Every rise of a band

A band is the stretch of rise around a handover on which a stem’s counted pair holds and its settled divergence stays flat, so that the only thing moving is which of the two contact steps is shorter. It is the design that isolates the step ordering from everything else the rise controls.

Six of them were grown and each was cut at nine rises, evenly spaced in the logarithm of the rise, always including both ends and the handover itself. Nine, because sixteen cut stems a rise at a hundred rises a band is hours for a quantity the design predicts to be constant — and a constant is tested at the extremes and at the crossing.

Where the design broke #

On the golden 8/13 band it is not constant. Three offsets change their surviving family somewhere inside, and at one of them the nine sampled rises came back 8, 4, 8, 4.

An alternation is the one thing a nine-rise design cannot interpret. It could be a period, it could be two changes that happen to be sampled either side of, or it could be a single rise doing something unusual. The design has no way to tell those apart and it was not built to.

The sample that found the alternation said so, in the sense that it reported it as an open question rather than as a finding. What it also did was quote the changes’ positions in its own steps, which reads as a measurement and is not one — and that is the part this sweep corrects rather than confirms.

The sweep #

Cut every rise of that band. It holds 126 rises at the step the geometry is swept at, and each rise is a control and fifteen or sixteen cut stems, so the sweep is 1,890 cut stems and it took sixty-seven minutes.

That is not cheap and it is not the several hours the nine-rise design was chosen to avoid. The estimate that produced nine was made when the bands were narrower and before the sweep’s underlying stems were kept on disk between runs.

Which band, and why that one #

The golden 8/13. It is the widest band on the ladder — 126 rises against 16 for the narrowest — and it covers seventy-two per cent of its own rung, spanning a factor of 1.28 in the rise.

Width is the whole reason. A band sixteen rises wide has no room for a feature two rises across, so the question this sweep is about could not be asked of it. The narrow bands could be cut whole for a tenth of the cost and would answer a smaller question.

The narrowest is the Lucas 3/4, and it is narrow because its two contact steps never separate — so it is both the band with least room for an interior and the band whose interior would say least.

The step is a ratio #

Two parts in a thousand between consecutive rises, which is the step the band’s geometry is swept at. It is a ratio rather than a fixed amount because the ladder is geometric: an absolute step is three different instruments on it, one per cent at one handover and half a per cent at another.

So the 126 rises are 126 multiplications rather than 126 additions, and a feature two rises wide is two parts in a thousand of the rise wherever on the band it sits.

What nine rises got right #

Both of the things the design was for.

The ends. The coarse end of the band keeps the 8 family at every offset that wrecks there, and the fine end keeps the 4 family at the three offsets that ever change. Both are confirmed at full resolution.

And the handover. Not one of the nineteen changes at full resolution sits at the rise where the two contact steps change places. That is the claim the whole thread rests on and the finer sweep does not touch it.

What it got wrong: how many #

Five changes in the sample; nineteen at full resolution. Thirteen of the nineteen are at one offset.

That is not a failure of the sample so much as a description of what a sample is. One step of the nine-rise design is sixteen rises of the sweep, and a change that happens and reverses inside sixteen rises is invisible to it.

Nineteen is also not a stable number. It counts transitions between consecutive wrecked rises, so an offset that recovers for a stretch and comes back with a different answer contributes one — and whether that is one change or two depends on a convention nobody has had to state before, because at nine rises the question does not arise.

The arithmetic of what the sample could have seen is worth doing, because it turns “the design cannot resolve this” from a statement about resolution into a prediction about what the design should have printed.

At the offset that alternated there are thirteen islands of the minority family among 126 rises, one to three rises wide, so the minority occupies something like a fifth of the band. Nine sampled rises drawn from a band a fifth of which is the minority family should land on it about twice. The sample landed on it twice: 8, 4, 8, 4.

So the alternation is not merely unresolvable by nine rises. It is precisely what nine rises drawn across this row would be expected to return by chance, with no period, no structure and no coincidence needed. Had the sample come back 8, 8, 8, 8 the reading would have been “no change here” and would have been equally wrong; had it come back 4, 4, 4, 4 it would have been “the fine end keeps the 4” and wrong again. Three plausible readings, one of which appeared, none of them supported.

That is a better statement of the failure than “the sample was too coarse”, because it says the sample’s output was uninformative rather than misleading — a uniform draw from a speckled row carries the row’s proportion and nothing about its arrangement.

And where #

The sample bracketed the changes two to four of its own steps below the crossing, and that was written down as a distance. At full resolution they sit 8 to 58 rises below it, and the three offsets that change cross at three different rises.

A position quoted in sampled steps is a position quoted in the sample. That sentence is the correction, it is worth more than the numbers on either side of it, and it applies to every distance this thread has ever quoted in steps of a sweep.

The three results #

They get essays of their own and it is worth naming them here. The alternation is not a period: it is thirteen short islands with uneven gaps, and a period fitted at its best phase buys exactly nothing over saying the commonest family and stopping.

The three offsets that change cross at three different rises, none of them the handover. And three of the six offsets that wreck never change at all, while which offsets wreck is itself a function of the rise.

The offsets that wreck, rise by rise #

One thing the sweep produces that the sample could not is a picture of which offsets wreck at all, across a whole band. It is not the same set at every rise: offset 7 wrecks at all 126, offset 8 at 123, offset 6 at 110, offset 4 at 98, offset 9 at 81 and offset 5 at only 22 of them.

So a census taken at one rise of a band and a census taken at another are censuses of different sizes. Every claim in this thread of the form “at every offset that wrecks” is quantified over a set that the rise decides, and nothing had drawn that set moving.

The sizes are worth adding up. Across the 126 rises there are 560 wrecked cells in all, so an average rise of this band wrecks at four and a half offsets rather than six. And all six wreck together only where the rarest of them does, which is at most twenty-two rises — one rise in six.

That has a consequence for how the thread’s own sentences should be read. “The six offsets that wreck on this band” describes at most a sixth of the band’s rises, and a census taken at a rise drawn at random from it would more often than not report four or five. The phrase is not wrong; it is a union over rises being read as a property of one, and a reader who took it as the latter would expect a census at any rise to reproduce six rows.

Which also prices the earlier design’s exposure. Nine rises drawn across this band would be expected to include one or two where all six wreck and seven or eight where they do not, so even the set the earlier claims were quantified over was being sampled rather than measured.

What a sample is for #

Not this. A nine-rise design answers “is the quantity the same at both ends and at the crossing”, and that is a real question with a real answer, and it got it right.

What it cannot answer is anything about the interior. The mistake was not in choosing nine; it was in reading the sample’s output as though it were the band’s. The 8, 4, 8, 4 was a genuine reading of four sampled rises and it was never a reading of a period.

The refusal that now sits on it #

Asking a nine-rise design for the islands is refused rather than answered. The islands are one to three rises wide and the design steps sixteen rises at a time, so it can only land on one by accident.

That is a piece of machinery rather than a caution in prose, and it is the piece that would have stopped the 8, 4, 8, 4 from being reported as an alternation in the first place. A design that cannot resolve a feature should decline to describe it.

What the sweep cannot do either #

It has a step of its own. Two parts in a thousand is fine enough to resolve islands of one rise and it is not infinitely fine: a feature at four parts in ten thousand would be invisible here exactly as these islands were invisible to nine rises.

Nothing in this sweep says the speckle has a floor. The honest form of the finding is that at a step of two parts in a thousand the transition region is speckled, and a finer step might show the speckle to be structure.

What was already on disk #

Most of the cost. The stems the sweep grows are shared with the census and with the band geometry, and the ones that had been grown before were read from disk rather than regrown.

That is why sixty-seven minutes rather than several hours, and it is the reason the “nine is enough” estimate was out of date. It was made against a cost that has since fallen by more than an order of magnitude, and nobody revisited it.

What the other five bands would cost #

Between a tenth and three quarters of this one each, and about two hours for all five. The narrow ones are cheap and would answer little; the Lucas 7/11 is 124 rises and would be the real second test, since it is nearly as wide as this one and on the other branch.

Not done. The reason is the ordinary one: the sweep was built to check one alternation on one band and it answered that, and extending it is a separate decision that has not been taken. It is recorded as an outstanding check rather than as a sentence about intending to.

The one thing the sample could not have been fixed to see #

Suppose the nine-rise design had been given sixteen rises, or thirty-two. The islands are one to three rises wide and the gaps between them run from one rise to forty-eight, so a design with a step of four would still be landing on islands by accident and reporting a sequence of coincidences.

What resolves speckle is not a denser sample; it is every rise. That is an unusual conclusion for this collection, which nearly always finds that a moderately denser sweep suffices, and it follows from the feature being one step wide at the sweep’s own step. There is nothing between “sample it” and “do all of it”.

What the sweep measures at each rise #

The same thing the census does. Every offset out to the front and two past it is cut in turn; each cut stem is grown and compared with a control that shares its history to the last digit; the surviving lag is the one whose hop is unchanged.

Nothing new is computed. The sweep is the existing measurement made 126 times instead of ten, which is what makes its results comparable with everything the thread already has — the same discipline the wider slot design follows on the other side of the site, where thirty lattices are a superset of six rather than a replacement for them.

Which is the argument for doing it #

A finer sweep that measures something new is two changes at once and its results are hard to attribute. This one changes the sampling and nothing else, so every difference between the nine-rise reading and the 126-rise reading is a difference the sampling made.

That is worth the sixty-seven minutes on its own, independently of what turned up. Most of the corrections in this collection have come from varying one instrument setting and leaving everything else alone — doubling a run length in the ablation thread is the same move made here on a different instrument, and it produced the same kind of result.

What a reader should carry #

That a band’s ends and its crossing are what nine rises measure, and that everything this thread has said about a band’s interior was a statement about ten sampled rises out of a hundred and twenty-six.

And that the correction runs in one direction. Nothing the sample said turned out to be false; several things it said turned out to be about the sample.

What the picture at the top shows #

Six rows and a hundred and twenty-six columns, coarse on the left. The dark cells are the 8 family kept, the mid cells the 4 family, and the pale ones are offsets that recover at that rise and have no survivor to report.

Read across the top three rows and the band is nearly uniform. Read the bottom three and there is a clean left half, a speckled middle and a clean right half — and the speckle is in a different place on each of the three. That is the whole of what this sweep found, and it is legible in one picture because there is nothing between the sampling and the band.

The one line #

The widest band on the ladder, cut at all 126 of its rises and 1,890 cut stems: the nine-rise design’s two claims — that the band’s ends differ and that no change of answer sits at the handover — both survive, and everything it said about where inside the band the changes are was a statement about a step of sixteen rises.

Nineteen changes rather than five, eight to fifty-eight rises below the crossing rather than two to four, and three offsets crossing at three different rises.