The offsets that never change

Six offsets wreck somewhere on the golden 8/13 band. Three of them change their surviving family somewhere inside it.

The other three do not. Offsets 4, 5 and 9 keep the 8 family at every rise where they wreck at all — 98, 22 and 81 rises of the 126 respectively — with no crossing, no island and no exception.

Which halves the finding #

The band does not move the survivor. It moves the survivor at half of the offsets that wreck, and the half it moves is the middle three.

That is a weaker statement than the thread has been making and a more specific one. A band being able to change the answer is a fact about bands; a band changing it at three offsets and not at three others is a fact with a shape, and the shape is the offsets.

Which is consistent with the census’s standing account: the offset mostly decides which family survives, and the rise modulates it. A band changing the answer at half its offsets is that account with a resolution attached.

And which offsets wreck is not fixed either #

This is the part no reading of a band had drawn. Offset 7 wrecks at every one of the 126 rises. Offset 8 wrecks at 123 and recovers at three. Offset 6 wrecks at 110, offset 4 at 98, offset 9 at 81 and offset 5 at 22.

So a census taken at one rise of this band and a census taken at another are censuses of different sizes, with different offsets in them. At the coarse end four offsets wreck; at the fine end six do, and in between the number moves about.

That is a second reading of the band and it uses the same 1,890 cut stems. It was not what the sweep was run for and it is the finding of the four that reaches furthest, because it is about the census rather than about this band — which offsets wreck at a lattice is one of the thread’s oldest tables.

Where each of them starts #

Offset 4 recovers across the coarse fifth of the band — the first twenty-seven rises, twenty-two per cent of the way down — and wrecks at every rise after that. Offset 9 recovers across the coarse third, the first forty-five rises, and wrecks throughout the rest. Offset 5 never settles into either: its twenty-two wrecked rises begin at the band’s midpoint and are scattered through the fine half in runs of one to ten.

Two of the three are clean transitions from healing to wrecking, at different places, and the third is speckled the way the changing offsets’ answers are.

Which is a second speckle #

Offset 5’s pattern of wrecking and healing has the same character as offset 8’s pattern of keeping one family or the other: short runs, uneven gaps, no clean edge. Two different quantities on one band, both speckled, both at the scale of a few rises.

That is worth noticing because it suggests the speckle is a property of the band rather than of the survivor. Something about this stretch of rise is finely balanced, and two independent readings of a cut stem both feel it.

The test that has not been run #

Score offset 5’s wrecking pattern for a period the same way the survivor’s was scored. It is the same four lines of arithmetic on a binary sequence, and it would say whether the two speckles are the same kind of thing.

Not done, and it is cheap. The sweep already holds the sequence; nothing has to be grown, which puts it in the same category as several other checks this round named and did not run: arithmetic on tables that are already on disk.

Two clean transitions and one flicker #

Offsets 4 and 9 each cross once, from healing to wrecking, and never go back. That is the shape a threshold produces: below some rise the cut recovers and above it does not, with one edge.

Offset 5 has no edge. It first wrecks at the band’s midpoint and then wrecks at twenty-two of the following sixty rises, in runs of one to ten with gaps of one to twelve. So of three offsets whose wrecking state changes across the band, two change sharply and one is speckled — which is the same two-to-one split the survivor’s readings show, on a different quantity.

What a census assumes #

Every census in this thread is taken at one rise per lattice. Which offsets wreck is reported as a property of the lattice — the golden 0.013 stem wrecks at these offsets and heals at those — and the sweep says it is a property of the lattice and the rise, at a scale of a few parts in a thousand.

The census’s lattices are far apart in rise, so nothing in it is called into question. What is called into question is the idea that a lattice has a set of wrecking offsets in the way it has a counted pair.

A counted pair is stable across a whole rung by construction — that is what a rung is. A wrecking set is stable across nothing in particular, and every claim quantified over “the offsets that wreck at this lattice” has been quantified over a set that a fifth of a per cent in the rise can change. None of those claims is wrong; all of them are narrower than they read.

How much a rise moves it #

Between adjacent rises of this sweep — two parts in a thousand apart — the set of wrecking offsets changes on 23 of the 125 steps. So about one step in five, moving the rise by a fifth of a per cent adds or removes an offset from the census.

That is a rate rather than an anecdote, and it is the number a reader should carry about how stable a wrecking set is. Nearly all of it is offset 5, which flickers; the other five offsets change state twice between them across the whole band.

What holds still #

The counted pair, exactly, at every rise: 8/13 throughout. The settled divergence, to within five hundredths of a degree. Those are what the band is built to hold and they hold.

So the picture is a band across which two quantities are constant to the limit of measurement and four are not: the survivor at three offsets, the wrecking set, the rise itself by twenty-eight per cent, and the two contact steps’ ordering.

Which is what a band is for and what it cannot do #

A band isolates the step ordering from the counted pair and the divergence, which is exactly what the thread’s rival accounts turn on. It does not isolate it from the rise, and it cannot: the rise is what is being varied.

So every negative this design produces — the ordering does not decide the survivor — is a negative against the ordering and not against everything. The list of things the band lets move is a list of candidates nobody has scored.

The offsets that never change are the strongest evidence #

Not the ones that do. Three offsets keeping one family across the whole band, at 98, 22 and 81 wrecked rises, is 201 cut stems agreeing — and every one of them spans the handover, so every one of them is a case where the two contact steps change places and the survivor does not.

The changing offsets are the interesting rows and the steady ones carry the claim. That inversion is worth stating because the essays about this band have been written about the changing three — including the one about where they cross, which is a study of the exceptions to a rule the steady rows establish.

What the nine-rise design saw of them #

The right thing. It reported offsets 4, 5 and 9 as keeping one family and offsets 6, 7 and 8 as changing, which is what the full sweep says.

So the sampling was adequate for the steady offsets and inadequate for the changing ones, which is the expected asymmetry: a sample can confirm a constant and cannot locate a transition. Where the earlier reading went wrong was entirely in the interior of the three changing rows.

Which is where the wrecking set’s edges are #

Both clean transitions are at the coarse end: offset 4 starts wrecking a fifth of the way down and offset 9 a third of the way. Nothing starts or stops wrecking in the fine half except offset 5’s flicker and offset 8’s three isolated recoveries.

So the band’s coarse end has four wrecking offsets and its fine end has five or six, and the growth is entirely at the coarse end. A finer rise wrecks at more offsets, which is what the census across lattices already says at a much coarser resolution — and this sweep shows it happening inside a stretch of rise a hundredth as wide.

What is not claimed #

That the three steady offsets would stay steady on a wider band or a finer sweep. A band 28 per cent wide in the rise is a small excursion, and an offset that keeps one family across it might not across a whole rung.

Nor that the three are steady for one reason. Offset 4 and offset 9 sit either side of the changing three, at a distance of nine and four organs from the band’s larger counted number, and nothing connects them except that neither changes.

And nothing here says the steady rows would stay steady at a finer step. The changing rows’ islands are one to three rises wide and only appeared when the step fell; a steady row could hold an island narrower than two parts in a thousand and this sweep would not see it.

The cheap extension #

Cut the same band at offsets past 9. They all recover at the rise the census was taken at, and the sweep shows the wrecking set moves — so an offset that heals everywhere at 0.005 might wreck somewhere on this band.

The sweep already cuts every offset out to the front and two past it, so the answer is in the data: offsets 10 and beyond recover at every one of the 126 rises. That is a negative and it is worth having, because it says the wrecking set moves within a range and does not simply drift outwards.

What a reader should carry #

That half the offsets on this band never change their answer, and that they are the rows carrying the thread’s central negative rather than the rows that make it interesting.

And that a census of which offsets wreck is a census taken at a rise. On this band the set changes on a third of the two-parts-in-a-thousand steps, so it is not a property of a lattice in the way a counted pair is.

The three steady offsets are not equally steady #

Offset 7 is not among them — it changes — but it is the only offset that wrecks at every single rise, so it is the most reliable row for a different reason. Of the three that never change their family, offset 4 has 98 rises behind it, offset 9 has 81 and offset 5 has 22.

Twenty-two rises spread over a flickering stretch is much weaker evidence than ninety-eight consecutive ones, and both are quoted here as “never changes”. Splitting them is worth doing: on the strong reading the claim rests on 179 cut stems from two offsets, and offset 5 is a bonus rather than a third row carrying it.

What this does to the other five bands #

They were cut at nine rises and four of them reported no change of answer anywhere. That reading now has two possible meanings and the design cannot separate them: the band genuinely does not change the survivor, or its changing offsets have transition regions narrower than the sampling.

The second is not far-fetched. This band’s three transitions occupy about a seventh of it; a band a fifth as wide with a transition a seventh of its length would have one about four rises across, and a nine-rise design on it steps three or four rises at a time. So the narrow bands’ clean readings are exactly the ones a sampling artefact would produce.

What the whole sweep adds up to #

One band, every rise, 1,890 cut stems, and four findings. The claim the design rests on survives at fourteen times the resolution. The alternation it was run to check is not a period. The three changing offsets cross at three separate rises rather than at one. And half the offsets never change at all, while the set of offsets that wreck moves across the band.

Of those, the first is the one that matters and it is the one that did not change. The other three are the interior of a band described for the first time.

The one line #

Three of the six offsets that wreck on the widest band keep one family at every rise they wreck at — 201 cut stems, all spanning the handover, all agreeing — and they are where the claim that the step ordering does not decide the survivor actually rests.

And which offsets wreck is not fixed: the set changes on 23 of the band’s 125 steps, so a lattice’s wrecking offsets are a property of the lattice and the rise rather than of the lattice alone.