The background is not one sample

The earlier work found that the second moment of the side-count distribution has a narrow deep dip wherever the divergence angle is rational, and that the dip’s half-width falls as the square of the head size. It measured three denominators — 8, 13 and 21 — and closed with a question: does the same law hold at 55 or 89, where the dip is much narrower?

Chasing it there turned up a defect in the measurement first, and the defect is worth more than the answer.

What a background is a claim about #

A dip is not a quantity. It is a comparison — this value, against what the curve does nearby — and “nearby” is doing all the work. Every number this thread reports about a dip is a difference between two measurements, one of which was never specified beyond a word.

For a small denominator the choice is unimportant. 3/8 is at 135°, the nearest other rational with a denominator of sixty or less is 22/59 at 0.763° away, and anywhere in between is flat. Take the background from any offset in a wide band and the answer is the same.

For a large denominator it is not flat, because the rationals crowd. The gap between neighbouring fractions of denominator at most Q near a point falls as 1/Q², so the distance to the nearest other rational with a denominator of sixty or less falls steadily: 0.763° at 3/8, 0.462° at 5/13, 0.312° at 8/21, 0.193° at 13/34, 0.193° at 21/55 and 0.074° at 34/89.

At 21/55 the fixed offset of two tenths of a degree is not in a flat stretch. It is 0.0074° from 13/34, which is inside 13/34’s dip at any head size that resolves it. So the background sample is a measurement of a different dip’s floor, it comes back below the floor it is meant to be a background for, and the verdict flips.

Nothing caught this. The inverted dips were never asserted about — no gate asks whether a dip is a dip — and the number that came out was a plausible number. It is the fourth defect on this site whose symptom was a value that looked reasonable rather than an error, and the fourth whose real symptom was absence: a dip that was there and was reported as not there.

Why an inversion is worse than a wrong number #

It is worth being precise about what the contaminated background did, because “the number was wrong” understates it in a way that matters for what a gate would have to check.

A background taken too high makes a dip look shallow. A background taken too low makes it look deep. Either is a wrong number and either would be caught by anybody comparing two head sizes, because the error would move with the sample rather than with the physics.

What happened at 21/55 is neither. The background came back below the floor, so the comparison did not merely mis-size the dip — it reversed its sign, and reported the rational as more disordered than its surroundings. That is a statement in the opposite direction from the one the whole thread rests on, arrived at by a method that is right at every smaller denominator, on a curve where the correct answer is plainly visible in the figure.

Which is the reason a gate is possible here at all. A wrong magnitude has no signature; a wrong sign does, and it is checkable in one line: a dip’s floor must be below its background. That assertion costs nothing, it would have failed on the day the 21/55 measurement was made, and nothing in the suite had it because nobody had thought a dip could come out upside down.

The general form is worth carrying, since this collection keeps meeting it. A comparison has an orientation as well as a value, and the orientation is the cheap thing to assert. Most of the checks here test that a number is in a range; testing that a difference has the sign its own definition requires is a different and usually simpler test, and it catches the failures where a plausible number is produced by a broken measurement.

The repair that was not enough #

The obvious fix is to take several offsets and use the median, which is robust to one contaminated sample. It was tried and it is only half a repair, and the half that failed is instructive.

At 13/34 two of six fixed offsets land on other rationals — 21/55 at 0.193° and 23/60 at 0.353° — and with two of six contaminated the median moves. The signature was visible in the sweep across head sizes: 13/34’s background came out rising with head size, 0.234 then 0.244 then 0.252, where every other denominator’s falls. A background that rises as the head grows is not a background; it is a neighbour’s dip getting deeper.

So the offsets are chosen rather than fixed. Lay down a candidate every hundredth of a degree from 0.10° to 0.45° on both sides of the rational, and keep the ones at least 0.03° from every fraction with a denominator of sixty or less. That leaves 72 clear offsets at 3/8 and 53 at 34/89, and the background is their median.

The clearance of 0.03° is the one number in the method and it is bounded from both sides by measurements already in hand: it has to be larger than the widest dip half-width here, which is 0.0077°, and smaller than the closest crowding, which is 0.074°. A factor of four of room at each end is not a lot and it is not a fit.

What the fix recovers #

With the background measured properly the dip is present at every denominator from 8 to 89, at every head size where the rows have organs in them. At the smallest head each of the three largest denominators is measured on: 13/34 gives a floor of 0.112 against a background of 0.381, 21/55 gives 0.148 against 0.339, and 34/89 gives 0.163 against 0.246.

The answer to that earlier work’s question is therefore yes, the dip survives, and the width law is the subject of the next essay. What is worth extracting here is the shape of the floor: it rises with the denominator at matched head size, because a head of a fixed number of organs has fewer organs in each of more rows.

Why the clearance cannot simply be made generous #

The natural instinct on discovering a contaminated sample is to move further away, and it does not work, because the neighbourhood has two ends.

Move the background band outward and it runs into other rationals — there are more of them further out, not fewer, since the band is wider. Move it inward and it runs into the dip itself. The clearance rule threads between the two: keep offsets that are at least 0.03° from any rational with a denominator of sixty or less, inside a band from 0.10° to 0.45°.

The band’s inner edge is set by the widest dip measured here, 0.0077° at 34/89, and by the shoulder above it — a dip does not stop at its half-width, and a tenth of a degree is more than ten half-widths out on every case measured. The outer edge is set by nothing sharp; 0.45° is where the curve starts responding to the next structure along, and the median over 53 to 72 samples is insensitive to where exactly it is cut.

What is not adjustable is the 0.03° clearance itself, and it is worth seeing why it has so little room. It must exceed the widest dip half-width, 0.0077°, or the excluded zone does not cover a neighbour’s dip. It must be less than the closest crowding, 0.074°, or every offset is excluded and there is no background at all. That is a factor of ten between the floor and the ceiling and the value sits near the geometric middle of it — which also says exactly where the method stops: at the denominator where a dip’s half-width and the distance to the next rational meet.

They do not meet within the range measured. At 34/89 the ratio is still about ten. But the ratio is falling, and both quantities have closed forms, so where they meet can be computed rather than guessed at.

The gap. Two fractions with denominators q and q′ are at least 1/(qq′) of a turn apart, and the nearest other fraction with a denominator of sixty or less is at q′ ≤ 60 — so the gap is about 6/q degrees. Checked against the measured list: 6/8 = 0.75° against 0.763°, 6/13 = 0.46° against 0.462°, 6/34 = 0.18° against 0.193°, 6/89 = 0.067° against 0.074°. The formula is the measurement.

The dip. Its half-width goes as about 150·q/n² degrees — falling with the square of the head and rising with the denominator. At q = 89 and n = 1,335 that is 0.0075°, against the 0.0077° the sweep reports.

Set them equal. 150q/n² = 6/q gives q² = n²/25, so the two meet at

q = n / 5.

Which is a much more useful ceiling than a number, because it is a statement about the arrangement rather than about the arithmetic: q = n/5 is the denominator at which each of the q rays holds five organs. Above it a dip is wider than the distance to its neighbour and there is nowhere clear to take a background; below it there is.

Every case measured here is comfortably inside it. At 34/89 on 1,335 organs each ray holds fifteen; at 21/55 on 825 it holds fifteen again. The method has a factor of three in hand at the hardest case it has been asked to do, and it runs out when the rays get down to five organs each — which is also, not coincidentally, about where a row of five points stops looking like a row at all.

That is a ceiling on the method, stated in advance rather than discovered when a number comes out strange, and it scales with the head rather than sitting at a fixed denominator.

What the repair does to that earlier work’s other numbers #

A correction to a background is a correction to every comparison made against it, so that work’s own results were re-read rather than assumed safe.

The dips are at rationals. Unaffected. That claim is about where the minima of the curve sit, which is a statement about the curve and not about any background.

μ₂ halves when the head doubles at an exact rational. Unaffected, and for the same reason: it is a statement about the floor, measured at the rational itself.

The dip’s width goes as the square of the head size. Affected, because a width is measured at a level a quarter of the way from the floor to the background, and the background moved. The numbers change and the exponent does not — which is the essay after that, and the reason it is a separate one is that the q-dependence it uncovers is a new result rather than a correction.

And the staircase — μ₂ flat in stretches with sharp steps between them. Unaffected. That was read off the curve directly.

And the golden angle is not the local maximum of the disorder. Unaffected, and worth naming because it is the one result here that a moving background could most easily have manufactured. That refutation is a comparison of one angle against sixty-six others on the same sweep, with no background subtracted from anything.

So one of five results moves and the movement does not change its conclusion. That is a better outcome than it might have been, and it is not luck: three of the four are statements about the curve at a point, and only the fourth is a comparison. A measurement that is a comparison carries the risk of its comparison; one that is a value does not. Which is a rule of thumb worth having, and it is why the site’s round trips — where a recovered parameter is checked against the parameter a lattice was built at — have never needed a correction of this kind.

The head has to have rows #

There is a floor below which the question is not askable, and it is worth stating because it decides which heads the measurement can be made on.

A head of n organs at a divergence of p/q has its organs on q radial rows, so each row holds n/q of them. Below about fifteen per row the arrangement at the exact rational is not visibly ordered at all: there are not enough organs in a row for the row to be a row, the cells around each one are cut by neighbours from other rows, and the second moment does not fall.

That is why every denominator here is measured at head sizes scaled to itself — max(300, 15q) and two multiples of it — rather than at a fixed list of head sizes. Comparing 3/8 at 300 organs with 34/89 at 300 organs is comparing a head with 37 organs in each row against one with three, which is not a comparison of denominators.

The gate this leaves behind #

A defect that nothing caught is worth a check, and the check is short enough to state in a sentence: the background must be flat, and the flatness is measured rather than assumed.

In practice that is two assertions, both now in the site’s own gate. The first is that the median background over the clear offsets exceeds the floor, which is the claim “this is a dip” made explicitly instead of being implied by a subtraction. The second is that the spread of the background samples is small compared with the depth being claimed — and at 34/89 it is not, which is recorded as a limit rather than repaired.

Both of those are cheap and neither existed. The reason they did not is worth naming because it is general: every gate this site had asked whether a number was computed correctly, and none asked whether the quantity it was compared against meant anything. A background is not a computation; it is a modelling assumption dressed as one, and modelling assumptions are exactly what a check on arithmetic cannot see.

The same shape has now appeared three times here. A recency cut-off that manufactured a lattice was a parameter of the program read as a property of the model. A control that held error independence fixed was an assumption in a null model read as a fact about arrangements. And a background taken at a fixed offset was a neighbourhood assumption read as a measurement. All three passed every gate, and all three were found by asking what the comparison was against.

The general form #

A background is a measurement standing in for a neighbourhood, and it is trustworthy exactly while the neighbourhood is featureless. That is a condition which can stop holding without anything changing in the code, and this is what that looks like: the same three lines of arithmetic, correct for a decade of small denominators, wrong from 34 upward, and the transition invisible because the output stayed plausible.

The check that would have caught it is cheap and is now in the site’s gate: a background must be flat. Measure it at several offsets and require the spread to be small compared with the depth being claimed. At 3/8 the six samples run 0.726 to 1.201 against a floor of 0.032 — a spread of a third of the depth. At 34/89 they run 0.042 to 0.333 against a floor of 0.163, which is a spread larger than the whole dip.