Three rows change sides

A wrecked stem’s displacement profile is called periodic when the widest spread inside any one residue class is under ten degrees. Over three hundred organs, twenty-five of the thirty cuts in the census pass. Over six hundred, twenty-six do.

The count moved by one and the membership moved by three. Two rows that failed the test at the shorter length pass at the longer, and one that passed fails.

The reading itself is the thread’s central result: a wrecked stem’s damage is not three hundred scattered numbers but a few levels, one per chain, repeating. What is at stake here is not that shape but which rows have it.

The two that join #

The golden 0.010 stem’s cuts four and five organs back. Over three hundred organs their widest spreads are 69.6° and 69.5°, which is not marginal — it is seven times the line. Over six hundred they are 7.9° and 8.1°.

A factor of nine, on two rows, from nothing but a longer run. Whatever those profiles are doing over the first three hundred organs, they are not doing it over the next three hundred. Neither row is near the line at either length, which is the useful part: this is not a marginal case tipping over, it is a reading that was wrong by an order of magnitude and is now right.

Which is a transient with a long tail #

The obvious reading, and it is the right one. The levels are measured over the last hundred and twenty organs of whatever run was grown. At three hundred organs that window sits between organs 180 and 300, which on these two rows is still inside the disturbance; at six hundred it sits between 480 and 600, which is past it.

So the shorter run was measuring a transient and calling it a failure of periodicity. The profile is periodic; the window was in the wrong place — which is the same diagnosis the onset reading got from the other direction, since both are computed against levels taken from the end of whatever run was grown.

Two rows or one stem #

The two are cuts four and five organs back on the same stem, so they share a control and differ only in which organ was removed. Their spreads at both lengths agree to a tenth of a degree: 69.6° and 69.5°, then 7.9° and 8.1°.

That is one lattice behaving one way, observed twice, rather than two independent confirmations. Counting them as two rows is what the census does everywhere and it is worth flagging here, because two rows changing sides sounds like a pattern and one stem changing sides is an anecdote.

And it is a selection effect in the census #

The five rows that never reached a pattern within three hundred organs were named as a group and treated as a category — the cuts whose damage never settles. Two of the five are now rows whose damage settles late.

That leaves three, and three is a small enough number that the category may not be a category at all. It might be one row, twice, plus a genuine case. And one of the three is the row whose onset turned out to be its run ending, so the group has been carrying an artefact as well as a duplicate.

The one that leaves #

The Lucas 0.013 stem’s cut four organs back. Over three hundred organs its widest spread is 6.1°, comfortably inside the line; over six hundred it is 171.2°, which is past a half turn and as far outside as a number can be.

That is not a transient ending. It is a profile that looks steady over one window and is not steady at all over a longer one.

What could make a profile look steady and not be #

Two things, and they are distinguishable. A profile that drifts slowly has classes whose means move together; over a short window each class looks tight and over a long one it does not. A profile that jumps — settles onto one set of levels and later onto another — looks tight on either side of the jump and ragged across it.

The row that leaves is the second kind. Its spread over six hundred organs is 171.2°, which is not a drift accumulating; it is two regimes with a large step between them. Nothing in this thread had a category for a wrecked stem that settles twice.

Which is the more awkward direction #

A row that joins is a window in the wrong place, and the fix is a longer run. A row that leaves is a window that was in a place where a non-periodic profile happened to look periodic, and the fix is not obvious: a longer run found it, and a longer run still might find others.

The two directions have different implications for the census. One says the count of non-periodic rows is too high; the other says the count of periodic rows is too high. Both cannot be corrected by growing runs until nothing changes, because nobody knows where that is.

The number that would settle it has not been read #

Drift and jump are distinguishable, and the distinguishing measurement is already inside the runs that were grown — it simply was not taken.

Read the widest within-class spread three ways on the same six-hundred-organ run: over the first three hundred organs, over the last three hundred, and over the whole six hundred. A drift gives a small number, a small number and a larger one, with the whole-run value scaling with the window in the way a linear slide does — roughly double. A jump gives a small number, a small number, and a value of the order of the step between the two regimes, which is far more than double.

Two of the three are known for this row: 6.1° over the first three hundred and 171.2° over the whole six hundred, a factor of twenty-eight rather than of two. That already excludes a drift. What the missing third number would add is the confirmation that the row is steady again after the step rather than ragged for the rest of the run — and those are different objects. A stem that settles twice has two destinations; a stem that comes apart and stays apart has none.

So the essay’s own reading, that this is two regimes with a large step between them, is supported by one comparison and would be established by a second that costs nothing, since the run exists. Saying which number is missing is the least a claim of this size should carry.

What a stem that settles twice costs the short list #

If the jump reading holds, the consequence reaches past this row, and it is worth stating before the number is confirmed rather than after.

Every destination this collection has recorded is a destination as of the end of a run. The short list of six and the census’s blocks and slips are all read over three hundred organs, and a stem that settles onto one arrangement and later onto another would have contributed its first arrangement to those tables and not its second.

The rate is measurable and small. Doubling the run turned up one such row in thirty, so about three per cent of wrecked stems reveal a second settling per doubling of the horizon. That is low enough that the short list is not in question and high enough that final is the wrong word for any entry on it. The honest term is the one the run length supplies: a destination observed over three hundred organs, with the horizon quoted beside it.

It also says what a third length would be for, which is not more precision on the rows already in hand. It would measure whether the three per cent is a rate that keeps producing rows or a one-off — and those two possibilities differ in whether the census converges at all.

The row that leaves is the only one of its kind #

Its surviving lag is 4, and it is the only wrecked cut in the whole census that keeps a four-hop. Every other row keeps a 5, a 7 or an 8.

That matters because a class of four residues is the shortest period in the census, so its classes have the most members and the least room to look accidentally steady. If any row were going to look periodic by luck, this is the one where luck should have had the least chance.

And a separate reading calls it out #

Four accounts of the exchanged pair’s size are scored on the seventeen rows that carry a clean exchange, and the best of them fits worst on exactly this row — 4.0 per cent against 2.3 for the next worst and 1.0 for the mean.

Two measurements, made for unrelated reasons, neither aware of the other, agreeing about which row of the census is least trustworthy. That agreement is worth more than either measurement, and it is the kind this collection looks for deliberately.

The gap narrows #

The ten-degree line was chosen because the two populations were separated by one. At three hundred organs the periodic rows measured up to 6.09° and the rest started at 10.28°, a factor of 1.69. At six hundred the periodic rows reach 8.11° and the rest start at 10.28°, a factor of 1.27.

Still a gap and a much smaller one. A threshold that sat in empty space now sits in nearly-empty space, which means it has started to matter where it is.

Which is a warning rather than a failure #

Nothing here is decided by the tenth of a degree either side of ten. The row that leaves is at 171.2° and the rows that join are at 7.9° and 8.1°, so a line anywhere between about nine and ten gives the same table.

What has changed is the margin. At three hundred organs the classification was robust to a factor of 1.69 in the threshold; at six hundred it is robust to 1.27, and at twelve hundred it might not be robust at all. That is a thing to check rather than a thing to worry about now.

The margin is also the reason the threshold was ever quotable. A tolerance chosen because it sits in empty space is a tolerance nobody had to argue about, and the arguing starts as soon as the space fills. Watching a gap close is the cheapest early warning a thread with stated tolerances can have, and it is only available because the gap was reported alongside the line rather than in place of it.

What the count hides #

Twenty-five and twenty-six is a comforting pair of numbers and it is the least informative reading of the comparison. Three rows changed sides and the count moved by one because two changes cancelled one.

A census reported as a count is a census whose membership can turn over without anybody noticing, and this one turned over by ten per cent of its periodic rows while the headline moved by four.

What a second settling would mean #

If a wrecked stem can settle onto one arrangement and later move to another, then the block a wrecked stem falls into is not a final state, and the census’s readings of what a cut leaves standing are readings of whatever the stem happened to be doing at organ three hundred.

That is a large claim to hang on one row and this essay does not hang it there. What it does is name the shape and say which measurement would settle it: the surviving lag, read separately over the first and second halves of a six-hundred-organ run. If the lag changes, the stem moved; if it does not, the profile drifted inside one arrangement.

Two readings on one row, on runs already grown. It is the cheapest open question in the thread.

What the shape survives #

The result the thread rests on — that a wrecked stem’s profile is a handful of levels rather than three hundred scattered numbers — is not in doubt at either length. Twenty-six of thirty rows have spreads under ten degrees against between-class differences past a hundred and fifty.

And the exchanged pair, its size and its direction, are unchanged on every row that carries one at both lengths. What moves is the classification of the marginal rows and every reading about timing.

What a third length would settle #

Twelve hundred organs on the thirty. Four minutes, and it answers two questions: are the two rows that joined still periodic, and does any further row change sides.

If nothing changes between six hundred and twelve hundred, the six-hundred classification is the one to quote and the three-hundred one was under-run. If more rows change, then the classification has no limit and the quantity needs a definition that does not depend on where a run was stopped.

That is the distinction the settling thread draws between a budget and a wall, and it is drawn there by exactly this comparison: grow to one length, grow to three times it, and see whether anything new arrives.

Why it has not been run #

The comparison in front of it was built to check one number — an onset reported at the last organ of its run — and finding a group of rows that change sides is something it did on the way. A sweep designed to check a number and then asked to characterise a group is always one length short.

That is not an excuse so much as a description of how this keeps happening. The cheapest useful thing is nearly always the next power of two.

What it means for the five #

The five rows that never settle are now three, and one of the three is the row whose onset was an artefact. So the category “damage that never becomes a pattern” has one or two members that anybody has confidence in, and it has been carried as a group of five through several essays.

Rewriting that is the correction this comparison forces. The group is smaller, its membership is different, and the two rows that left it did so by becoming periodic rather than by being reclassified on a technicality.

The threshold’s other half #

Ten degrees is one of two tolerances in this reading and the other has not moved. A class counts as an exception when it sits more than ten degrees from the level the rest share, and over the whole census the classes at the common level are within 5.55° of it while the exceptions are 12.4° and up.

That gap is a factor of 2.2 and the doubling does not narrow it, because it is a gap between levels rather than a gap in how steady a level is. So of the two tolerances in the periodicity reading, one is being squeezed by the run length and the other is not, and the essays quoting them have treated the pair as one number.

Why a classification is worth this much attention #

Because everything downstream is counted over it. The exchange thread works on the seventeen rows that carry a clean balanced pair, and whether a row carries one is decided on the same levels the periodicity reading uses. A row that changes sides here can change the denominator there.

In this case it does not: the row that leaves still carries a balanced pair at the shorter length, which is where the exchange was read. But it means the two threads share an instrument, and a change to the instrument moves both.

The one line #

Doubling every run moves three rows of thirty across the line between a periodic profile and a ragged one — two into the periodic group by a factor of nine, one out of it by a factor of twenty-eight — while the count moves by one because the changes partly cancel.

The row that leaves is the census’s only four-hop survivor and is independently the worst fit of a scoring made for an unrelated reason, and the gap the threshold sits in narrows from a factor of 1.69 to 1.27.