A band that moves nothing

A band is supposed to move exactly one thing. The counted pair is held because a rise returning a different pair ends the band; the settled divergence is held to a twentieth of a degree because that is what the band’s extent is defined by; and the step ordering reverses, because a band is grown around the rise where the two contact steps change places.

Five of the six do that. The sixth holds all three, and this essay is about why it is in the table rather than out of it.

The band #

The Lucas 3/4 rung runs from a rise of 0.0640 down to 0.02390 and its handover — the rise at which its 3-step and its 4-step change places — sits at 0.04299, which is 40.4 per cent of the way down it. That is the deepest handover on the ladder and still in the coarse half.

Grown outwards at two parts in a thousand — the ratio step the four new bands needed — the band around it carries 16 rises and spans a factor of 1.030 in the rise. That is three per cent of the rung, against 14.6, 23.2, 72.1, 18.7 and 48.4 per cent for the other five. It is an order of magnitude narrower than anything else here.

Sixteen rises is not too few to sweep. It is too few for the thing being swept: the band’s whole purpose is to put two orderings at its two ends, and sixteen steps of two parts in a thousand do not carry the rise far enough from the crossing for the two steps to come apart.

What is wrong with it #

Two things, and they are the same thing.

The ordering changes hands three times inside those sixteen rises rather than once. And at the two ends the two contact steps differ in length by 0.03 per cent and 0.43 per cent — against 3.4 to 8.3 per cent at the ends of the other five bands, and against a threshold of four parts in a thousand below which an ordering between two steps is not an ordering.

A quantity that flips three times over sixteen samples, while the difference it is the sign of is a few parts in a thousand, is a quantity that is not doing anything. It is the sign of a number that is zero to the resolution available.

Three is worth putting between the two things it could have been, because it sits between them and the position is informative. A clean crossing gives one flip: the sign is negative on one side and positive on the other, and there is nothing else to report. A sign with no information in it at all — a coin at every rise — gives on average seven and a half over sixteen samples, since there are fifteen places for it to change and each is even money.

Three is neither. It is well above one and well below seven and a half, which is what the sign of a smooth quantity looks like when the quantity is wandering near zero at a scale comparable to how finely it is resolved. The runs are too long for noise and too many for a crossing.

That reading has a cause available. The settled divergence is read off a grid of 1,536 candidate azimuths, so as the rise steps by two parts in a thousand the divergence moves between neighbouring grid values in runs rather than independently — and a step-length difference of three parts in ten thousand inherits those runs. What the three flips report is therefore the grid’s own structure showing through a quantity that has none of its own, which is the same diagnosis the rest of this section reaches and is arrived at from the count rather than from the size.

It also says what a finer grid would do, which is not what a reader might expect. It would not resolve the ordering; it would raise the flip count towards seven and a half, because the runs would shorten while the quantity underneath stayed zero.

Why the two steps stay together #

Because the band is narrow, and the band is narrow because the divergence is not flat there.

Fitted a quadratic across the band’s own rises, five of the six bands have a slope at their handover of between 0.005 and 0.13 degrees per unit of log rise — which is stationary to the resolution available. The Lucas 3/4 band has 2.19, a factor of seventeen larger than the next.

That is a measurement rather than an explanation. Why the divergence should have a turning point at a handover on five rungs and not on the sixth is not answered anywhere here; the two curves whose crossing makes a handover are the balanced divergence and the rule’s own, and nothing says the second has to be stationary where they meet.

So on five rungs the divergence has a genuine turning point at the handover and the curve stays inside a twentieth of a degree for a long way either side. On this one the divergence is running downhill through the crossing, hits the bound after eight steps, and the band ends before the two steps have had room to separate.

The two facts are one fact #

The two contact steps are equal at the handover — that is what a handover is — and they separate as the rise moves away from it. How far the rise can move is what the band’s width is. So a narrow band is a band whose ends are close to the crossing, and ends close to a crossing are ends where the two steps are nearly equal.

Stated that way it stops being a coincidence and becomes arithmetic. A band is wide when the divergence is flat; the ordering is meaningful at a band’s ends when the band is wide; so the same property of the divergence curve decides both.

The Lucas 3/4 rung fails on both counts because it fails on one.

What it would have taken to hide this #

Very little, and that is the reason to write it down.

The first version of the check over the six bands required every one of them to change its ordering exactly once. That version fails, and the obvious repair is to loosen it to “most of them” — at which point a table of six reports five matched pairs and one unremarked row, and nothing in the output says a band exists that holds all three quantities.

The second obvious repair is worse: drop the row. Six handovers, five bands, and a sentence saying the sixth was unsuitable. That is a design reporting five for five.

What the check does instead is require that exactly one band fail, and that the one that fails do so because its steps are never separated rather than because its divergence moved. Both halves are asserted, so a future change that broke a different band in a different way would not pass.

It is a control #

Having one is useful. Five bands hold two quantities and move a third, and the answer does not change; one band holds all three, and the answer does not change either. The second is what a null control looks like.

Without it, “the surviving family does not change across a band” is a statement about bands that move the ordering, and the natural worry is that a band is simply a stretch of rises too narrow for anything to change across. The Lucas 3/4 band is narrower than any of the others and its cuts behave the same way, which is a small piece of evidence that the constancy is not about width.

Small, because that band wrecks at no offset at all — so what it controls for is the geometry rather than the cut.

There is a second reason to keep a row that measures nothing. The census this thread’s readings were scored on turned out to have been very nearly holding the quantity it was thought to be varying, and that was found by computing a column nobody had computed. A table that lists the cases where a design does not apply is a table someone can compute such a column from later; a table that lists only the cases where it worked is not.

It wrecks at nothing #

The Lucas 3/4 band sits at rises around 0.043, which is the coarse end of the Lucas branch. Cut a stem there at every offset the front reaches and every cut repairs.

That is the coarse end behaving as it has been measured: a front of four organs is its own two edges with no middle, and a single removal has nothing to sever. The golden 3/5 band, also coarse, does the same thing — two of the six bands wreck at no offset, and both are the coarse ones.

So the Lucas 3/4 band is doubly not an experiment: it does not move the ordering, and it produces no cut to see the effect of moving it.

Two coarse bands, two different failures #

The golden 3/5 band and the Lucas 3/4 band both wreck at nothing, and they are not the same case.

The golden 3/5 band carries 70 rises, spans 1.148 in the rise, reverses its ordering exactly once, and has ends separated by 3.9 and 4.7 per cent. It is a perfectly good matched pair with nothing to measure on it. If a way were found to wreck a coarse stem — a removal of two organs, say — that band would be ready.

The Lucas 3/4 band would not. Its ends are not a matched pair whatever is done to them, because its two steps are the same length at both.

That is worth separating because the two rows look identical in a column headed “wrecked cuts: 0”.

The distinction has a use. Removing two organs wrecks a coarse stem where removing one cannot, so the golden 3/5 band is a design waiting for an experiment that now exists. The Lucas 3/4 band is not: a two-organ cut across it would still be a cut across a band whose two ends carry the same ordering.

What the grid can and cannot resolve #

The two steps at this band’s ends differ by 0.03 and 0.43 per cent in length. The azimuths those lengths are computed from are read on a grid of 1,536 candidate positions, so a divergence is quantised at 0.234° — and a difference of three parts in ten thousand in a step length is well below what that grid can settle.

Which means the sign of the difference at this band’s ends is not a measurement. It has a value, the arithmetic produces it, and it would change if the grid were finer. Calling one step shorter there is reporting the last digit of a number whose earlier digits are all zero.

The other five bands are not in that position: 3.4 per cent is more than a hundred times the resolution.

This site has been caught by exactly this before, in the other direction. A period the grid invented was a structure read off a quantity the azimuth grid had quantised, and the check that found it was re-measuring on a finer grid. The same check is available here and is not worth running: a difference of three parts in ten thousand would move on a finer grid, and the point is that it is not a difference worth having a sign for.

Where the width was checked #

The band’s extent is found by stepping outwards until the divergence leaves the bound, and the crossing is located to within one step — two parts in a thousand. On a band spanning a factor of 1.030 that is about a tenth of the whole width.

So the width quoted for this band, unlike the others, is good to roughly ten per cent rather than one. That does not change any of the conclusions — three flips in sixteen rises is three flips whatever the ends are, and 0.03 per cent is 0.03 per cent — but it is the one number here worth a caveat, and it is the number the essay leans on least.

What this says about the other five #

Nothing, directly, and something indirectly. The five that work do so because their divergence has a stationary point at their handover, and this one does not. Nobody chose which rungs had one.

That is a small warning about the design. A band is available where the divergence happens to be flat, and the divergence is flat at five of six handovers for reasons nothing here explains. If it were flat at two of six, the ordering result would rest on two bands and the same argument would be made about it — and the number of bands available would be a fact about the curve rather than about the effort.

The rung it belongs to #

Worth a paragraph, because the Lucas 3/4 rung is the widest on the ladder and its band is the narrowest.

It runs from 0.0640 to 0.02390, a factor of 2.68 in the rise, carrying a hundred rises at the ladder’s one per cent step — more than any other rung on either branch. Its divergence slides only 0.645° across the whole of that, from 102.422° to 101.777°, which is the smallest end-to-end slide of any rung except the fine 8/13.

So it is a wide, flat rung with a narrow band on it. That combination is what the width prediction gets most wrong, and it is the reason: a rung-average slope of −0.654 says the band should be wide, and the curve is doing something local that the average cannot see.

What it does to the count of bands #

Six handovers, six bands, five usable. That is the number to quote, and quoting “six bands” without the qualifier would be the ordinary way for a table to overstate itself.

It also changes the arithmetic of the ordering result. That claim is now carried by the four bands that wreck, of which this is not one, so the Lucas 3/4 band contributes nothing to it at all — neither support nor a counter-example. A row that contributes nothing is worth listing precisely because a reader counting rows will otherwise count it.

What a reader should take from a row like this #

That a design’s coverage and a design’s applicability are different numbers.

Six handovers exist and six bands were built, so the coverage is complete: nothing was skipped, nothing was chosen. The applicability is five, because one of the six turns out to hold the quantity it was built to move. Both numbers are true and only the second belongs in a claim.

The same distinction runs through the other design built beside it. Five matched pairs exist and four produce an experiment, because the fifth sits at the coarse end and neither of its stems wrecks. In both cases the unusable cell is unusable for a reason the ladder supplies rather than for a reason the experimenter chose, which is why both are worth listing.

The one line #

The Lucas 3/4 band carries sixteen rises, spans three per cent of its rung, holds its counted pair and its divergence, and does not move the ordering: the two contact steps differ by 0.03 and 0.43 per cent at its ends and change places three times inside it. It is the one rung whose divergence is not stationary at its handover, it wrecks at no offset, and it is in the table because five for five is not what happened.