A wrecking set with a range

One result has replicated on every band cut at every rise it holds: which offsets wreck is a function of the rise, not of the lattice. A cut taken behind the front at one rise wrecks the stem; the same cut, on the same band, a few rises along, recovers.

Four of the ladder’s six bands wreck at all. On all four the wrecking set moves, and with four of them the finding stops being a replication and becomes a quantity with a range.

What a wrecking set is #

A cut removes one organ from a growing stem at a chosen offset behind the front and the stem is grown on. Sometimes it closes the hole and carries on, and sometimes the lattice never recovers and settles to a different contact family — that second case is a wreck, and the family it settles to is the survivor.

The offsets that wreck at a given rise are the ones a census can ask anything about. An offset that recovers has no survivor to report, so it contributes nothing.

Every claim in the ablation thread of the form at every offset that wrecks is therefore quantified over that set, and until a band was cut whole nobody had drawn the set moving.

The measure #

For each offset on each band: how many of the band’s rises it wrecks at, and how many separate unbroken stretches those rises fall into. The ratio is rises per stretch, and it is the fragmentation — a small number means a wrecking that comes and goes and a large one means a wrecking that holds.

Placing all four bands on one logarithmic axis is what turns four separate observations into one reading, and the reading is that the axis is nearly two decades wide.

The range #

From 2.43 rises a stretch to 124, a factor of fifty-one.

The low end is offset 3 of the golden 5/8 band: seventeen wrecking rises broken into seven stretches. The high end is the Lucas 7/11 band, where three separate offsets each wreck at all 124 of its rises without a single break.

That is the same measurement, taken on four objects of the same kind, spanning a factor of fifty-one. It is a wider spread than any other quantity the ladder is described by: the spans run from three per cent of a rung to seventy-two, a factor of twenty-four, the cut stems a band costs run from ninety-six to 1,890, and the rises a band holds run from sixteen to 126.

Two of the four have no offset that wrecks everywhere #

The golden 5/8 and the Lucas 4/7. Every one of their offsets comes and goes, so on those two bands there is no cut at all whose outcome can be quoted without naming the rise it was taken at.

The golden 8/13 has one such offset and the Lucas 7/11 has three. So of the sixteen offsets that wreck anywhere on the ladder, four wreck at every rise of their band and twelve do not.

Neither branch owns the property. One band of each has an offset that never lets go and one of each does not, which is the first thing to say to anyone reaching for the account that survived the ladder as an explanation for this as well.

The most broken band #

The golden 5/8, and it is broken in an unusual shape. Offset 3 wrecks at 17 of its 112 rises in seven stretches; offset 5 at 28 in eleven; offset 4 at 111 of 112 in two, with a gap one rise wide in the middle of it.

So the band has one offset that behaves almost like a property of the lattice and two that behave almost like noise, and nothing in between. Two of its three offsets sit almost on top of each other on the axis, at 2.43 and 2.55, while the third sits at 55.5.

On the all-bands view those two marks are about five pixels apart, so the drawing clusters them and labels the cluster. That two of a band’s three offsets fall that close together in fragmentation is a fact about the band rather than a convenience of the picture.

The least broken band #

The Lucas 7/11. Offsets 5, 6 and 7 wreck at all 124 rises in one stretch each; offset 8 wrecks at 88 of them, also in a single unbroken stretch; and only offset 4, at 27 rises in five stretches, comes and goes at all.

Four of its five offsets are therefore intervals rather than speckle. A census taken anywhere in the middle of that band would find the same four offsets wrecking, and a census taken at the edges would find some of them missing — which is a much milder version of the problem than the golden bands present, and still not nothing.

It is also the band that changes its surviving family nowhere, so on it the wrecking set is the only thing that moves at all.

The widest band holds most of the spread #

The golden 8/13 has six wrecking offsets and they run from 3.14 rises a stretch to 81 — almost the whole of the ladder’s range inside one band. Offset 5 wrecks at 22 of 126 rises in seven stretches, offset 9 at 81 rises in one, and the four between them are spread across the axis with no clustering worth the word.

That matters for how the range should be read. A factor of fifty-one across four bands would be unremarkable if each band were internally tight and the four sat at different places; it is a different statement when one band spans most of it on its own.

What the spread is not is a function of how often an offset wrecks. Offset 9 wrecks at 81 rises in one stretch and offset 6 at 110 in six, so the offset that wrecks more often is the more broken of the two.

And the narrowest of the four is the tidiest #

The Lucas 4/7 wrecks at two offsets only. Offset 4 wrecks at 83 of its 86 rises in two stretches and offset 5 at 55 in two, so neither wrecks everywhere and only one comes close.

Two offsets is the smallest wrecking set on the ladder, and it is the band that was cut to test the account left standing rather than to say anything about wrecking. It contributes two marks to the axis and both land well inside it.

The two bands below it on the ladder wreck nothing at all: ninety-six and four hundred and ninety cut stems with not one wrecked cut between them, which is why the range is measured over four bands and not six.

One offset across four bands #

Offset 5 wrecks somewhere on all four bands, which makes it the one cut that can be compared across the whole ladder. It reads 2.55 on the golden 5/8, 3.14 on the golden 8/13, 27.5 on the Lucas 4/7 and 124 on the Lucas 7/11.

So a single named offset spans nearly the whole range on its own. Whatever decides fragmentation, it is not the offset — the same cut, taken the same distance behind the front, is speckle on two bands and an unbroken interval on a third.

That is the sharpest form of the finding. It was already known that the wrecking set moves across a band; what four bands add is that how much it moves is decided by the band and not by the cut.

What a census is, given this #

A sample of a rise. A census taken at one rise of the golden 5/8 cuts three offsets; a census taken fourteen rises along cuts one, because two of the three are between stretches there.

The censuses are not of different quality. They are of different sizes, and neither knows it, because a census reads the offsets that wreck at the rise it is taken at and has no way to ask what the neighbouring rises would have given.

The rise the census sits at was itself never chosen for this, which is the part that makes this an instrument fact rather than a curiosity. It is also the one quantity on a band that a coarse sample reads correctly, since a texture several rises across is visible to any step, and that is the half of the coarse design that survives.

What that costs the published results #

Less than it looks, and the reason is worth stating precisely rather than reassuringly.

Nearly every claim in the ablation thread is about what a wrecking cut keeps rather than about how many cuts wreck. A claim of the first kind is evaluated cut by cut: each wrecked stem has a survivor, the survivor is read, and a claim about survivors is true or false of the stems that exist. Whether three or six offsets wrecked at that rise changes the sample size and not the verdict.

A claim of the second kind is different, and there are fewer of them than the phrasing suggests. Six offsets wreck on this band is a union over 126 rises being read as a property of one, and a census at a rise drawn at random from that band would report four or five.

The claim this does bite #

Scoring a rule over a census. The offset mostly decides which family survives, and that score is taken over the offsets that happened to wreck at the rises the census holds.

On a band where an offset wrecks at 17 rises of 112 in seven stretches, the offset is a property of a rise as much as of a lattice, and a rule indexed by it is being asked to work on a quantity that is only sometimes defined. The score is not wrong; its denominator is a draw.

The repair is not a better census. It is quoting the rise, which is what a band cut whole does by construction, and it is why the four whole cuts are the evidence rather than the censuses that preceded them.

And the claim it does not bite #

Anything read at a band’s ends, because both ends are cut directly and the offsets that wreck there are the offsets that wreck there.

Anything about the handover, because a handover is geometry computed from hop lengths rather than from cut stems. And anything about the family a cut keeps, which is the question the whole ladder was cut to answer and which is evaluated on the stems that wrecked rather than on a count of them.

Three of the four kinds of claim in the thread are therefore untouched. The fourth is the one that counts.

A stretch count counts two things #

The honest limit, and it is in the instrument rather than in the object. A stretch here is a run of rises with one survivor, so the count is broken by a recovery and by a change of surviving family.

The tell is on the widest band. Offset 7 of the golden 8/13 wrecks at all 126 of its rises — it never recovers anywhere — and the count reports four stretches, which can only be the three changes of family recorded at that offset. Its fragmentation as wrecking is one stretch of 126 rises, and the figure reads 31.5.

So four of the sixteen numbers on the axis are mixed: offset 5 of the golden 5/8, whose eleven includes that band’s two changes, and offsets 6, 7 and 8 of the golden 8/13.

Which the range survives #

Both ends sit on offsets that never change their surviving family. Offset 3 of the golden 5/8 changes nothing across its seventeen wrecking rises, and the three Lucas 7/11 offsets at 124 sit on a band that changes nothing anywhere.

So 2.43 and 124 are both pure fragmentation, and the factor of fifty-one is a statement about wrecking rather than about surviving. The four mixed numbers all sit in the interior of the axis, where they move nothing.

It is worth recording as a defect anyway, because a later band with a changing offset at one end of the range would make the headline number mean two things at once and nothing would say so.

The branch does not predict it #

The golden bands span 2.43 to 81 between them and the Lucas bands span 5.4 to 124. Those two intervals overlap over almost their whole length.

That is worth saying because the branch is the account that survived every other question the ladder was asked, and it is tempting to reach for it here. It does not work: whether a band changes its surviving family looks like a property of the branch on four bands, and how broken its wrecking is does not.

Nor does anything else measured here #

Not the width of the band: the widest band holds both a heavily broken offset and an unbroken one. Not the size of the wrecking set: the band with the fewest offsets is neither the most broken nor the least, and the band with the most holds the highest mark on the axis and one of the lowest. Not the counted pair, which orders the bands the same way the three eliminated accounts did.

Nothing available orders the four bands the way fragmentation orders them, and with four bands there is no room to test a fifth account against them anyway. That is the honest position: a measured range with no account attached.

Saying so is better than fitting one. Three accounts were eliminated on this ladder by being wrong on a band each, and each was written down before the band that killed it was cut.

What would refute the reading #

A band on which every offset wrecks at every rise. That would be a wrecking set that is a property of the lattice after all, and it is what the thread expected before any band was cut whole.

The Lucas 7/11 comes closest and does not manage it: four of five offsets unbroken and the fifth at 27 rises in five stretches. One band with all its offsets unbroken would not undo the other three, but it would put a boundary on the finding and give the fragmentation something to correlate with.

Nothing on this ladder can supply it, because all six of its bands are cut.

What the ladder cannot say next #

There are no more bands. Six rungs with a handover, all six cut at every rise they hold, 534 rises and 5,982 cut stems, and the four that wreck are the four that carry this measurement.

A fifth mark on the axis would need a rung the ladder does not currently hold, and the rungs it does not hold are at the coarse end, where nothing behind the front wrecks at all. So the cheapest extension is also the one most likely to add another silence.

That is a limit on the measurement rather than a limit on the argument. Four bands is four, and a range read off four objects is a range read off four objects.

What is not measured here #

Why an offset stops wrecking. A stretch ends because the stem recovers at the next rise, and nothing in the sweep says what changed about the lattice between two rises two parts in a thousand apart.

Nor is there anything about the arrangement of the stretches beyond their count. Seventeen rises in seven stretches could be seven short stretches spread evenly or one long stretch and six singletons, and the mean says nothing about which — the same limitation the islands on the widest band have, where a period fitted at its best phase bought nothing over naming the commoner family.

And the sweep’s own step has never been varied, so a stretch one rise long is one rise long at two parts in a thousand and might be shorter.

The one line #

Sixteen wrecking offsets across four bands, every one of them cut at every rise its band holds: fragmentation runs from a stretch every 2.43 wrecking rises to one every 124, four offsets wreck at every rise of their band and twelve do not, and two of the four bands have no unbroken offset at all.

So a census of a band is a census of a rise, every claim quantified over the offsets that wreck is quantified over a set the rise decides, and the claims that survive it are the ones about what a cut keeps rather than about how many cut.