The last unmoved setting

The band sweep’s step is a ratio: each rise is the one before it multiplied by 1.002, which is two parts in a thousand. That was chosen deliberately over a fixed amount, because the ladder the bands sit on is geometric and an absolute step is a different instrument at each end of it.

Halving it is the first time the setting has moved, and the first thing moving it found was that the sweep has never stepped by 1.002 anywhere.

The grid under the grid #

A band’s rises are computed by multiplying outwards from the handover, and each one is then rounded to five decimal places before anything is cut at it. The rounding is not a display convention; it is the value the sweep uses, so it is the grid the sweep actually walks on.

At the widest band’s handover of 0.00605, one unit of that fifth place is 1.653 parts per thousand of the rise. The nominal step is 2.00. A setting only 1.21 times its own grain is not a step size in any useful sense — it is a request that the rounding rounds up or down as it pleases.

Ninety-nine of a hundred and twenty-five #

Which is what it does. Of the band’s 125 steps, 99 are a single unit of the fifth decimal place and 26 are two units. There is no third size.

So a sweep whose step was set to two parts in a thousand takes the smallest step its grid allows four times in five, and takes a double step the rest of the time. That is worth holding beside the finding that a band of stuck rises can sit inside a rung a coarse ladder never samples: the two are the same hazard, one at the scale of the ladder and one at the scale of a single band. The pattern of which is which is not a property of the band; it is where the multiplication happens to fall relative to a decimal boundary.

The slots it skips #

Another way to see the same thing: count the values the fifth decimal place can take between the band’s lowest and highest rise. There are 152 of them, the band occupies 126, and it skips 26.

The next band along skips far more — 351 slots across its range and 112 used — because its handover sits at a larger rise, where the same decimal place is a finer grain. A sweep that uses five slots in six on one band and one in three on another is two instruments, and the sentence swept at two parts in a thousand covers both of them without distinguishing them.

What the steps actually are #

Read as ratios in the rise, the band’s steps run 1.468 to 3.752 parts per thousand, with a mean of 2.004. The mean is the number the setting asked for, and it is the only place the setting appears.

That is a spread of a factor of two and a half within one band, arranged by arithmetic rather than by geometry. An instrument reporting a resolution of two parts in a thousand while resolving between 1.47 and 3.75 is reporting a setting rather than a measurement, and every width this thread has quoted in steps of it inherits that.

The thing the ratio was adopted to remove #

The point of a ratio step was that a band sweep should be the same instrument at both ends of the ladder. The rounding puts the problem straight back in, because a fixed decimal place is a fixed amount and the handovers are at different rises.

One unit of the fifth place is 1.653 parts per thousand at the widest band’s handover and 0.642 at the next band’s, which sits at 0.01558. The grain is therefore 2.6 times coarser relative to the rise on one band than on the other, and that is exactly the disparity the geometric ladder was swept by ratio to avoid.

Why one part in a thousand cannot be asked for #

The obvious way to halve a sweep’s resolution is to halve its step: ask for 1.001 instead of 1.002. On the widest band that produces duplicate rises, because a step of one part in a thousand at a grain of 1.653 rounds to the same five-decimal value twice in a row.

A sweep asked for that way would have returned a grid with repeated entries, the same answers at them, and no more resolution than before — which is a failure that looks exactly like a finding of nothing.

So the finer grid was laid the other way #

By insertion: one rise at the geometric midpoint of each existing gap, held to seven decimal places. That is why the fine grid contains the coarse one by construction, and it is also the only way it could have been made finer at all.

The resulting steps run 0.734 to 1.876 parts per thousand about a mean of 1.002. The irregularity is inherited from the coarse grid’s own irregularity, halved along with everything else.

An island has a size #

With that established, the question the finer grid was built for can be asked properly. An enclosed run — a stretch where an offset keeps the minority family with the majority on both sides — has a width, and there are two ways to quote it: how many rises it occupies, and how much rise it occupies.

The sixteen enclosed runs of the widest band occupy 81 rises at the coarse step and 162 at the fine one.

The null result a count of rises always gives #

Eighty-one to a hundred and sixty-two is a ratio of 2.000, and it means nothing whatever. A grid with twice as many rises in it counts twice as many rises in any fixed stretch, so the number would have come out at two whether the runs were intervals, points, or invented by the refinement.

That is worth dwelling on, because the ratio is exact to three decimal places and an exact number is the easiest kind to mistake for a measurement. Nothing about the band is in it.

The measurement that is not the null result #

The same sixteen runs occupy 0.161387 of log rise at the coarse step and 0.161752 at the fine one, a ratio of 1.0023.

That is the result. A stretch of rise that holds still while the number of rises across it doubles is a stretch the grid is reading; a stretch that halves is one the grid is imposing. Eleven of the sixteen do the first.

The widest island and the longest run #

Two extremes make the holding concrete. The widest enclosed run under the collection’s rule is 3 coarse rises and 5.540 parts per thousand; at half the step it is 6 rises and 5.538. Nothing about it moved except the number of cells it is drawn in.

The band’s longest run of a single family is offset 7’s stretch of the 4 family: 48 rises and 100.645 parts per thousand, then 96 rises and 100.231. Across a stretch two orders of magnitude wider than the narrowest island the agreement is the same, which is what says the 1.0023 is a property of the reading rather than an accident of small numbers.

Eleven measured, five at the floor #

The other five are one rise wide at both steps, so their extent halves by construction: 1.726 to 0.856, 1.753 to 0.876, 2.651 to 1.325, 2.738 to 1.368 and 1.837 to 0.918 parts per thousand.

Five unresolved objects out of sixteen is also the reason the whole-band sweep’s caution about its own step was the right one to write. An object that is one cell across at every resolution it has been looked at is detected rather than measured, and these five are in the same position the new island is — visible, located, and without a width.

What the five cost the total #

Together they carry 10.7 of the 161.4 milli-units of log extent, which is 6.6 per cent. The eleven sized runs gain slightly more than the five lose, which is why the total moves up by two parts in a thousand rather than down.

So the headline ratio of 1.0023 is a sum over two populations behaving differently, and quoting it alone would hide the more interesting half. The mean island under the collection’s own rule goes from 1.538 rises to 2.533 and from 3.180 parts per thousand to 2.655 — which is a mean over a set whose membership changes, and that is the next problem.

A threshold that is a count #

The collection defines an island as an enclosed run under three rises wide. At the coarse step the band has thirteen of them.

At the fine step it has eleven, and not one run is lost. A run three coarse rises wide is a run of six fine ones occupying exactly the rise it always did, and it stops being called an island because the rule counts rises and the rises got smaller.

The stated direction is wrong #

A finer grid finds more, never fewer is the sentence this thread has been operating under, and it is a theorem for changes: refinement cannot lose one. For islands as this collection defines them it is false, and it is false in the plainest possible way — thirteen become eleven with nothing removed and nothing merged.

Read against a threshold that is a ratio in the rise, 6.012 parts per thousand, the same runs go the other way: thirteen to fifteen.

Which of the two counts is right #

Neither, in the sense that both are conventions. But only one of them is a convention about the band: three rises is a statement about the instrument, and a ratio in the rise is a statement about the object, and the whole argument for sweeping by ratio in the first place was that the object is what the sweep is for.

This is the same error the azimuth grid produced when a repeating block of three angles turned out to be three consecutive samples, arriving from the opposite direction. There the grid put structure into a reading; here it takes structure out of a tally, and in both cases the giveaway is that the number moves when only the instrument moves.

The fix, and why it is not made here #

One line: the island threshold should be a ratio in the rise rather than a count of rises. It is not changed here, because the rule is used by every band sweep this collection holds and changing it discards all of them.

That is a real cost and it is recorded rather than paid quietly. Every island count published before this measurement is a count of enclosed runs under three rises of a grid whose steps run 1.47 to 3.75 parts per thousand, and comparisons between two bands read at grids of different grain are comparisons of two thresholds.

The share is the band; the fragmentation is the grid #

The control that says the finer grid reads the same band is how often each offset wrecks: the worst drift is 0.85 percentage points on the widest band and 1.23 on the next, across eighteen offset-band pairs. That is a property of the band and it comes back.

How many separate stretches an offset wrecks over does not. The stretch counts go 36 to 42 on one band and 20 to 25 on the other, because a stretch is a feature and a grid holds the features it can resolve.

What that costs a comparison already made #

The fragmentation figures this thread has compared bands with — a stretch every so many wrecking rises, one offset broken into sixteen pieces — are therefore statements about the step as much as about the band. That is not an error in the essay that measured them: the counts are exactly what the sweep found, and the claim they support, that which offsets wreck is a function of the rise, rests on the shares rather than on the stretch counts.

The distinction is the point. A wrecking share survives a change of grid and a stretch count does not, so the first can be compared between two bands and the second can only be compared between two readings of one band at the same grain — and the two bands’ grains differ by that factor of 2.6.

What a quarter of the step would find #

Two laws fit the two resolutions this band has been cut at. As a power law the count grows as the grid to the 0.1444, which gives 23.2 changes at a quarter of the coarse step; as a constant two changes per halving it gives 23.

They agree at 23 and separate only far out — 28.4 against 27 at a sixteenth, 42.3 against 35 at a two hundred and fifty-sixth. A speckle with no scale at all would give 76 at a quarter step and pure point events would give 19, so both fits sit much nearer the second than the first.

Two laws, two points #

The weakness is arithmetic rather than judgement: two resolutions are two numbers, both laws are fitted to the same two numbers, and their agreement at a quarter step is a consequence of that rather than evidence for either.

The increment of two rests on a single newly visible island, and five runs are still one rise wide, so neither law is being fitted to a finished count. A third band that gained two would be worth more than a third resolution of this one.

The refusal on the second band #

Asking the same extrapolation of the band that gained nothing is refused rather than answered. Its fitted exponent is zero, because two resolutions reporting the same count fit no law, and an exponent of zero is the absence of a fit rather than a finding about how a speckle scales.

The refusal was found by drawing the picture anyway: it came out as a flat rule with a 23 per cent empty band above it, which is a drawing with nothing in it. A prediction that renders as an empty box is a prediction nobody should have been offered, and declining is cheaper than explaining.

What is not established #

That the speckle has a floor. At half the step it looks like itself with two more events in it, its islands at the same widths in the rise, and five of them still unresolved.

That the second band’s zero generalises. It is one band at one halving, and it refused the assertion written to make it agree with the first.

And that the fine grid is fine enough. Its own steps run 0.734 to 1.876 parts per thousand, which is the same irregularity one level down, and a feature narrower than 0.734 is invisible to it exactly as the island at 0.86 was invisible to the coarse grid.

Where the rounding should be a setting #

The rounding is not wrong; it is unstated. A sweep needs some precision and five decimal places is a reasonable default, and the defect is that the number is fixed in the routine that lays a band out rather than being a setting a sweep can ask for — so that asking for a finer step silently asks for something the grid cannot deliver.

Made an option defaulting to five places, no existing sweep moves and a finer one becomes possible to request honestly. That is the same correction a sampled azimuth grid needed and for the same reason: a constant nobody can vary is a constant nobody can test, and this thread has now had two of them turn out to be carrying results.

What a reader should carry #

That the last unmoved setting in this thread was not the number it was called. A nominal step of two parts in a thousand, a grain of 1.653, four steps in five taken at the grain, and a spread of two and a half within one band — none of which is visible in any output the sweep produces, because a grid never reports what falls between its rises.

And that a width quoted in rises is a width quoted in the grid. Eighty-one rises became a hundred and sixty-two and the same stretch of rise stayed where it was, which is the whole distinction between an object and a reading of it.

The one line #

The band’s rises are rounded to five decimal places, so its nominal step of 2.00 parts per thousand sits against a grain of 1.65 and 99 of its 125 steps are a single unit of that grain — and measured on that instrument, sixteen islands hold 1.0023 of their extent while doubling their count of rises, eleven of them having a width and five still at the floor, while the collection’s own three-rise threshold counts thirteen islands at the coarse step and eleven at the fine one with nothing lost.