A count or a floor

A band is the stretch of rise around a handover over which a stem’s counted pair holds and its settled divergence stays flat, and cutting one at every rise it holds is how this collection stopped sampling a band’s interior and started measuring it. The band sweep has a step of its own: two parts in a thousand between one rise and the next. That number has been the same since bands were first grown, and until now it had never been moved.

So the nineteen changes of surviving family found on the widest band are a number with an instrument setting inside them, and nobody knew which kind of number they were.

The setting that had never moved #

Everything else in this thread has been varied and reported. Which band is cut, which offsets are followed, how many rises are sampled, whether the ends or the interior are being asked about — all of them have been changed on purpose at some point, and the changes are what the thread is made of.

The step is the exception. Cutting every rise of the widest band replaced a nine-rise sample with the whole band and left the grid the band’s rises sit on exactly where it was. That was the right thing to do at the time, because changing two settings at once produces a difference nobody can attribute. It also means the finest thing the thread has ever measured was measured at one resolution.

The step is a ratio, not an amount #

Two parts in a thousand is a multiplication rather than an addition, and that was a deliberate choice: the ladder these bands sit on is geometric, so a fixed step is three different instruments along it, one per cent of the rise at one handover and half a per cent at another.

The refinement inherits that. A rise inserted at the geometric midpoint of a gap divides it into two equal ratios, so the fine grid is the coarse one with its step square-rooted everywhere rather than halved at one end and not the other. A feature two rises wide at the coarse step is two parts in a thousand of the rise wherever on the band it sits, and four fine rises wide wherever it sits.

What a floor would mean #

A count is a total: this is how many changes there are, and a finer instrument would find the same ones. A floor is a lower bound: this is how many changes are wide enough to be seen, and a finer instrument would find these and others.

The difference is not academic, because the two readings support different sentences. If nineteen is a total, the transition region has been described. If it is a floor, then every statement about how speckled that region is — how many islands, how wide, how spaced — is a statement about the grid the region was read on, and the fitted period that was rejected there was fitted to a sampling.

An instrument setting is a claim #

Reporting a count from a sweep asserts, silently, that the sweep can see what it is counting. Nothing in the coarse reading made that assertion out loud, and nothing could have: a grid cannot report what falls between its own rises, so the only way to know whether it is missing anything is to lay a finer one and look.

That is the whole design here. One setting changes and every other setting is held, so any difference between the two readings is a difference the setting made — the same discipline cutting a whole band instead of nine rises followed, one level down.

Insert, do not relay #

The comparison is only a comparison of resolutions if the two grids are grids of the same thing. So the finer grid is the coarse one with rises inserted: one new rise at the geometric midpoint of every gap, which is the midpoint in the logarithm of the rise because the sweep steps by a ratio rather than by an amount.

That construction makes every coarse rise a rise of the fine grid by definition. A grid laid down independently at half the step, outwards from the handover, covers almost the same stretch of rise — 1.28516 times against 1.28437 — and shares 2 of the band’s 126 rises. It is refused rather than tabulated, because a table of gains between two grids that barely intersect is a table of two different measurements.

What was cut #

The widest band goes from 126 rises to 251 and the next band along from 112 to 223, which is 125 and 111 inserted rises and 236 new rises in all. Each is cut at every offset that wrecks somewhere on its own band, so the new work is 1,083 cut stems — 750 on the wide band and 333 on the narrow one — and the whole fine reading stands at 1,506 and 669.

Twelve minutes and seventeen seconds across six workers, or seventeen and a half when the machine was busier. The sweep is keyed per rise rather than per band, so a partly computed grid keeps what it cost.

The offsets followed #

Six on the wide band and three on the narrow, being those that wreck somewhere at the coarse step. The front reaches fifteen and ten respectively, so most of what could be cut was not.

That is a decision about cost with a consequence attached, and the consequence is named rather than buried: an offset that wrecks only at an inserted rise is invisible to this reading. Nothing here says there is no such offset. It says the question was not asked.

Nineteen becomes twenty-one #

On the wide band the count goes 19 to 21. On the narrow one it goes 2 to 2.

Two extra changes is a factor of 1.105, and the size of that factor is most of the finding. An order-of-magnitude gain would have meant the coarse reading described nothing; a gain of zero would have meant the step was never the constraint. A tenth means the coarse reading was very nearly right and was not exactly right, which is the least comfortable of the three answers and the only one that requires the sentence to be rewritten rather than kept or thrown away.

Where the two extra changes are #

All of the gain is at offset 6, which goes from three changes to five. Offset 7 carries three and gains none; offset 8 carries thirteen and gains none; offsets 4, 5 and 9 change their answer nowhere at either step.

Not the speckled row #

That is the result inside the result. The row a reader would have bet on is offset 8: it carries thirteen of the nineteen, it is the row whose islands were counted and whose period was rejected, and it is the reason the word speckle is in this thread at all. It gains nothing.

The row that was under-resolved is the quiet one, with three changes on it. A grid too coarse for a feature does not preferentially blur the busy places; it misses whatever happens to be narrower than a step, and where that is has nothing to do with how much else is going on nearby.

The band that gains nothing #

The second band cut at half the step is the golden 5/8, which is the band that was cut whole to settle which of four accounts survived. It has two changes, both at offset 5, and it has two changes at half the step.

Not one offset of it gains anything. It is the same machinery, the same halving, the same branch and the same kind of object, and the answer it returns is that there was nothing there to find.

The assertion that failed #

This is worth stating in the order it happened, because the order is the argument.

The check written to guard the finding said that the gain would replicate: that a second band, cut the same way at the same halving, would also find changes the coarse grid had missed. That is the sensible expectation. If a step of two parts in a thousand is too coarse for the structures these bands carry, it is too coarse on both of them.

It failed. The second band gained nothing at all, 2 to 2, and the assertion refused the reading rather than the reading refusing the assertion.

What was changed, and what was not #

The claim. The measurement stands exactly as the machinery produced it, and the sentence that was going to be written about it does not.

What replaced it is narrower and is the actual finding: the gain belongs to one band rather than to the sweep. That is a weaker statement than the one that was being guarded, and it is the one the two bands support.

An assertion that has never rejected anything proves nothing, and this thread says so often enough that it is worth pointing at an occasion when one did. The cost of that rejection was a paragraph; the cost of not having written it would have been a generalisation from one band, which is the mistake this thread has already made once.

So a floor is a property of the band #

A tally is a floor where the band carries features narrower than the step, and exact where it does not. The wide band holds sixteen enclosed runs at the coarse step — stretches where an offset keeps the minority family with the majority on both sides — and the narrow band holds none.

That is not a subtle difference between two bands. One has a speckled interior and the other has two isolated crossings, and the finer grid finds more only in the first.

Why the narrow band has nothing to find #

Its two changes sit at the ends of long quiet stretches. Bracketed on the fine grid they are 37 and 69 rises wide, which is to say offset 5 does not wreck anywhere across those stretches and there is no cell between them for a finer grid to fill in with anything.

Halving the step in a place where the instrument reports nothing gives twice as many readings of nothing.

The cost of asking #

Halving a step doubles a sweep, and this one was affordable because most of its cost was already on disk. An inserted rise on the wide band is six cuts and a control and takes twenty-one to twenty-eight seconds under contention; on the narrow band it is three cuts and seven or eight seconds.

That arithmetic matters for what happens next. A quarter of the coarse step is another 472 inserted rises on the two bands and roughly four times this run, which is affordable; a sixteenth is not, and the interesting question about the speckle sits somewhere below both.

The transition region is the transition region #

The consequence for what has already been written is small and specific. The three offsets that change and the three different rises they cross at survive: none of the nineteen coarse changes is lost, none of them turns out to have been in the wrong place, and the claim that no change of survivor sits at the handover is untouched.

What moves is the count, by two, and the description of the region as speckled — which was always the honest reading — now has a second resolution behind it rather than one.

The control that says it is the same band #

How often an offset wrecks should not depend on the grid, and it does not. Across the six offsets of the wide band the worst drift between the two readings is 0.85 percentage points; on the narrow band it is 1.23. Offset 7 wrecks at every rise of the wide band at both steps.

Eighteen offset-band pairs coming back within about a point is the control this comparison needed. The finer grid is reading the same object, so the two extra changes are something found rather than something introduced.

What refinement forces #

That no coarse change can be lost is not an observation. The fine grid holds every coarse rise with the same answer at it, and a change is counted between consecutive wrecked rises, so a stretch that changes at the coarse step must contain at least one change at the fine one.

The direction is therefore guaranteed and carries no information. What carries information is the size of the gain, and on one of these bands it is zero.

Which is why the guard is load-bearing #

Only because the fine grid contains the coarse one. Take that away — lay a second grid independently at half the step, from the handover outwards — and the two readings share almost no rises, so a change present in one and absent in the other says nothing about resolution at all.

The comparison refuses such a pair rather than returning a difference. A refusal is cheap and a plausible table of gains between two unrelated samples is not, which is the same argument as refusing to describe islands a design cannot resolve.

Two readings of the same two changes #

There are two quite different things the extra pair could be, and this essay does not separate them.

They could be edges: known islands whose boundaries the coarse grid put in the wrong place, so that one coarse change turns out on inspection to have been two. Or they could be events: structures the coarse grid stepped clean over, present nowhere in the coarse reading at all.

The first would mean the coarse tally located things imprecisely; the second would mean it undercounted. Matching every coarse change against the fine ones that carry it is what tells them apart, and the answer is not the one the arithmetic alone would suggest.

What the coarse designs still say #

The nine-rise sample found five of the nineteen changes and is not made worse or better by this. It remains right about the ends and about the handover and wrong about the interior, which is what cutting the whole band established and what the third band’s two missed changes confirmed.

The one thing that has changed is the reference the sample is scored against. It was five of nineteen and it is five of twenty-one, and a design measured against a moving denominator is worth knowing about even when the movement is a tenth.

What half a step cannot see #

Three things, and they are worth listing because the temptation after a result like this is to treat the fine reading as the truth.

Offsets that wreck only at inserted rises are outside it by construction. Features narrower than the fine step are outside it exactly as these two were outside the coarse one. And the narrow band’s answer of nothing is an answer about that band at this halving, not a proof that its interior is empty at every resolution.

No floor to the speckle #

Nothing here says the structures have stopped arriving. At half the step the speckle looks like the speckle: a little more of it, its islands in the same places, and several of them still one rise wide at both readings — which is what an unresolved event looks like, not what a resolved one does.

The finding is a comparison of two resolutions, and two resolutions can say that a count moved. They cannot say it has stopped moving.

What a reader should carry #

That nineteen was a floor and the floor is nearly the number. The transition region reported from the coarse sweep is the transition region, its busiest offset is exactly as busy as it was said to be, and the correction is two changes on the quietest row that changes at all.

And that the second band’s zero is the more useful half. One band would have given a resolution effect; two give a property of a band, and the difference between those was bought by an assertion that was allowed to fail.

The one line #

Two hundred and thirty-six inserted rises and 1,083 cut stems: 19 changes become 21 on the band with islands in it and 2 stay 2 on the band without, no coarse change is lost on either, and the wrecking shares hold to a point — so whether a tally of this kind is a count or a floor is decided by the band and not by the sweep.