A dip with no outer edge

Build a head at a divergence of exactly p/q and its organs fall into q rows. Rows are tidy: the cells come out more nearly equal in area than at a nearby irrational angle, so the second moment of the cell areas — the statistic this part of the site measures disorder with — has a dip at every rational.

How wide that dip is has been doing a great deal of work. It goes as the inverse square of the head size, its coefficient carries the denominator rather than the quality of the approximation, and what is left over after both of those has been read as an effect of how crowded the fraction’s neighbourhood is.

All of it rests on a half-width read off a curve, and the curve is a staircase.

What the profile actually looks like #

It is worth walking out from a rational once, slowly, because the shape is not what “a dip” suggests and every difficulty below follows from it.

Take 5/13 on a head of three hundred organs. At the rational itself the second moment is 0.118 of its background. It stays at 0.118 for the first three thousandths of a degree — not approximately, exactly, because within that range the tessellation has not changed at all and neither has any cell area. Then it steps to 0.169, and holds. Then 0.220, and holds for four samples. Then 0.222, 0.223, 0.225 — three treads within a hundredth of each other — and then 0.362, 0.480, 0.517, 0.654, 0.697, 0.864, and it is still climbing at 0.913 when the sampling stops.

Two things about that sequence matter.

The first is that it is flat in places and steep in places, which is what makes a half-width a lottery. A level at half the background lands between 0.480 and 0.517 here — inside a riser, so this fraction is measurable — but a level a tenth lower lands on the tread at 0.220 and reports the tread’s edge.

The second is that the profile has not returned to its background by the end of the range. At the widest offset sampled it is at 91% of it and climbing, and the background band itself starts twice as far out again. There is no place where the curve arrives and stays.

Why a level crossing was the wrong instrument #

A half-width is measured by walking out from the floor of a dip until the curve has climbed half way to its background, and interpolating. On a smooth curve that is exact. On a staircase it reports the edge of whichever tread the level lands on, which is a fact about the treads and not about the dip.

Two of the twelve fractions had to be refused rather than measured, on a test that compares the same quantity at two head sizes: 9/34 gives a scaled width of 2,665 at one head and 6,810 at another, and 24/55 does the same. Ten agree within a few per cent and two disagree by factors of two and a half.

Refusing them was the right thing to do — a method that returned a number for a fraction it cannot measure would be reporting the position of a combinatorial flip as a width. But a method that refuses a sixth of its cases is a method with a resolution, and the repair suggests itself: integrate the profile rather than finding a crossing on it. An area has no level in it. It cannot land on a tread.

The quantity, and the axis it has to be measured on #

The area of what, exactly, needs stating before anything is measured.

The deficit at an offset δ from the rational is the background minus the second moment there. Integrating that over δ gives an area whose units are degrees times a second moment, which is not comparable between fractions of different depths — so it is divided by the dip’s own depth, giving an equivalent width: the width of a rectangular dip, of the same depth, holding the same missing disorder.

The offsets themselves cannot be measured in degrees either, because the dip shrinks as the head grows. A head twice as large has a dip a quarter as wide, so sampling both at the same angles measures the floor of one and the shoulder of the other. The axis used here is therefore scaled: u = δ·n²/q, in which the dip is the same size at every head and every denominator, near enough to compare.

It fixes what it was meant to fix #

On that axis the equivalent width reproduces, and it reproduces well.

At a window of 200 scaled units, the four fractions of denominator 34 give 324.6, 318.4, 343.2 and 357.7 at a head of 791 organs, and 349.8, 337.7, 353.1 and 358.6 at a head of 1,224. The largest disagreement between the two heads is 8% and the smallest is under 1% — against a level-crossing method whose worst case in the same family was 160%.

And the two fractions the level method refused come back with numbers. 9/34 gives 318.4 and 337.7, a disagreement of 6%; 24/55 gives 348.0 and 361.8, a disagreement of 4%. Both are as reproducible as anybody else in their family.

That is the repair working. The staircase’s resolution is gone, because an integral over a staircase is a perfectly well-behaved thing: the treads contribute their exact areas and nothing depends on where a level lands.

And it never settles #

Then comes the question every integral has to answer, which is where to stop.

For 13/34 the equivalent width is 100 scaled units at a window of 50, then 152, then 325, then 445, and then 242 at a window of 800. Not a plateau anywhere: the largest step between consecutive windows is a factor of 2.1 and the smallest is 45%.

The turnover is not the window running out of dip. It is the window running into the neighbour. At a window of 800 scaled units the integral has reached 23% of the way to 21/55, whose own dip pulls the profile below the background that was measured near this rational — so the deficit goes negative, the integral starts subtracting, and a wide enough window returns a dip of negative width.

Where a window can be put, and it is not many places #

The integral has two conditions on its limit and they pull in opposite directions.

It has to be wide enough to contain the dip, or the number is a statement about the floor. At a window of 50 scaled units every fraction of denominator 34 gives about 88, which is essentially the window itself: the deficit is still near its full depth out there, so the integral is depth times width and the width is the limit.

It has to be narrow enough to stay away from the neighbour. At a window of 800 the integral has reached 23% of the way to the next rational and has already turned over.

Between those two there is no stretch where the answer holds still. The windows of 100, 200 and 400 give 152, 325 and 445 — each one is inside both conditions, and each one gives a different answer.

That is the difference between this and an ordinary choice of numerical parameter. A well-posed integral has a range of limits over which the answer is stable, and choosing inside that range is bookkeeping. Here the answer is proportional to the limit over the whole safe range, so the choice is the measurement.

Which means the dip has no width #

That is the finding, and it is worth putting flatly. The disorder dip at a rational has:

• a depth, which is well defined and measurable — the second moment at the rational against the level nearby;
• a scale, which is well defined and is the whole content of the 1/n² law — the dip shrinks as q/n²;
• and no outer edge, so no width.

A quantity with no outer edge does not have an integral either. What an integral of it reports is a joint statement about the profile and the limit, and here the limit contributes more than the profile: over the sixteenfold range of windows measured, the number moves by a factor of four and a half.

The half-width was never a width either, and this is the clearer way to see why. It was a level crossing on a curve that goes on climbing: choose a different level and get a different number, with no level at which the answer stops moving. The area makes the same problem visible because the window is a knob one can turn, and the level was a knob nobody turned.

An objection, and what it costs to meet it #

The obvious objection is that this is a failure of the background rather than of the dip. If the background were modelled — a smooth curve fitted through the shoulders, say, and subtracted — then the deficit would go to zero properly and the integral would converge.

It would, and the number would then be a statement about the fitted background. That is not an improvement; it is the same free parameter wearing a different hat, and a worse hat, because a window is a number one can print and a fitted background is a decision buried in a procedure. The shoulders being integrated over belong to other rationals, and a smooth curve through them is a claim that they are noise.

There is a stronger version of the objection: use a local background, measured just outside the window, and let the window and the background move together. That is exactly what makes the answer move with the window, since a background taken further out is measured on a lower part of somebody else’s shoulder.

The honest end of it is that the second moment against the angle is not a curve with features on a flat ground. It is a hierarchy: dips at the rationals of every denominator, each one sitting on the shoulders of the shallower ones around it, with no scale at which the picture becomes simple. Measuring “the width of one dip” presupposes a background that the object does not have.

What survives #

Three things, and they are the things the thread was actually about.

The 1/n² scaling survives, because it is about the shape of the dip rather than its extent. On the scaled axis the profiles at two head sizes lie on top of each other, which is a stronger statement of the same law than a pair of widths agreeing.

The denominator result survives, and is now testable on more fractions. The coefficient belongs to q, and the twelve fractions of three denominators can now be twelve rather than ten.

The refusals survive as well, in a better form. The level-crossing method refused two fractions because it could not measure them; the area measures them and reports that every fraction’s number depends on the window. The first refusal was about two fractions and the second is about the quantity.

What this does not say #

It does not say the dips are not real. They are large, they are at exactly the rationals, and they deepen with the head size in the way rows should. Nothing here touches that.

It does not say the earlier numbers were wrong. The level-crossing widths were measured consistently at one level and compared with each other, and comparisons of that kind are legitimate as long as nobody promises the number means something on its own. What has changed is that the promise has been withdrawn.

And it does not say the ordering results are wrong. It says they need re-testing on a quantity that does not move with the window, which is the subject of a separate essay and does not come out well.

The check #

The area is asserted to do the thing it was built for and to fail at the thing it was hoped to do, and both are checks that can go red.

The first requires the equivalent width at a stated window to agree between two head sizes to within a fifth, across a whole family — which would fail if the scaled axis were the wrong axis, or if the background were being taken somewhere it should not be.

The second requires the two fractions the level method refused to come back with agreeing numbers, so the repair cannot quietly stop working.

The third requires that no two windows in the sweep agree — that the width moves by more than 15% at every step — and that the largest value is at a window inside the range rather than at its edge. Together those say there is no plateau and no asymptote, which is exactly the claim, and a fraction that did settle would stop the build and take this essay with it.