A band that holds the angle still

Two quantities move inside a rung that no counter can read. The settled divergence slides. The two contact steps change places. Sweeping the rise moves both at once, so a result that changes along a rung cannot be attributed to either — and that is exactly what a sweep of one rung found, leaving the attribution open and saying so.

Separating them needs a place where they come apart: rises with the same counted pair, the same settled divergence, and opposite step orderings. This essay is about finding two such places and checking that they are real.

The floor #

The settled divergence does not slide monotonically down a rung. Across the golden 5/8 rung it climbs from 136.64° to 137.87° — but not smoothly. In the middle it flattens, and around the flattening it takes the same value at two different rises, one either side.

That flattening is where the handover is. Both facts have the same cause: the rule’s own divergence crosses the curve on which the two contact steps are equal, and near a crossing of two lines with similar slopes the difference between them is stationary.

So the band is not a lucky feature of one rung. It is a consequence of the arithmetic and it is available on every rung with a handover — which is six of the eight this ladder carries.

The two bands #

A band is defined by a bound rather than by hand: it is every rise whose settled divergence sits within a twentieth of a degree of the divergence at the crossing, grown outwards from the crossing until the bound or the rung’s edge stops it. That way the extent is measured and the only judgement in it is the bound.

On the golden branch the 5/8 band runs from 0.0176 down to 0.0142 — eighteen rises at a step of 0.0002, over which the settled divergence moves 0.047°, with the five-step giving way to the eight-step at 0.0156.

On the Lucas branch the 4/7 band runs from 0.0237 down to 0.0201 — nineteen rises, over which the divergence moves 0.020°, with the four-step giving way to the seven-step at 0.0225.

Two hundredths of a degree is a fifth of the run-to-run scatter of a settled divergence and a twelfth of the azimuth grid step. Against the 1.262° the same quantity travels across the whole rung, it is a hold rather than a slide.

What “the same divergence” is being claimed #

A settled divergence here is the mean of a run’s last sixty divergences, and two runs reporting the same mean is not the same thing as two runs having the same angle. It is worth separating the two, because the whole design rests on the distinction.

What is claimed is that the lattice is the same to within a twentieth of a degree — that if the arrangement of points at the coarse end of the band and the arrangement at the fine end were each described by a divergence and a rise, the divergences would agree far more closely than either differs from anything else on the rung. That is a claim about a summary of each run.

What is not claimed is that the two runs are the same run. They differ in the rise by a fifth, so their organs sit at different heights, their fronts contain slightly different numbers of organs, and their scatters differ a little. A band is a matched comparison in one quantity, not a pair of identical stems.

The distinction matters most for a null result. If cutting an organ out at both ends of a band gives the same answer, the honest statement is that the answer did not move when the ordering reversed and the rise moved by a fifth and the front changed by an organ. Those are three held-or-varied quantities, and only the first is the one the design was built for. Reading such a null as “the ordering is irrelevant” is stronger than the evidence; reading it as “the ordering is not what was doing the work in the sweep that raised this” is exactly right.

The check the band needed #

A quantity that comes out constant is the first thing a discretised measurement should be suspected of. Every azimuth here lands on a grid of 1,536 steps, a quarter of a degree apart; a settled divergence is a mean of sixty such angles, so it is not itself confined to the grid, but a stem that locks onto exactly the same sequence of grid values at two rises will report exactly the same mean at both. A flat that is really a grid artefact would look identical to a flat that is real.

So both bands were grown again on a grid four times finer — 6,144 steps, 0.059° apart — and the answer is that the flat is not exactly flat and the band survives anyway.

At the finer resolution the golden band shows a minimum 0.082° deep, with its floor near the handover, and the Lucas band one 0.054° deep. That is real structure the coarse grid was hiding, and it is exactly what the arithmetic predicts a crossing of two nearly parallel lines should look like.

But the two ends of each band — which is where the claim lives, because that is where the two step lengths are far enough apart for the ordering to mean anything — still agree. On the golden band they agree to 0.000°, and on the Lucas band to 0.020°.

So the matched pair is a matched pair at four times the resolution that would have broken it, which is the only reason to believe it at the original one.

Why the ends and not the middle #

At the handover itself the two step lengths are equal, so there is no ordering to speak of. This collection’s own bound for calling two steps ordered is one per cent, and within about a per cent of the handover the gap is under it.

That is why the design puts its weight on the ends. The coarse end of the golden band has its two steps four per cent apart with the five first; the fine end has them three per cent apart with the eight first. Both are properly ordered, they are ordered oppositely, and their settled divergences agree to a thousandth of a degree.

The rises in between are not wasted. They are the evidence that the two ends are connected by a continuum rather than being two arbitrary lattices that happen to share an angle, and they are where the handover is located.

What is held and what is not #

Worth listing, because a matched design is only as good as its list.

The counted pair is held. Every rise on each band returns the same pair from a counter shown nothing but positions, and that is asserted rather than assumed.

The settled divergence is held, to 0.047° and 0.020° on the two bands.

The branch is held. Each band is one branch, seeded from one angle.

The front depth is nearly held. The number of offsets that wreck at all changes by one or two across a band, which is the front deepening as the rise falls, and it is the reason the survivor grids have a ragged edge rather than a straight one.

The rise itself is not held, and cannot be: it is the knob the band is swept with. The golden band spans a factor of 1.24 in the rise and the Lucas one a factor of 1.18. Anything that depends smoothly and strongly on the rise is not controlled here, and the honest reading of any null result from a band is that the ordering does not act and that the rise does not act over a range of a fifth.

That last one is the design’s real limitation and it is not fixable by a narrower band: narrowing it holds the rise better and shrinks the ordering difference at the ends, until at the handover itself there is no ordering left to compare.

The two do not trade off one for one, though, and the asymmetry between them is what makes the design work at all. The divergence has a minimum at the handover — the finer grid resolves it, eight hundredths of a degree deep — so its departure from the floor grows as the square of the distance from the crossing. The step-length gap passes through zero there and grows linearly. Halving the tolerance on the divergence therefore narrows the band by a factor of √2 and costs only √2 of the ordering, not two.

Run that forward and the geometry is generous. To bring the coarse end’s four per cent gap down to the one per cent floor takes a band four times narrower in rise, which is a tolerance sixteen times tighter — three thousandths of a degree rather than fifty. So the arithmetic would permit a band holding the divergence ten to fifteen times more tightly than this one does while still carrying an ordering at both ends.

What stops it is not geometry but measurement, and saying so locates the real bound. A settled divergence’s run-to-run scatter is about a tenth to a fifth of a degree, so the hold this band already claims — 0.047° on one and 0.020° on the other — is tighter than a single run reproduces. That is legitimate because the quantity being held is a mean of sixty angles rather than one angle, and a mean of sixty has a standard error something like an eighth of the scatter. It is also the ceiling: a tolerance of three thousandths of a degree is far below what a mean of sixty can distinguish, so the band cannot be tightened to where the geometry would allow without either much longer runs or many seeds per rise.

That reverses which half of the design is the limiting one. The obvious reading is that a band is limited by having to keep an ordering to compare; the arithmetic says it is limited by how precisely a settled divergence can be measured, and that the ordering has an order of magnitude in hand. Anyone building a tighter band should spend on run length rather than on narrowing the bound, and the bound chosen here — a twentieth of a degree — is roughly where the two limits meet at the run lengths this collection uses.

What it is for #

The design exists because an earlier sweep found the family a wrecked stem keeps changing along a rung, and could not say which of the two movers was responsible. A band holds one and varies the other, which turns an ambiguous sweep into a test.

Run on both bands, at every offset that wrecks, the answer is that the survivor does not move. That is a negative result and it is the strongest kind available here: not a low score on a census but a quantity reversed under a controlled comparison with the outcome unchanged.

The cost of it #

Worth recording, because a design that is cheap gets used and one that is not does not.

The geometry of both bands — thirty-seven rises, each grown once to read its settled divergence and its hop ranking — is under a minute of work. The check on the finer grid is four times that, because the placement scan is linear in the number of azimuth samples, so a few minutes. Neither is a consideration.

The ablation is the expensive half: one cut stem per offset per rise, each grown to settle and compared against a control that shares its history. Fifty-five wrecked cuts came out of about a hundred and fifty grown ones, since most offsets repair, and the whole of it is a few minutes on one core.

It is also worth reading the two halves against each other. The geometry costs under a minute and the ablation costs a few, so the design is about five parts question to one part setting up the question — where the census it replaces spent most of its budget on covering pairs it did not need to cover. That ratio is the practical argument for computing a crossing rather than searching for one.

Against the census this replaces the design is smaller in every direction — one rung instead of a ladder, one branch per band instead of two, and no attempt to span the counted pairs. That is the point. A comparison that varies one thing does not need to cover a subject; it needs to cover the one thing, twice.

What else a band is good for #

Anything stated over the step ordering. Two candidates are already in this collection.

The first is the front. Its width is the larger of the two counted numbers, which is a statement about the pair rather than the ordering, so a band should leave it alone — and that is a prediction rather than a hope, because the front depth across a band changes by one or two offsets rather than by the several a reversal would imply.

The second is the rigid hop. Which lag a wrecked stem holds steady is the block it repeats, and the block is measured against each stem’s own control, so nothing about a comparison between two rises enters it. A band should leave that alone too.

Neither is a hard test, and that is the point of writing them down: a design that changes nothing it should not change is a design worth reusing, and the way to find out is to state in advance which quantities are supposed to be inert.

Why this was not available before #

Because the flat is only visible once the rung has been swept finely enough, and the rungs were swept at one rise each for three rounds.

A census taking one rise per rung cannot see a flat: a flat is a property of a run of rises, and one rise is not a run. The sweep that first went along a rung finely enough went at a step of 0.001, which puts eleven rises on the golden 5/8 rung and about five inside the flat — enough to notice, and not enough to know whether the flat had structure or whether the grid had made it.

What made the band a design rather than an observation is the arithmetic: once the crossing is computable, the flat has a predicted location and a predicted reason, and a band can be grown around a computed rise rather than searched for. Both bands here were located that way, and neither cost a search.

What is left #

Four rungs carry a handover and have no band grown around them. The 8/13 golden rung and the 7/11 Lucas rung are narrow — thirty-six and fifty-two sampled rises — so their bands would be narrow too, and whether the divergence flattens enough inside them to give a usable band is a question of arithmetic rather than of taste. The 3/5 golden and 3/4 Lucas rungs are wide and their bands should be comfortable.

The more interesting extension is a band at a transition rather than at a handover. A transition is where the counted pair changes, so no band can hold the pair across it — but two rises either side share a great deal else, and the pair changing is precisely the quantity a great many readings here are stated over. A matched comparison across a transition is a different design from this one and it would test different claims — and it would have no floor to build on, since the divergence has no stationary point at a transition, so its tolerance and its contrast would trade off one for one rather than as a square root.