Nine rises were not enough

The design that cuts a band takes nine rises, evenly spaced in the logarithm of the rise, plus both ends and the handover itself. Nine was chosen for a quantity the design predicted to be constant, and a constant is tested at its extremes and at the crossing rather than by sampling the middle more finely.

The quantity turned out not to be constant. Since then the nine-rise design has been run beside three full sweeps, and the interesting number is not how many changes it finds but how often it is right about whether there are any.

The record it had #

On the golden 8/13 band the sample found five changes where the sweep found nineteen. Wrong about how many, right that something changes.

On the Lucas 7/11 band it found none where the sweep found none. Right about how many and right that nothing changes — and that was the more valuable of the two, because a negative from a sample is worth nothing unless the sample can be trusted to find a positive.

So after two bands the record read: right twice about whether, right once about how many. That reading is what let the design say a negative from nine rises means something, and the sentence was written down in those words.

What the third band did to it #

The golden 5/8 band holds 112 rises. The nine-rise design visits ten of them at a step of fourteen. It finds no change.

The full sweep finds two: offset 5 keeps a family off its counted pair at rises 0.01625 and 0.01605, and keeps the ordinary one at the other twenty-six rises it wrecks at.

So the record is now two of three on the question the design exists to answer, and the band it is wrong on is the one it was asked about most recently.

Why it missed them #

Both changes sit at single rises with offset 5 recovering on either side. A feature one rise wide, sampled at a step of fourteen, is hit about one time in fourteen; two of them, at rises twenty apart in a band of 112, are missed together most of the time.

That is not a criticism of the sample. It is arithmetic, and the same arithmetic was already written down: the first band’s islands are one and two rises wide and the sample hit two of thirteen of them.

What is new is that on that band there were thirteen chances and on this one there were two. A design that undercounts thirteen features to five is a design that will report zero of two sooner or later, and this is sooner.

The two questions come apart #

Does anything change? and how much? are different questions and the design answers them with different reliability, which is why they have been kept apart since the second band.

On how much it has always been poor: five of nineteen, none of none, none of two. The only band it gets exactly right is the one with nothing to count.

On whether it was perfect and is now not. And the failure mode is asymmetric in a way that matters: the design can only ever undercount, because a change it reports is a change between two rises it actually cut. So a positive from nine rises is sound and a negative is a bound.

What a bound is worth #

Something, and it is worth stating rather than dismissing. Ten rises finding nothing on a band of 112 rules out any feature wide enough to have been stepped on — anything eight rises across or more would have been hit twice.

So a negative from nine rises is not nothing changes; it is nothing changes over a stretch wider than about seven rises. On the Lucas band the full sweep then confirmed the stronger statement, and on this band it did not.

The difference between the two readings is exactly the class of feature this thread keeps finding: islands one and two rises wide, with nothing continuous distinguishing them from their neighbours.

The cost of the alternative #

A full sweep of a band is between 1,120 and 1,890 cut stems, which is seventeen to thirty-eight minutes. The nine-rise design is about 140 stems and two minutes.

Three bands have been swept whole and three have not. Of the three that have not, two wreck nothing at all — nothing behind the front wrecks at the coarse end — so the real remainder is the Lucas 4/7 band at 86 rises and about half an hour.

That is affordable, and after it the ladder has no more bands. So the question of what a coarse sample is worth on a band is about to become a question about no bands, which is a reason to answer it now rather than later.

Where the design is still right #

At the ends. The coarse design reports what each end of a band keeps, and on all three bands that report matches the sweep exactly.

That is not trivial. A band’s two ends are the two lattices the whole handover thread compares, and everything said about the ordering of the two contact steps changing places rests on reading them correctly. Nine rises reads them correctly on every band tried.

So the design has a part that works and a part that does not, and they are separable: the ends are two readings and the middle is seven.

And where it was never asked to work #

The design was built to test a constant. If the surviving family had been constant across every band — which is what everything before the first full sweep suggested — then nine rises would have been the right number and sampling the middle finely would have been waste.

It is worth being fair about that. The design is not a bad sample of a varying quantity; it is a good sample of a constant, used on a quantity that turned out to vary. The mistake was in the model, and the design followed from it correctly.

What follows is that the design should not be reused unchanged on the fourth band, because the model it was built for is now known to be wrong.

What a better design would look like #

Not more rises evenly spaced. A step of seven instead of fourteen doubles the cost and still misses a feature one rise wide thirteen times in fourteen.

The feature that matters is narrow, so the only sampling that finds it reliably is one that does not skip. Between a full sweep and a nine-rise sample there is nothing useful: the quantity is an integer read off a run, it changes between adjacent rises with nothing continuous moving, and no interpolation is available.

That is a real conclusion about the object rather than a complaint about resources. A discrete answer that changes at single rises can only be found by cutting every rise, and the design should be either the full sweep or an honest bound.

The same shape, elsewhere on this site #

This is the third time a sampled instrument here has been swept and found to have been reporting its own step.

The reference search window decided answers when it was varied. The run length moved nineteen of twenty-four onsets when it was doubled. The reading window turned out to be measuring a drift rather than a level.

Four instrument settings have now been varied and all four decided something. That is a run long enough to be worth stating as a working assumption: a setting nobody has moved is a setting that is deciding an answer, until it has been moved.

What the coarse design has decided elsewhere #

Its step is also what the earlier location claims rested on. The changes on the first band were reported as sitting two to four sweep steps below the crossing, and one of its steps is fifteen rises — so that sentence located nineteen changes to within about sixty rises.

At full resolution they sit seven to fifty-seven rises below the handover and the three offsets cross at three different rises. The coarse statement was not wrong; it was a statement about the sample, phrased as a statement about the band.

That is the more insidious of the two failure modes, because it produces a number rather than a silence.

What is not in doubt #

Every claim the sweeps make about what a band’s ends keep, because those are read directly.

Every claim about the wrecking set, because that is read at every rise of every band and does not depend on the sampling at all — which offsets wreck is a function of the rise on all three bands cut whole, and the coarse design would have found that too.

And every claim about a handover’s position, because a handover is geometry: it is computed from hop lengths rather than from cut stems, and it can be walked at whatever resolution is wanted for nothing.

The negative that is now weaker #

No change of surviving family sits at a handover. That claim is scored over nineteen located changes on one band and none on another, and it stands.

But it stands on changes that were found, and the count of changes found is now known to be a lower bound on two bands rather than one. If the golden 8/13 band holds changes that even a full sweep at its own step missed — features narrower than two parts in a thousand in the rise — then the tally of nineteen is a floor as well.

Nothing suggests it does. The point is that the sweep’s own step is now the only thing standing between the tally and that possibility, and the sweep’s step has never been varied.

A setting that was never moved #

The band sweep’s step is two parts in a thousand between consecutive rises. It was chosen because the ladder is geometric and a fixed step in the rise is a step that means different things at the two ends of it.

Nobody has cut a band at one part in a thousand. That would double the cost of every sweep already made and would answer one question: whether the features this thread calls one rise wide are one rise wide or narrower than the grid.

Given the last four settings varied, the expected answer is that it matters. That is a prediction with a direction attached — a finer grid finds more changes, never fewer — and it is the cheapest thing that would test the tally of nineteen.

What to do with the fourth band #

Sweep it whole. The nine-rise design would cost two minutes and produce a number nobody could use: a positive would be sound and a negative would be a bound that this essay has just shown to be violated once in three.

The fourth band is the Lucas 4/7, and it is the test the surviving account of the three bands most needs. Running it at the coarse design and getting a negative would leave the account resting on a reading that has recently been wrong.

Half an hour, once, and the ladder’s bands are done.

What the sample would have to be to work #

Consider what it would take for nine rises to find a feature one rise wide with any reliability. A step of fourteen hits a given rise with probability about one in fourteen; to be more likely than not to hit at least one of two such features, the step has to come down to about three, which is 37 rises of 112 and a third of the cost of the full sweep.

At that point the argument for sampling has gone. A third of the cost for a coin-flip on the question, against the whole cost for the answer, is not a trade anybody would take once it is written down as numbers.

That is the general shape of sampling a discrete quantity with narrow features: there is no useful middle. Either the sample is dense enough that it is nearly a sweep, or it is a bound whose width is its own step.

What the sweeps have cost, and what they bought #

Three bands, 4,622 cut stems and about ninety minutes of machine time in total. What they bought is four results that no sample produced.

That the alternation on the first band is speckle rather than a period; that its three changing offsets cross at three different rises; that a second band has no changes at all; and that a third has two, which is what makes the branch the only account left standing.

Not one of those is available from nine rises. The first two need the changes located, the third needs the negative at full strength, and the fourth needs a positive the sample missed.

And what a sample is still the right tool for #

The wrecking set, which the sweeps found moves on every band. A sample of nine rises would have found that too, because an offset that wrecks at 17 rises of 112 in seven stretches is not a narrow feature — it is a coarse texture, and any sampling sees it.

So the design is not wrong about everything. It is wrong about the one quantity that changes at single rises, and it is right about the one that changes over stretches.

The lesson generalises past this band: the resolution a design needs is set by the narrowest feature of the quantity it is reading, and nothing about the design says what that is until something is swept whole.

What the record is worth as a record #

Three bands is a small denominator, and the design’s record on them — two of three about whether, one of three about how many — is not a probability. It is three cases with names.

What makes it usable is that the failure is understood rather than merely counted. The design missed this band’s changes because they are one rise wide and its step is fourteen, and the same arithmetic says it will miss any feature of that width about thirteen times in fourteen. That is a prediction about the next band, and it does not depend on how many bands have been cut.

So the honest summary is not the design is right two thirds of the time. It is: the design finds features wider than its own step and misses narrower ones, which is what a sampling does, and this thread’s features are narrower than its step.

Where that leaves the uncut bands #

The Lucas 4/7 is the only one left that wrecks, and it should be swept whole rather than sampled. The argument is now arithmetic rather than precautionary: a negative from nine rises on that band would be a bound of about seven rises, and the features this thread finds are one and two rises wide, so the bound would exclude nothing anybody wants excluded.

Half an hour buys the answer instead. That is the same trade the previous three bands took, and the only reason it was ever in question is that the coarse design’s record made a negative from it look like a result.

Stating the record #

Three bands cut at every rise they hold. The nine-rise design run beside each of them.

Right about whether anything changes on two of three, right about how many on one of three, right about both ends on three of three. It cannot overcount, so a positive from it is sound and a negative is a bound of about seven rises.

That is what a coarse sample of a band is worth, measured rather than assumed, and the measurement took three full sweeps.