The window nobody varied

Every counted pair in this collection’s settling work was read over the top two hundred organs of a twelve-hundred-organ stem. That is one number, it was never chosen against an alternative, and every arrangement and every additive sequence the table reports stands on it.

It has now been varied. Seventeen windows from twenty organs to eight hundred, a factor of forty, on all 384 settled runs at forty starting angles. Not one run changes its answer above twenty-five organs. Not one of the nineteen destinations is window-dependent at any of the three samplings. The census of destinations, arrangements and additive sequences is the same read at twenty-five organs and read at eight hundred.

What the counter reads through #

The count is taken by cylCount, which is handed the positions of the organs in a band of the stem and returns the two contact families that cross. The band is the top two hundred organs, and the wrapper that fixes it at two hundred is the one every count on the settling table goes through.

Two hundred is a default. It is not a measurement, nothing was compared against it, and it was inherited by eight further libraries that take a count without ever restating it. A setting in that position is worth the twelve minutes it costs to move.

What made the question sharp #

One destination in the table gave a reason to move it. At 79.2° the counter returns three pairs — 4/5, 4/9 and 5/9 — and 4/9 skips a term of the sequence the other two sit on.

A pair that skips a term of its own sequence is exactly what a count taken at the wrong place produces, and the collection said so at the time and could not test it. On a disc a count moves with the radius it is taken at, reliably and by a whole rung, so the same suspicion on a stem was neither idle nor cheap to dismiss.

The published grouping of that table into ten additive sequences rested on the answer. If 4/9 were a window artefact the table had nine sequences, and no reading of the runs already grown could say which.

What was swept #

Seventeen windows on every settled run at every starting angle the table has: eight rises from 0.030 to 0.003, four falloff exponents, forty starting angles, 1,200 organs a stem. Twelve minutes on a quiet machine, of which about three per cent is spent counting rather than growing — a single count of a single window is three milliseconds.

The widest window is eight hundred organs and the exploratory scan behind it goes to eleven hundred, at thirty-nine windows rather than seventeen. Both are reported below, because a sweep that finds nothing is only as good as the range it looked over.

What an honest window is #

A window that reaches below the organ where a run settled is reading the transient rather than the lattice, so it is dropped rather than counted. The largest honest window anywhere in the table is 1,199 organs and the smallest is 391, and every one of the 384 runs carries at least 13 of the seventeen.

That rule is what makes the wide end of the sweep readable at all. Without it the eight- hundred-organ column would be a mixture of settled patches and unsettled ones, and a difference there would say nothing about the window.

The answer, in one row #

Runs whose pair changes with the window, above the floor their own pair sets: 0 of 384. Destinations returning a window-dependent pair: 0 of 19 at forty starting angles, 0 of 18 at twenty, 0 of 15 at nine. Runs whose answer at the narrowest window that can hold their own pair differs from their answer at two hundred organs: 0 of 384.

Over the wider scan the same statement is stronger. Across thirty-nine windows from thirty to eleven hundred organs, every one of the 384 runs returns exactly one distinct pair.

The destination the question came from #

At 79.2° the runs sort into eleven rows by the rise and the falloff exponent they were grown at, and none of the eleven moves at any window.

The three pairs are three stems. 4/5 comes from the two coarsest rises; 5/9 comes from a rise of 0.013 at exponents 3, 4 and 5; 4/9 comes from the same rise at the shallowest exponent and from nowhere else. Which pair a run returns is decided by the rise and the falloff it was grown at, which is what a destination has always been said to be.

So 4/9 is not one stem read at three places. Whatever is wrong with it, the place it was read is not it.

The other destination that could have moved #

If a window were going to decide anything it would decide it where the counted numbers are largest, because a narrow window runs out of room for a large family first. The largest counted number anywhere in the table is 19, and one of the destinations carrying it is 132.3°.

Eight rows, twelve runs, two pairs, and the only column that differs is the twenty-organ one — which is a window that cannot hold 19 at all. From twenty-five organs to eight hundred the eight rows are flat.

What the census does #

The three numbers this collection actually quotes from the table are how many destinations it holds, how many distinct arrangements those destinations show, and how many additive sequences the arrangements sit on. All three are the same at every window that can hold what it is being asked to count.

At forty starting angles the table holds 19 destinations, 31 arrangements and 15 additive sequences, and five of the destinations sit on a ladder sequence. Every one of those numbers is unchanged from twenty-five organs to eight hundred, and unchanged again when each run is read at the narrowest window that can hold its own pair rather than at two hundred.

Which windows do not give it, and why #

Twelve of the seventeen return the published census outright. The other five divide into two kinds and neither is a counter-example.

The twenty-organ column is below the floor for 37 of the 384 runs, and it returns a census that is genuinely different — 26 arrangements against 31, and 14 sequences against 15. That is a bound on the instrument rather than a property of the window, and it is the subject of the essay after this one.

The four widest columns lose runs rather than changing them. At four hundred organs and above a handful of runs have their settling organ inside the window, so those runs are dropped and the census is taken over fewer of them. Nothing there returns a different pair.

What the one differing column reports #

The twenty-organ column is worth reading rather than dismissing, because it is the only place in the sweep where the census moves at all.

At nine starting angles it reports 18 arrangements against 20 and the same ten sequences; at twenty, 22 against 24; at forty, 26 arrangements against 31 and 14 sequences against 15. Every one of those readings is a legible pair. Nothing refuses, nothing widens an error bar, and no number in that column looks out of place beside the others.

So the census is robust to the window over more than a factor of thirty and wrong outside it, with no signal anywhere marking the boundary. That is the reason the essay after this one is about the bound rather than about the setting.

A shrinking set is not a change #

The distinction in that last paragraph is not a nicety, and the first version of the drawing did not make it.

A destination was called window-dependent when the set of pairs its runs returned differed from the set at two hundred organs. At eight hundred organs a few runs reach below their own settling organ and are dropped, so the set shrinks — and a comparison of sets reads a smaller set as a different one. It flagged 157.7° at nine starting angles, where nothing moved and one run was simply not counted.

A change is now a pair the standing window never returns, or a set that differs in size with nothing dropped. A window read in part carries its own colour instead. The defect and the finding have the same shape: an instrument reporting a difference that belongs to how much it was given rather than to what it was looking at.

Three samplings, read separately #

The nine starting angles the table was grown from are a biased list — they settle half as often again as angles placed between them — so a result read only at nine would carry that bias into a claim about an instrument.

All three samplings were read, and they disagree about the census exactly as they should: 15, 18 and 19 destinations, 20, 24 and 31 arrangements, 10, 13 and 15 sequences, at nine, twenty and forty starting angles. What none of them does is disagree with itself across the windows. The sampling moves every number in the census and the window moves none.

Why two windows rather than one #

The design choice worth naming is that the re-reading never takes a pair from a single width, even though the sweep shows a single width would have done.

That is deliberate and it is not redundancy. A count agreed by two windows is a count that has survived a change in the thing being tested, and the whole point of the exercise is that nobody knew in advance whether the width mattered. Reading twice costs three milliseconds and turns an assumption into a measurement — the same argument that says a third counted family is a free check rather than a better number.

What the null is a null about #

Narrowly: on this table, at these rises and exponents, with these stems, the counting window decides nothing between twenty-five organs and eleven hundred.

Not: that a counting window is harmless. The reading window in the ablation work was varied and it decided three rows, and that is the same class of setting doing the opposite thing. The difference is measurable and it is arithmetic rather than luck.

Why the table is inert #

Because its largest counted number is small. A window of w organs can hold a family of at most w − 6, so two hundred organs allows 194, and the largest number the table ever returns is 19. The counter also refuses a band under twenty nodes, which puts the floor for almost every run in the table at twenty organs.

The standing window therefore sits between eight and ten times above the narrowest window that would do, with nothing in between for it to trip over. That is a plateau, and a setting anywhere on a plateau is a setting that cannot matter.

The reading has its own refusals #

A sweep that answers whatever it is asked is a sweep that cannot report a problem, so the re-reading refuses four things and answers a fifth.

It refuses a pair claimed from one window, because a single reading has nothing to agree with. It refuses a window narrower than the floor for the family it claims. It refuses two windows that disagree. And it refuses two families that wind the same way, which is the condition that turned out to matter. What it answers is a pair two wide-enough windows agree on whose families cross, and it returns the sequence that pair sits on with it.

Ten assertions stand behind that, the refusal last. A refusal rule that never fires on anything is decoration, and this one fires on five runs of 384.

What would have refuted it #

One run returning two different pairs at two windows both wide enough to hold either of them. That is a single row of a single figure and it does not occur anywhere in 384 runs across seventeen windows, or in the same runs across thirty-nine.

A weaker refutation was also available and also absent: a destination whose set of pairs gains a member at some window. That would not have shown a run changing its mind, but it would have shown the grouping into arrangements depending on the reading, which is the thing the published census is made of.

What it cost #

Twelve minutes to grow and count, six tenths of a second to read back afterwards, and about three per cent of the total spent on the counting rather than on the stems. The counting is free; the stems are the expense, and they were already grown.

That ratio is the argument for varying a setting as a matter of course rather than when something looks wrong. A sweep that is three per cent of a table’s cost and that could have overturned a published grouping is not a luxury.

The suspicion was worth having #

It is easy to read a null as evidence that the question should not have been asked. The opposite holds here, and the reason is that the alternative was live.

Ten additive sequences or nine was not a matter of taste. One of the ten rested on a single reading, the arithmetic of the limit angles had already argued for nine, and no measurement had been able to separate an artefact from a rung. A test that could have gone either way went one way, and the reason a null is worth more here than a found anomaly is that a found anomaly would have needed the same sweep to be believed.

What actually produced 4/9 #

The counter computes a flag saying whether the two families it picked wind opposite ways, and the wrapper that every count on this site goes through takes the pair and drops the flag. Five of the 384 runs return two families of one handedness, which on a cylinder is what two counted numbers two terms apart in a sequence look like.

Three of them are the 4/9 readings. That is a different essay’s result, and what belongs here is only its shape: the account being tested was refuted and the conclusion it was reaching for was confirmed, by a quantity the instrument had computed all along.

Which is the useful kind of null #

A null that leaves a phenomenon unexplained is a dead end. This one closes a route and hands the question to a quantity already on the table, so the anomaly is no less real and the collection knows one more thing about where it is not.

That distinction is worth holding onto when reading any negative result about an instrument. The question is this an artefact of the setting has a good answer here only because the same runs could be re-read at seventeen settings for three per cent of what they cost to grow.

Why a fixed window survived this long #

Nothing complained. Every count it produced was a legible pair, every pair sat on a sequence, and the arrangements it grouped into were arrangements a reader could believe.

An instrument setting is only visible when something downstream refuses, and this one never produced anything a refusal could catch. That is the general shape: a default survives because it is inside the range where it does not matter, and its being inside that range is exactly what nothing on the page can show. The count itself has been priced this carefully; the width it was taken over had not been priced at all.

Where the setting reaches #

Eight libraries in this collection take a count through the same wrapper at the same default, and every one of them discards the same flag. The null measured here is a null about the settling table and it does not transfer: a library counting larger families or shorter stems is a library whose floor sits somewhere else.

What does transfer is the rate. Five same-handed readings in 384 is the rate to expect in each of them until each is checked, and none has been.

What a reader should carry #

That the two-hundred-organ window is right, that it is right by luck rather than by design, and that the difference is not academic. An instrument that can refuse is worth more than one that cannot, and the window is a setting that cannot refuse: it returns a pair at every width, and the pair it returns below the floor is a perfectly legible one.

The one line #

Seventeen windows, a factor of forty, 384 settled runs: not one changes its pair, no destination is window-dependent at any sampling, and the census read at twenty-five organs is the census read at eight hundred.