The coarse design scored

The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, plus both ends and the handover itself. It is the instrument every statement about the interior of an uncut band has come from, and until the bands began to be swept whole there was nothing to score it against.

Six bands are now cut at every rise they hold. That closes the record, and a closed record can be read rather than extrapolated: right about whether a band changes on five of six, wrong on the golden 5/8, where it finds none of two.

What the design was built to test #

A quantity predicted to be constant. If the family a wrecking cut leaves standing is the same at every rise of a band, then the informative rises are the two ends and the crossing, and sampling the middle finely buys nothing at all.

That is a good design for that prediction. It is the design the six bands were first read with, and on five of those six it reported no change at all, which is what the prediction says it should.

The quantity is not constant. The widest band changes its answer nineteen times, which is what turned a sampling decision into a question about what a sample is worth.

Five of six, and the denominator is wrong #

The headline is five of six, and it is the number a reader would quote. It is also the number that most needs qualifying, because two of the five are wins over bands with nothing on them.

The Lucas 3/4 wrecks nothing and the golden 3/5 wrecks nothing. Ninety-six cut stems on one and four hundred and ninety on the other, six and seven distinct offsets tried, and not one wrecked cut anywhere on either. A band with no wrecked cut has no surviving family, so it has no answer to change.

The coarse design says nothing changes on them. Nothing does. It is right, and being right about a band where the only available answer is nothing is not evidence that it can find something.

Three of four #

On the bands that can answer, the record is three right and one wrong. That is the number this essay is about and it is the number the design should be quoted at.

Dropping two trivial wins from a score of six looks like pedantry until the arithmetic is done. Five of six reads as an error rate of about one in six; three of four reads as one in four, and the difference between them is entirely a matter of which bands were allowed into the denominator.

The same correction has already been forced on the accounts of why bands change: scoring them over all six eliminates every one of them and scoring them over the four that can answer leaves one standing. Counting a silence as a no is one mistake, and it changes two different tables.

What the two quiet bands do test #

One thing, and it is worth having. The design does not invent a change where there is none.

Neither quiet band produced a false positive, and a false positive from this design would be a serious defect rather than a resolution limit: the design can only report a change between two rises it actually cut, so a change it reports is a change that is there. The two silent bands confirm that the reporting side is sound.

That is a property of the instrument and not a property of the ladder, so it generalises where the rest of the score does not.

The one band it is wrong on #

The golden 5/8. It holds 112 rises, the design visits ten of them at a step of fourteen, and it finds no change. The full sweep finds two.

Both are one rise wide, with the offset that carries them recovering on either side. A feature one rise wide, sampled at a step of fourteen, is hit about one time in fourteen, and two of them are missed together most of the time.

The arithmetic of that miss was worked out when it happened. What is new is that it is now the design’s only error across a completed ladder, and that the ladder has no further band on which it could be repeated or contradicted.

The step is a different instrument on every band #

Two rises on the Lucas 3/4, nine on the golden 3/5, eleven on the Lucas 4/7, fourteen on the golden 5/8, sixteen on the golden 8/13, fifteen on the Lucas 7/11.

The design’s name is a count of samples, so its resolution is set by the width of whatever band it is pointed at. On the narrowest band it steps two rises and on the widest it steps sixteen, and the widest band is the one with the most to find.

That is the wrong way round. An instrument whose resolution falls as the object gets richer is an instrument tuned against itself, and nothing in the design notices, because nine is a count of cuts rather than a statement about what lies between them.

The half that has never been wrong #

The value at a band’s two ends. On all six bands the coarse design reports what each end keeps, and on all six the full sweep agrees.

That half carries more of the thread than the interior does. A band’s two ends are the two lattices the whole handover argument compares, and the ordering of the two contact steps changing places is read off them. Nothing in that argument has ever depended on an interior rise.

So the design has a part that works and a part that does not, and they are cleanly separable: the ends are two readings taken directly, and the middle is seven readings used as though they were a hundred.

And the half that is wrong is wrong in one direction #

The design can only undercount. Every change it reports sits between two rises it cut, so a positive from it is sound; a negative is a bound whose width is its own step.

On the golden 8/13 it found five changes where the sweep found nineteen. On the golden 5/8 it found none where the sweep found two. On the two testable Lucas bands it found none and there were none.

Never once has it reported a change the sweep does not hold. That is not luck — it is what the failure mode allows — but it is worth stating, because it is the difference between an instrument that is coarse and an instrument that is unreliable.

What it gets wrong is anything narrower than its own step #

Sixteen rises on the golden 8/13, and the features on that band are one to three rises across. Thirteen islands of the minority family, none of them a period, and the sample landed on two of them.

Fourteen rises on the golden 5/8, and the features there are one rise wide. The sample landed on none.

That is one rule rather than two results. The design finds features wider than its step and misses narrower ones, which is what a sampling does; this thread’s features happen to be narrower than its step on both bands that have any.

Five of nineteen is not a record about how many #

On the one band where it finds anything it finds five of nineteen. That is the entirety of its evidence on the second question the design gets asked.

A single case is not a record, and a single case at about a quarter is not a rate. What can be said is that the number it returns on a band with a transition region is not the band’s number, and that the reasons are arithmetic rather than accidental: sixteen rises a step across features one to three rises wide will return roughly the minority family’s share of the band, which is what a uniform draw returns.

So the design has one question it answers and one it does not, and only the first has a record worth the word.

The Lucas column decides nothing #

Three Lucas bands, three correct verdicts, and not one of them a test.

The Lucas 3/4 wrecks nothing. The Lucas 4/7 wrecks at two offsets and changes its answer nowhere. The Lucas 7/11 wrecks at five offsets and changes its answer nowhere. On a band with no change in it, a design that cannot see narrow features returns the same verdict as one that can.

Half the ladder is therefore silent about the instrument. That is a selection effect built into the ladder rather than into the design, and it means the design’s whole record rests on three bands.

Which leaves the golden bands as the score #

Three of them, and the design is right on two. The golden 3/5 is silent, the golden 8/13 it gets right about whether and badly wrong about how many, and the golden 5/8 it gets wrong.

So the strongest honest statement is narrow: on the two bands where anything at all happened, the coarse design found it once. Everything else in the record is a band that offered it nothing to find.

That is a small denominator with names attached rather than a probability, and it should be read as two cases rather than as a proportion.

What a negative from it is worth #

A bound, and the width of the bound is the step. Ten rises finding nothing on a band of 112 rules out any feature wide enough to have been stepped on, which is anything about seven rises across or more.

That is a real statement and it is not the statement anybody wanted. The features this thread keeps finding are one to three rises wide, so the bound excludes nothing of the class that is actually there.

Before the ladder closed, the argument for reading a negative as nothing changes was the design’s own record. The record now says the design has been asked that question three times on bands that could answer and got it wrong once.

What a positive from it is worth #

The change it names, and nothing about the count. The five changes it found on the golden 8/13 are five real changes, and the nineteen the sweep found include them.

Its positions are a different matter, and that correction has already been made once: positions quoted in sampled steps located nineteen changes to within about sixty rises, where at full resolution they sit eight to fifty-eight rises below the handover with three offsets crossing at three different rises.

A positive is therefore a witness rather than a measurement — it establishes that something is there and it says nothing reliable about how much or where.

The claim it was reading for still stands #

No change of surviving family sits at the rise where the two contact steps change places. Nothing in the coarse design’s failures touches that, because the handover is one of the rises the design always cuts and the sweeps read it directly.

The one time a change has been flagged at a handover it was flagged across a bracket nothing wrecks in, which is a limitation of that band rather than of any sampling.

So the design’s poor half and the thread’s central claim do not overlap. That is fortunate rather than designed, and it is worth writing down because it is not obvious from the score.

Nine was not the mistake #

The count followed from the model, and the model was that the quantity is constant. Nine rises is a reasonable test of a constant and an unreasonable test of a quantity that changes at single rises, and nothing about the design says which case it is in.

That last clause is the whole of it. The second band read the design as exactly right and the reading was correct — on that band it was exactly right, and it could not have known it.

An instrument that cannot report its own adequacy is one that has to be checked from outside, and checking it from outside means cutting a band whole.

What a denser sample would have cost #

A step of seven instead of fourteen doubles the cost and still misses a one-rise feature thirteen times in fourteen. To be more likely than not to hit one of the two changes on the golden 5/8, the step has to come down to about three, which is a third of the cost of the full sweep for a coin-flip on the answer.

There is no useful middle. The quantity is an integer read off a run, it changes between adjacent rises with nothing continuous moving, and no interpolation is available between two sampled rises.

That is a conclusion about the object rather than a complaint about the budget, and it is why the six bands were swept rather than sampled more densely.

What the whole ladder cost #

534 rises and 5,982 cut stems, across six bands. Read at the coarse design’s resolution the same ladder is nine or ten rises a band, which is about a ninth of the work.

The ninth that was skipped is where every result in this thread came from. Ten rises of the golden 8/13 return five changes; its 126 return nineteen, thirteen islands, and three offsets crossing at three different rises.

The trade was worth taking once, on a ladder with six rungs, to price the instrument that would otherwise have been used on all of them. It is not obviously worth taking again, and there is nothing left on this ladder to take it on.

Where the design should still be used #

On the wrecking set, which is coarse. An offset that wrecks at seventeen rises of 112 in seven separate stretches is a texture several rises across, and any sampling sees it — the fragmentation of the wrecking set spans a factor of fifty-one and none of its features is one rise wide.

And on the ends, always. Both readings there are direct and both have been confirmed six times.

What it should not be used for is a negative about a band’s interior, which is the one thing it was most often used for.

What this score does not support #

A rate. Four testable bands is four cases, and quoting three of four as a reliability is the same error as quoting five of six, one denominator further along.

It does not support any claim about the design on a band unlike these six, and it says nothing about a rung the ladder does not hold. Six bands is the whole population here rather than a sample of one, which is unusually good for a denominator and does not turn four into many.

Nor does it price the sweep’s own step. Two parts in a thousand between consecutive rises has never been varied, so every feature this thread calls one rise wide is one rise wide at that grid and might be narrower.

The asymmetry that makes the score readable #

The design cannot overcount and cannot invent. That means the score has only one kind of error in it, and a score with one kind of error is far easier to read than one with two.

Every wrong verdict on the ladder is a miss. Every right verdict is either a genuine find, a genuine confirmed absence, or a band with nothing on it — and the three are distinguishable by looking at the band rather than at the design.

So the record can be decomposed cleanly, which is why it can be scored at all. An instrument with false positives in it would need a much larger ladder before anything could be said.

What replaces it #

Nothing, on this ladder, because the ladder is finished. On any rung added later the answer is the full sweep or an honest bound, and the bound should be quoted with its width.

The general rule that comes out of it is worth more than the score: the resolution a design needs is set by the narrowest feature of the quantity it reads, and nothing about the design says what that is until something has been swept whole.

It is also the finding four instrument settings on this site have now produced between them: a setting nobody has moved is a setting that is deciding an answer, until it has been moved.

The record, closed #

Six bands, 534 rises, all cut at every rise they hold. The coarse design run beside each of them.

Right about whether a band changes on five of six, three of four once the silent bands are set aside, and one of two on the bands where anything happened. Right about how many on two of four, and both of those are bands with nothing to count. Right about both ends on six of six. It has never reported a change that is not there.

That is what a nine-rise sample of a band is worth, measured on every band there is.