The window is the neighbour

The area under a dip is a well-defined number once somebody says where to stop integrating. This essay is about where that is, and the answer turns out to have a name already: it is the distance to the nearest other rational — the crowding — which is the quantity this thread has spent its whole length trying to measure the effect of.

Two conditions, pulling opposite ways #

The window has to satisfy two requirements and they are not compatible.

It has to scale with the dip. A head of 1,224 organs has a dip a quarter the width of a head of 612’s, so a window fixed in degrees contains the whole of one dip and a sixteenth of the other. Measuring both at the same angular limit reports mostly-dip for one head and mostly-shoulder for the other, and the comparison between head sizes — the one thing the width law is about — becomes a comparison of two different measurements.

Measured: with a window fixed at a tenth of the way to the neighbour — a natural choice, and fixed in degrees — the equivalent widths of the four fractions of denominator 21 disagree between two head sizes by factors of 2.0, 2.3, 2.6 and 2.1, and at wider settings the larger head returns negative widths while the smaller one is still positive.

It has to stay clear of the neighbour. Every rational has a dip; the nearest other fraction to 23/55 is 18/43, a seventh of a degree away. Integrate far enough and the profile being subtracted from is somebody else’s shoulder, the deficit turns negative, and the integral eats the dip it just measured.

The first condition wants a limit proportional to q/n². The second wants a limit proportional to the crowding, which is a property of arithmetic and does not know how many organs the head has. A single window cannot satisfy both across a range of head sizes, and the whole of what follows is what happens when one of them is chosen.

What a fixed-degree window does, measured #

The abstract statement above — that a window fixed in degrees is not comparable between head sizes — is worth putting numbers on, because the size of the failure decides whether it is a nuisance or a wall.

Take the window at a fixed fraction of the distance to each fraction’s nearest neighbour, which is a defensible choice: it is the same relative position in every neighbourhood, and it is well clear of the neighbour’s own dip. Then measure each fraction’s equivalent width at two head sizes.

A fifth of the way to the neighbour is already a wall. At that setting every one of the eight fractions of denominators 21 and 34 gives a positive equivalent width at the smaller head and a negative one at the larger: 437 against −202, 383 against −75, 318 against −578, and so on through the table. The two heads are not disagreeing about a number there; they are disagreeing about whether there is a dip.

A tenth of the way is unreliable rather than impossible. Three of the four fractions of denominator 21 agree between heads to within 22%, and the fourth disagrees by 50%. In the denominator-34 family two agree within 20% and two disagree by factors of 4.1 and 9.4.

The reason is the same in both cases. At the larger head the dip is 2.4 times narrower in degrees, so a window fixed in degrees puts most of its length on the shoulder — where the profile is crossing its own background and the deficit is small and signed — and the integral is dominated by the part of the range that has nothing to do with the dip.

That is what makes the scaled axis compulsory rather than elegant. Without it the head-size comparison, which is the whole of the 1/n² law, cannot be made with an area at all.

Choosing the scaled window, which is the right choice #

The scaled axis is the one that makes the measurement reproducible, so it is what the area is measured on: u = δ·n²/q, with the window stated in those units.

It works. Across the eight fractions of denominators 34 and 55, at a window of 200 scaled units, the two head sizes agree to between 0.3% and 8%.

But now look at what a common window means in each fraction’s own neighbourhood. At 200 scaled units, on a head of 791 organs, the window reaches:

The most crowded fraction’s window reaches 1.6 times further into its clear space than the loneliest one’s. The window is common in units of the dip and uncommon in units of the neighbourhood — and it is the neighbourhood that decides how much of the neighbour’s shoulder has been integrated.

The mismatch has one source and it is not the head size #

The two conditions are incompatible in general, and it is worth working out where the incompatibility actually bites in this table, because it is narrower than the general statement.

Put the neighbour on the scaled axis. If the nearest other rational is g degrees away, its position in scaled units is g·n²/q. For 13/34 on 791 organs that is 0.1925 × 791² ÷ 34, which is about 3,500 — and a window of 200 reaching 5.6 per cent of the way there is exactly 200 ÷ 3,500. So the dip’s half-width sits near 150 scaled units and its neighbour near 3,500: a factor of twenty-three of clear space, in the units the dip is measured in.

Now vary things one at a time. Across head sizes at fixed fraction, g is fixed and the neighbour’s scaled position moves as n² — so a common scaled window is not a common fraction of the neighbourhood between two heads either, and the larger head’s window reaches proportionally less far. Across fractions of one denominator, n and q are both held, so the scaled neighbour position moves only with g, which is what makes the reach run from 3.8 to 6.1 per cent in the table above.

That last one is the whole of the within-family bias, and it has a single cause: the nearest neighbour’s own denominator. 11/34’s neighbour is far because the simplest fraction near it happens to have a large denominator; 15/34’s is close because the simplest fraction near it does not. Nothing about the head, the denominator or the dip is involved.

Which suggests the repair the thread has not made #

If the contamination is the neighbour’s shoulder, and the neighbour’s position and denominator are both known, then the shoulder can be modelled and subtracted rather than avoided.

Every dip on this ladder has the same shape once it is put on the scaled axis — that is what the whole scaling argument establishes — so a neighbour at a known distance with a known denominator has a computable profile, of computable depth, which can be evaluated at each offset of the window and removed before the integral is taken. What is left is this fraction’s own deficit against a flat background, over a window that can then be as wide as one likes.

That is not a small proposal and it is not a large one either. It needs the dip’s scaled shape, which the sweeps already contain, and it needs the depth law, which is measured. It would fail if the shape were not universal, and whether it is universal is checkable on the same data by overlaying four fractions’ profiles on the scaled axis and asking whether they lie on one another.

Which is worth saying because this essay otherwise ends in a refusal, and a refusal with a stated repair beside it is a different object from one without. Nothing here has run the subtraction; what is established is that the obstacle is a known quantity rather than an unknown one, and that is usually the difference between a method that stops and a method that continues.

Which is why the residual cannot be tested this way #

The residual claim was this: after the 1/n² law and the denominator’s coefficient are taken out, what is left is ordered by crowding, with the most crowded fraction giving the widest dip.

An area measured at a common scaled window integrates more of the neighbour’s shoulder for a crowded fraction than for a lonely one. The neighbour’s shoulder lowers the profile, which reduces the deficit, which makes a crowded fraction’s area come out smaller — the opposite direction to the claim, by construction, before any measurement.

So there is a systematic effect running the other way, of unknown size, built into the instrument. Which means neither an agreement with the claim nor a disagreement with it settles anything: an agreement would be the residual winning against the instrument, and a disagreement could be either the instrument or the absence of a residual.

Why this is not a complaint about precision #

A reader who has followed the thread this far might reasonably say: every measurement has a free parameter somewhere, and choosing one and stating it is what a method is. That is right, and it is worth being exact about why this case is different.

A well-posed integral has a range of limits over which the answer is stable. Choosing inside that range is bookkeeping: the number does not move, so the choice is not a measurement. That is what makes an integrated line strength in spectroscopy, or a total under a peak in a diffraction pattern, a quantity rather than a convention — the peak returns to a background, and any limit past the return gives the same total.

Here there is no such range. The equivalent width at windows of 100, 200 and 400 scaled units is 152, 325 and 445 for one fraction, and every one of those windows satisfies both conditions. The answer is roughly proportional to the limit over the whole safe range, which means the limit is not a tolerance — it is the measurement’s dominant term.

The distinction has a practical form. If somebody publishes a dip width for a rational divergence, the question to ask is not “at what precision” but “out to what offset”, and the answer will be doing most of the work. This site has published widths, and this essay and the one before it are the correction: the numbers were level crossings, they were compared with each other consistently, and none of them means anything on its own.

The measurement that would settle it, and what it costs #

There is a version of this that would work, and it is worth setting out because it says why nobody has run it.

Hold the window at a fixed fraction of each fraction’s own crowding — so that every dip is integrated over the same share of its own clear space — and then restore comparability across head sizes by measuring each fraction at the head size that makes its dip the same size relative to that window. That means a different head size for every fraction, chosen so that q/n² times the window equals a fixed share of the crowding.

The heads that come out are not free. For the four fractions of denominator 34, matching the loneliest member’s ratio would require the most crowded one to be measured at a head about 25% larger, and cell areas on a head that size cost what they cost. The whole table is six denominators times four fractions times a head size that is now a function of the fraction rather than of the denominator — and every one of them needs its own background band re-chosen, because the band is defined relative to the neighbours.

That is not an unreasonable experiment. It is a bigger one than the repair was supposed to be, and its result would be a residual measured against a control that is itself a function of the quantity being tested — which is the shape that has gone wrong twice already in this thread.

The same difficulty, one level up #

It is worth noticing that this is the third time this thread has met the same shape, because that is what says it is a property of the object rather than a run of bad luck.

The first was a background taken as a single sample. At 21/55 the sample sat seven thousandths of a degree from 13/34 — inside the neighbour’s dip — and reported a background below the floor, which inverts the verdict and makes a dip look like a bump. The repair was to take a band and choose the offsets in it that are clear of every other rational.

The second was the width read as a level crossing. The level lands on a tread of a staircase and the number reported is the tread’s edge; two fractions of twelve had to be refused.

The third is this one: a window whose outer bound is the distance to a neighbour.

Each repair fixed the previous instrument and inherited the same difficulty in a new place, and the difficulty is that the object has no clear space. A sample can land in a neighbour, a level can land on a neighbour’s shoulder, and a window can reach a neighbour. The dips at the rationals are not features on a ground; they are the ground.

What the thread is left with #

Three statements, in decreasing order of confidence.

The dip’s depth and scale are measurable and stand. The second moment at a rational, and the way the whole profile shrinks as q/n², are properties of the head and not of any window.

The dip’s width is not a quantity. Whatever number is reported for it is a joint statement about the profile and about where the measurement stopped, and the two cannot be separated because the profile has no outer edge.

The residual ordering is not testable with the instruments here. Not “is false” — untestable, which is a weaker and more useful thing to say. The level crossing has a resolution set by the treads of a staircase; the area has a limit set by the crowding. Both instruments’ free parameters are entangled with the quantity the residual is about.

What this does not say #

It does not say crowding has no effect. It plainly does have one — it is why a background sample can land inside a neighbouring dip and invert a verdict, which is a defect this thread found and repaired. What is untestable is the residual claim: crowding as an ordering of dip widths after the main laws are removed.

It does not say the area was a wasted repair. It measures two fractions the level crossing could not, it is reproducible where the level crossing was a lottery, and it makes the shape of the difficulty visible. A knob one can turn is better than a knob nobody knew was there.

And it does not say the dips are shallow or unimportant. A rational divergence really does make a tidier head, and the tidiness is large: at 5/13 the second moment at the rational is under an eighth of its level nearby. Nothing in this essay is about the size of the effect. It is about the shape of the thing whose size is being asked for.

The check #

The window’s outer condition is asserted rather than described, in two parts.

A window wider than the profile that was sampled is refused outright, so an extrapolated area cannot be reported by accident. And the window every comparison here uses is required to sit inside a third of the way to the nearest other rational — checked against the crowding computed from the arithmetic, for the most crowded fraction in the set, so that the constraint is tested where it binds.

The reproducibility claim is asserted too: at the stated window the equivalent width must agree between two head sizes to within a fifth, over a whole family. That one exists to keep the essay honest in the other direction — the area’s failure is specific, and it is not a failure to measure anything.

The window can be made common after all #

This essay’s difficulty is that a window has two conditions on it that pull opposite ways, and that the limit deciding it is the crowding. Both hold. What they do not prevent is a comparison in which the crowding is the same for every object being compared.

On such a set the window is simultaneously the same in scaled units, the same in degrees, and the same fraction of the way to the neighbour — 5.1% to 5.2% across seven fractions, against 3.8% to 6.1% for an unmatched comparison at the same nominal window.