset applied to all of them.

The answer, and it is not one number #

The backward-displaced chain is the hole’s own chain — chain 0 — on ten of the seventeen. On four more it is chain 2. On the remaining three it is chain 4, chain 5 and chain 6, once each.

Against a chain drawn at random from each row’s own period, ten at one value is against expectation of 2.8. So the exchange favours the hole’s own chain strongly and does not sit there always, and neither half of that is worth rounding away.

What a tendency is worth #

A law would be better and this is not one. Ten of seventeen at a single value out of four to eight is a real concentration — it would happen by chance about once in a hundred thousand tables — and it leaves seven rows that need an account nobody has.

The temptation is to describe the ten and call the seven noise. They are not noise: their exchanges are as clean as the others, equal and opposite to within a twentieth, on adjacent chains. They are somewhere else.

And where they are is worth reading rather than counting. The seven sit at chains 2, 2, 2, 2, 4, 5 and 6 — six of the seven at an even chain, with a single five. Add the ten at chain 0, which is also even, and sixteen of the seventeen rows put the exchange at an even-numbered chain.

The raw sixteen-of-seventeen is not the number to quote, because ten of those sixteen are the chain-0 concentration already reported and would be even whatever else were true. The informative version conditions on that: given that a row is not at chain 0, is it more often even than chance allows? Setting aside chain 0, a little over two fifths of the remaining chains are even at the periods this census carries, so six of seven coming out even is about one chance in forty.

That is suggestive and it is not a finding, and the distance between those two is exactly the point of computing it. One in forty on seven rows is the kind of pattern that appears in every table somebody looks at long enough, and it is also the kind that turns out to be real about half the time. What makes it worth recording rather than discarding is that a second parity pattern has already turned up in this thread from a different quantity — the sign of the exchange’s own size sorts by whether the surviving lag is odd or even — and two parity observations in one census are either one structure seen twice or one artefact seen twice.

Which of those it is has a cheap test, and it is the same test both times: a row whose surviving lag or exchange chain breaks the pattern. The thirteen rows this essay does not score have chains of their own that nobody has tallied, and they are where a seventeen-row parity claim would first meet something it could not absorb.

It does not follow the offset #

The obvious candidate is the offset that was cut — how far back the removed organ was. It fails. The 0.010 stem is cut six, seven and eight organs back and all three put the exchange at chain 0; the 0.008 stem is cut four, five and seven organs back and puts it at chains 2, 0 and 0.

So the exchange is not at a fixed place relative to the cut, and it is not at a place the cut’s own offset predicts. Two lattices, six cuts, and the two answers do not agree about which quantity is doing the work.

That is worth contrasting with what the offset does decide. It decides whether the stem wrecks at all, it decides how much the next organ moves, and on most lattices it decides which family survives. It is the strongest column in this whole thread and it is silent here.

Nor the surviving lag #

The second candidate is the lag whose hop the stem kept, which is the period the profile is folded on. The chains at 0 come from cuts with surviving lags of 4, 5, 7 and 8; the chains elsewhere come from cuts with lags of 4, 5, 7 and 8.

Both lists are the same list. Whatever puts an exchange at chain 2 rather than chain 0 is not the period it happens inside.

Nor the branch, nor the rise #

Six of the ten chain-0 rows are golden and four are Lucas. Four of the seven others are golden and three are Lucas. The rises on both lists run from 0.005 to 0.026.

Four columns tried and four columns failing is the same shape the slot interaction produced on six rows before it was run on thirty, which is a reason to be careful: a quantity that sorts nothing on seventeen rows may be a quantity that has been sampled seventeen times.

It is also worth saying what four failed columns do establish, since it is not nothing. The position of the exchange is not the offset, not the lag, not the branch and not the rise — which between them are every column the census carries. So whatever decides it is a quantity the census does not record, and the next move is to compute a new column rather than to re-sort the existing ones.

What would settle it #

More rows. Seventeen is what thirty wrecked cuts yield once the rows whose exceptions are not a balanced pair are set aside, and thirty wrecked cuts is what twelve lattices yield. Twenty-four lattices would roughly double it.

That is a sweep this thread has already learned to want. The design that produced these thirty was built when the question was whether the damage had a shape at all, and it is now being asked a question about where inside that shape something sits.

The rows that carry no pair #

Thirteen of the thirty do not have one exceptional pair. They have three exceptional chains, or four, or two that are not equal and opposite. Those rows are refused rather than reduced to their two largest exceptions.

Refusing them costs the census nearly half its rows and it is the right cost. A row with three exceptions is a different shape, and taking the largest two of three would manufacture a pair on every row that has one available — which is the failure mode a summary statistic computed past its own scope always has: it keeps returning a number after the thing it was a number for has gone.

And they are not a separate population #

The thirteen are not the ragged rows. Their profiles are as periodic as the seventeen — the widest spread inside a chain is under six degrees on most of them — and their exceptions are as far from the common level.

So the shape “one balanced pair” is not what damage does, with exceptions. It is one of at least two things damage does, and this thread has an account of the first and none of the second.

What the thirteen would need #

An account of a profile with three exceptional chains is a different piece of work from an account of one with two, and this thread has not started it. The obvious first question is whether three exceptions are a pair plus something, or a three-cycle — three chains rotating rather than two swapping.

That is answerable from numbers already computed: a three-cycle displaces its members by amounts summing to zero, and a pair plus a stray does not. Nobody has added them up.

Which makes the ten a smaller claim #

Ten of seventeen, out of thirty. Read against the whole census that is ten rows in three, and the sentence “the exchange is usually at the hole’s own chain” is a sentence about a subset chosen for having a clean exchange in the first place.

The subset is chosen by a property that has nothing to do with position — whether the two exceptions balance — so it is not obviously a biased sample. It is a selected one, and the difference between those two is exactly what cannot be checked without the rows that were dropped.

What the position would mean #

If the exchange were always at the hole’s own chain, the account would write itself: the chain that lost an organ is short by one, so the organs above the gap shift up a place and the chain next to it takes the strain. Ten rows out of seventeen is that account working most of the time.

The seven that put it two, four, five or six chains along have no such story. The organ that was removed is not on those chains and the exchange is not near it, so whatever propagates does so without leaving a mark on the chains in between.

The chains in between are undisturbed #

That is the part worth pausing on. On a row whose exchange sits at chain 4, chains 1, 2 and 3 are at the common level to within a few degrees, and so are chains 5 and onwards.

So the disturbance does not spread from the hole and stop; it appears at a distance with nothing between. Every organ between the hole and the exchange is exactly where it would have been, and the two chains that moved are four chains away.

Which is a mechanism-shaped fact #

A displacement that appears at a distance with nothing in between is not what a local disturbance propagating outwards looks like. It is what a relabelling looks like: the arrangement above the hole is the control’s arrangement with two of its chains swapped, and where the swap happens is a property of the whole arrangement rather than of the hole.

That is a candidate account and it predicts something checkable: the position should depend on the arrangement’s own structure — its counted pair, its offsets — rather than on the cut. Four such columns have been tried here and none of them works, which is evidence against it.

The instrument is not in doubt #

Every number here comes out of the same two runs the rest of the thread uses, compared organ by organ at the same indices, over a window of a hundred and twenty organs at the top of a three-hundred-organ run. Nothing about the position reading is new machinery.

What is new is only the labelling, and the labelling is arithmetic that could have been done on the day the exceptional pair was found. It was not, which is the ordinary reason a leaving exists.

Except that the window is a choice #

The levels are read over the last hundred and twenty organs of the run, and a longer run moves them. One of the seventeen rows here is a row that stops looking periodic when the run is doubled, which means its exchange is a reading taken inside a transient.

Removing it leaves sixteen rows and nine at chain 0, which changes nothing about the shape of the answer and is worth stating rather than leaving for a reader to work out.

What carries forward #

An exchange has a definite location, it is at the hole’s own chain more often than anywhere else, and it is not always there. Four candidate accounts of the exception have been tried and none of them sorts the rows.

That is a partial regularity with an honest denominator, and it sits beside a result on the same seventeen rows that has no exceptions at all — the direction the exchange runs, which is the same on every one of them.

A reading that is available and was not taken #

The chains could have been labelled by azimuth rather than by index — ordered round the stem by where they sit, rather than by how many places above the hole their members are. The two orders are different: chains adjacent in index are one divergence step apart in azimuth, which is most of a turn.

Labelled that way, “adjacent chains” would become “chains a divergence apart”, and the exchange would look like a swap between two chains on opposite sides of the stem. The same measurement, described two ways, and the index labelling is chosen because the surviving hop is a lag and not a direction — every other reading in this thread is indexed the same way.

What a reader should hold on to #

That the exchange has a place, that the place is measured rather than assumed, and that it is at the hole’s own chain more often than anywhere else without being there always. Everything past that is a list of columns that do not sort the exceptions.

The list is short — the offset, the surviving lag, the branch, the rise — and it is the list anybody would have tried. What is missing is a column nobody has thought of, and this collection’s usual response to that is to widen the census rather than to think harder.

The one line #

The two chains that change places in a wrecked stem sit at the hole’s own chain on ten of the seventeen cuts that carry a clean exchange, at chain 2 on four more, and elsewhere on three — against 2.8 rows at any one chain by chance.

Which offset was cut, which lag survived, which branch and which rise all fail to sort the seven that are not at the hole, and the chains between the hole and a distant exchange are undisturbed.