New islands or old edges

Halving the step the widest band is swept at takes its changes of surviving family from nineteen to twenty-one. That is the arithmetic, and the arithmetic is the least interesting part of it, because two extra changes can arrive by two mechanisms that have nothing in common.

Separating them is a matching problem rather than a counting one, and it has an unambiguous answer here. Every coarse change is carried by exactly one fine change; every coarse island is carried by exactly one fine island; nothing splits. The whole gain is a single rise the coarse grid does not hold.

Two things a finer grid can find #

The first is an edge. A change is counted between two consecutive rises at which an offset wrecks, so it is really a bracket: somewhere between these two rises the family the cut leaves standing swaps. A finer grid inside that bracket can discover that the answer went from one family to the other and back, so what was recorded as one change was two.

The second is an event. A structure narrower than any coarse step falls entirely between two coarse rises, contributes nothing to the coarse reading at all, and appears from nowhere when a rise is inserted where it sits.

What each would mean for a tally #

They license opposite sentences about the coarse number.

If the extra changes are edges, the coarse tally located things imprecisely: it knew about every feature the band has and drew some of their boundaries in the wrong place, and a finer sweep is a refinement of positions. If they are events, the coarse tally undercounted: there are features it has no record of whatsoever, and a finer sweep is a census of a larger population.

Only the second is a reason to doubt a description of the interior. A misplaced boundary inside a bracket eight rises wide changes no claim this thread has made, because positions here are quoted as brackets rather than as locations and always were.

How a match is made #

A fine object is matched to a coarse one when it lies inside it: a fine change whose bracket sits within a coarse bracket at the same offset, a fine run of the minority family whose extent sits within a coarse run at the same offset. The match runs both ways, so three numbers come out of it — how many coarse objects have exactly one fine partner, how many have none, and how many fine objects have no coarse partner at all.

Those three numbers are the whole test. A world of edges puts coarse objects with two partners in the first; a world of events puts fine objects with no partner in the third.

Nineteen of nineteen #

Every one of the nineteen coarse changes is bracketed by exactly one fine change. Not one of them turns out to have been two, and not one of them goes missing.

Losing one was never possible — the fine grid holds every coarse rise with the same answer at it, so a coarse transition must contain a fine one — but having a coarse change carry two fine ones was entirely possible and does not happen anywhere on the band.

Nothing splits #

The same holds for the objects the changes come in pairs around. An enclosed run is a stretch where an offset keeps the minority family with the majority family on both sides, which is what this thread has been calling an island since the widest band was first cut whole. There are sixteen of them at the coarse step.

Every one of the sixteen is carried by exactly one fine run. None becomes a pair, none disappears, and every one of them stays at the offset and in the stretch of rise it was found in.

Sixteen of sixteen, and two over #

So the fine reading holds eighteen enclosed runs and sixteen of them are the coarse ones. The remaining two overlap no coarse run at all.

That is the third number, and it is where the extra changes come from. Both readings agree about every object the coarse grid could see, and the fine one holds two objects the coarse grid has no record of.

One rise, at 0.005845 #

The two new runs are adjacent and they are one event. At the rise 0.005845, offset 6 keeps the family 4 with 8 on either side — a run one rise wide. The rise immediately below it, 0.00584, is then a run of 8 one rise wide, because a single cell of a different family dropped into a stretch cuts what was there into an island of its own.

Both are inserted rises. Neither exists on the coarse grid, and neither has any trace in the coarse reading.

The picture that decides it #

The same window with the fine grid drawn beneath the coarse one is the whole argument in one drawing. The coarse strip runs uniform through the window; the fine strip carries two cells of different colour, and the ticks along the bottom are the coarse grid’s own rises, so that 0.005845 is visibly between 0.00585 and 0.00584 rather than asserted to be.

A cell in either strip is drawn as the interval running to the midpoint of the gap on each side, so a coarse cell and the two fine cells inside it cover exactly the same rise. The strips line up because the grids do, not because they were drawn to.

Narrower than the grid’s smallest step #

The island is 0.86 parts per thousand of the rise wide, and the one it creates below it is 1.29. The smallest step the coarse grid takes anywhere on this band is 1.468 parts per thousand.

So this is not a feature the coarse sweep was unlucky with. There is no rise it could have placed anywhere on the band that would have landed inside the narrower of the two, because its finest available step steps over it. The coarse reading could not have found this island by sampling differently; only by sampling more finely.

Two changes and two runs, from one rise #

The bookkeeping is worth doing slowly, because two changes sounds like two independent discoveries and it is one.

A single cell of the minority family with the majority on both sides is one change entering it and one change leaving it. The same cell also cuts the surrounding stretch, so the rise below becomes a majority island inside what is now a minority neighbourhood. One inserted rise, two changes, two enclosed runs — which is exactly the increment the band shows, 19 to 21 and 16 to 18.

Why the busiest row gains nothing #

Offset 8 carries thirteen of the nineteen coarse changes and gains none of the two. That looks backwards until the match is read: a row with thirteen changes on it has thirteen brackets, and every one of them turns out to hold exactly one fine change.

So the busiest row is not the least resolved row. It is a row whose structure is all wider than a coarse step, which is precisely why the coarse grid found so much of it in the first place. Three of the band’s six wrecking offsets change nothing anywhere, one carries most of what does change, and the one that gains is the quiet one in between — an arrangement no reading of the coarse tally would have predicted and none contradicts.

The whole offset, for scale #

Offset 6 across the band is three changes at the coarse step and five at the fine one, and the window above is a few dozen rises out of 251. Drawn whole, the row is a long quiet stretch of one family, a short speckled region, and a long quiet stretch of the other, with the new island sitting inside the speckle rather than at either edge of it.

The two readings are one measurement #

None of the matching above would mean anything if the two grids were merely similar. A fine change lying inside a coarse bracket is evidence only when the coarse bracket’s own endpoints are rises of the fine grid too, so that inside is a statement about the same object rather than about two overlapping samples.

The fine grid is the coarse one with a rise inserted in every gap, so containment holds by construction. A grid laid down independently at half the step covers nearly the same stretch of rise and shares two of the band’s 126 rises, and the comparison declines to match against it — which is a refusal in the same family as declining to describe islands a nine-rise design cannot resolve.

The band with no such rise #

The other band cut at half the step has no enclosed runs at either resolution, so the matching has nothing to do on it: no coarse object with a missing partner, no fine object with none above. Its two changes are single crossings, each of them the boundary of a long stretch where the offset does not wreck at all.

That is the same result read from the other side. A band gains what its interior holds, and an interior with no islands narrower than the step holds nothing for a finer grid to uncover.

What undercounting licenses #

One sentence, and it is about a population rather than about a number. The coarse tally of nineteen was missing an event, so the population of events on this band is larger than the coarse reading of it — and by an amount the coarse reading gave no way to estimate, since a grid says nothing about what falls between its rises.

That is a different and worse kind of error than a bracket in the wrong place, which is why it was worth separating them rather than reporting a gain of two and moving on.

And what it does not #

It does not say the coarse description of the interior was wrong. Thirteen islands at the busiest offset, uneven gaps, no period that a fitted period improves — all of that is measured on objects both readings agree about, and the new island is at a different offset entirely.

It also says nothing about how many more such rises there are. One event found at one halving is one event, and a second band found none at all.

Does the island have a width #

The new run is one rise wide at the only grid that holds it, which means its measured extent is the width of one fine cell and nothing more. An object one cell across has not been measured; it has been detected.

That is a different question from the one this essay settles, and it is asked of every enclosed run on the band rather than of this one — eleven of the sixteen hold their extent in the rise while their count of rises doubles, and five do not. The new island joins the second group by default, since it has been seen at exactly one resolution.

What the coarse grid’s grain permits #

One number above deserves suspicion rather than trust: the smallest coarse step of 1.468 parts per thousand, against a nominal step of two. A sweep whose steps run well below and well above what it was asked for is not stepping the way its setting says, and the reason is in how the band’s rises are laid down.

It does not weaken the argument here; it strengthens it. The comparison against the island’s 0.86 uses the smallest step the coarse grid takes anywhere on the band, which is the most favourable number the coarse reading could be given, and the island is narrower than it regardless.

A bracket is still not a location #

The nineteen coarse changes each having one fine partner means their brackets tightened and their contents did not move. It does not mean they are located: a change bracketed between two rises at half the step is bracketed twice as tightly and is still a bracket.

That distinction is the standing habit here rather than a new caution. A change bracketed across a hundred grid steps was pinned to one by cutting the rises in between, and the pinning was worth doing precisely because the wide bracket had been quoted as though it were a position.

What would have made this an edge result #

Worth stating, because a test that could not have come out the other way is not a test.

Any coarse change carrying two fine changes would have done it. So would any coarse run carrying two fine runs — an island that turns out to be a pair with a gap the coarse grid could not see. Either would have appeared in the match as a coarse object with two partners, and the drawing joins partners with a line, so a fork in the joining lines is what an edge result looks like.

There are no forks. Every joining line in the picture runs one to one.

The reading that would have been cheaper to believe #

There is a third account of two extra changes that neither of the two above covers, and it should be dismissed on the evidence rather than by preference: that the island is an artefact of the finer grid. A cell appearing where a grid gains rises is exactly what a quantisation artefact looks like, and this collection has reported three consecutive samples of a grid as a period once already.

What rules it out is that the fine reading reproduces the coarse one everywhere else. If inserting rises manufactured cells, it would manufacture them across a band 251 rises wide rather than at two of them, the wrecking shares would move rather than holding to under a point, and the sixteen coarse islands would not each come back as exactly one. An artefact that appears twice and nowhere else, at one offset, in one stretch, is doing a poor impression of an artefact.

What one island is worth #

As evidence about the speckle, very little on its own, and the honest reading is arithmetic rather than a story: one newly visible island, at one offset, on one band, at one halving of one step.

As evidence about the kind of error the coarse grid makes, it is decisive, because the two candidate kinds make incompatible predictions about the match and only one of them occurs. That is the difference between a result and a measurement of a difference.

The control, again #

Offset 6 wrecks at 87.3 per cent of the coarse band’s rises and 86.5 per cent of the fine one’s. The row that gains two changes is not a row that has started behaving differently under refinement; it wrecks in the same places, keeps the same families, and has two extra cells in the middle of it.

A row whose wrecking share had moved by ten points would have made every claim above suspect, because then the fine grid would be reading something other than the coarse grid’s band.

What it cost to look #

An inserted rise on this band is six cuts and a control, twenty-one to twenty-eight seconds under contention, and there are 125 of them. The whole of the difference reported here comes from two of those 125.

That is the ordinary shape of a resolution result and it is worth stating plainly, because a sweep that doubles in size to move a count by two invites the question of whether it was worth running. It was, and not because of the two: the nineteen of nineteen and the sixteen of sixteen are what the run bought, and those could not have been known without cutting every inserted rise rather than the interesting ones.

What a reader should carry #

That the extra changes are new objects rather than corrected boundaries, and that the distinction was settled by matching every old object against the new reading rather than by comparing two totals.

And that the object itself is small enough to be worth naming: one rise, at 0.005845, where a cut at offset 6 keeps the 4 family — 0.86 parts per thousand of the rise wide, on a grid whose smallest step is 1.468.

The one line #

Nineteen of nineteen coarse changes and sixteen of sixteen coarse islands are each carried by exactly one fine object, two fine islands overlap nothing at all, and both of those come from a single inserted rise — so halving the step of this band’s sweep found an event it had never seen rather than moved an edge it had drawn badly.