Four crossings nobody visited

A handover is the rise inside a rung at which the two contact steps change places. Six of the ladder’s eight rungs carry one, each carries exactly one, and every one of the six sits in the coarse half of its own rung.

Two of the six had a band built around them — a run of rises over which the counted pair and the settled divergence are both held while the ordering reverses. The other four were left as a note saying each was a few minutes of geometry and a few of ablation. This essay is those four, and the note was optimistic about one thing.

What a band is for #

The rise moves three things at once: the pair a counter returns, the divergence the rule settles on, and which of the two contact steps is shorter. A result that changes along a rung cannot be attributed to any one of them.

A band is the design that separates the third from the other two. Near a handover the settled divergence is flat, so a stretch of rises either side of it share an angle to a twentieth of a degree while the counted pair holds — and the ordering, which is what changes hands at the handover, is reversed between the two ends.

Cut at both ends and anything that changes is attributable to the ordering. Nothing does, which is what two bands established and what four more are built to test on four more counted pairs.

The step has to be a ratio #

The first version of this swept a band at an absolute step of 0.0002 in the rise. At the two handovers it was written for — 0.0156 and 0.0225 — that is about one per cent, which is a reasonable step.

At the 8/13 handover of 0.00605 the same 0.0002 is three per cent. At the 3/5 handover of 0.04172 it is half a per cent. So one absolute step is three different instruments at three different places on a ladder whose rungs span a factor of fifteen in the rise, and the coarse end is over-sampled while the fine end is stepped over.

That is the same argument the ladder itself is swept with, made one level down. The rungs are geometric — consecutive transitions sit at a ratio of about 1/φ² — so everything measured on them has to be measured in ratios. The bands here are swept at two parts in a thousand.

How a band’s extent is found #

Not stated. Grown outwards from the handover, one step at a time in each direction, while three conditions hold: the rise stays inside the rung, the counter returns the rung’s own pair, and the settled divergence stays within a twentieth of a degree of its value at the handover.

The first rise that fails any of them ends the band on that side, and which condition ended it is recorded. Three of the six bands are ended on both sides by the divergence; two are ended on their coarse side by the rung’s own transition, which is to say the band would have run further if the rung had.

That distinction matters for reading the widths. A band bounded by the divergence is a measurement of how flat the curve is; a band bounded by the transition is a measurement of how much rung there was.

The four new bands #

The golden 3/5 band sits around 0.04172 and carries 70 rises, spanning a factor of 1.148 in the rise, which is 14.6 per cent of its rung. Its divergence moves 0.0391° across the whole of it.

The golden 8/13 band sits around 0.00605 and carries 126 rises, spanning 1.284 — and that is 72.1 per cent of its rung, because the 8/13 rung is the narrowest on the ladder and the flat runs across most of it. Its coarse end is where the rung ends rather than where the divergence leaves the bound.

The Lucas 7/11 band sits around 0.00800 and carries 124 rises, spanning 1.279, which is 48.4 per cent of its rung. And the Lucas 3/4 band sits around 0.04299 and carries 16 rises, spanning 1.030 — three per cent of its rung, an order of magnitude narrower than any of the others.

Why the fine bands are wide #

The 8/13 band covers nearly three quarters of its rung and the 7/11 band nearly half, while the two coarse bands cover a seventh and a thirtieth. Read as a fraction of the rung that is a large difference; read as a factor in the rise it is not — 1.28 against 1.15 and 1.03.

Both readings are worth having and they say different things. The fraction says how much of a rung’s interior a band occupies, which is what decides whether a band can be sampled at all. The factor says how far the rise has to move before the divergence has slid a twentieth of a degree, which is a property of the curve rather than of where the transitions happen to be.

The fine rungs are narrow and their divergence is flat across them. The 8/13 rung slides only 0.117° end to end — less than three times the band’s own tolerance — so most of it qualifies.

Five of the six are matched pairs #

A band is only useful if its two ends are a matched pair: the same counted pair, the same divergence, and opposite orderings that are far enough apart to be orderings at all.

Five of the six qualify. The ordering changes hands exactly once inside each of them, and at both ends the two contact steps differ by between 3.4 and 8.3 per cent in length — comfortably more than the four parts in a thousand below which an ordering is not an ordering.

The sixth is the Lucas 3/4 band, and it is a band that moves nothing. It is in the table.

What the bands hold #

The counted pair, by construction: a rise whose counter returns a different pair ends the band.

The divergence, to a twentieth of a degree at each end from the handover, which means at most a tenth across a whole band. Measured, the six slide 0.0391°, 0.0469°, 0.0430°, 0.0703°, 0.0391° and 0.0430° — five inside a twentieth and one, the Lucas 3/4, at seven hundredths because its two halves happen to slide in opposite directions.

And nothing else. The rise moves by 3 to 28 per cent across a band, and everything the rise controls moves with it: the neighbourhood size, the step lengths, the front’s depth. A band is a narrow experiment, not a small one.

Where the handovers sit #

All six sit in the coarse half of their rungs: 11.6, 14.6 and 34.5 per cent on the golden branch, 40.4, 5.5 and 33.2 on the Lucas one. Never past the middle, which is a result the previous round established and did not explain.

Having six bands rather than two does not change that number — the positions come from the ladder sweep, not from the bands — but it does change what can be said about the consequence. A person growing a stem counted 5/8 picks a rise comfortably inside the rung, which is past the handover; so does a person growing one counted 4/7 or 8/13. The census’s sampling bias is a property of all six rungs and not of the two that had been looked at.

The two rungs with no handover #

The coarsest rung on each branch — golden 2/3 and Lucas 1/3 — has none. Their two contact steps never change places anywhere inside them, so there is no crossing to build a band around and asking for one is refused rather than answered with the rung’s coarse end.

That refusal is asserted rather than left to chance, because the failure mode is quiet: a function that returned the nearest thing it could find would produce a band with no handover in it, whose two ends have the same ordering, and every other check on it would pass.

The reason those two rungs have none is that their crossing sits above the range the ladder is swept over. The sweep stops at 0.0700 because above that the rise is no longer a small quantity and a stem’s front is its own two edges.

What it cost #

The geometry is cheap and the cuts are not. Sweeping a band’s rises costs one grown stem each — 70, 112, 126, 16, 86 and 124 of them, plus the ladder sweep underneath — and each stem is a few hundred organs placed one at a time.

The ablation is sixteen cut stems per rise, each grown three hundred organs past the cut with a control beside it. Cutting every rise of every band would be several hours for a quantity the design predicts to be constant, so the cuts are made at nine rises spread through each band plus both ends and the handover itself.

That is a choice and it is the right one for the shape of the claim. A constant is tested at the extremes and at the crossing; sampling the middle more finely buys resolution in the one place where nothing is supposed to happen.

What a finer sweep would add #

Nothing to the claim and something to the picture. At two parts in a thousand the widest band carries 126 rises, which is enough that the divergence’s shape inside it is drawn rather than sketched. Halving the step would double the stems and change no number here.

Where a finer step would matter is at the ends. A band’s extent is where the divergence crosses a bound, and the crossing is located to within one step — two parts in a thousand in the rise, which is a tenth of the narrowest band’s whole width and a hundredth of the widest’s. The widths quoted here are therefore good to about one per cent on the wide bands and ten on the Lucas 3/4.

That last figure is worth carrying, because the Lucas 3/4 band is the one whose width is most surprising and the one whose width is least well measured.

What the four new bands are for #

Two things. The first is weight: the ordering result rested on two counted pairs and now rests on four, with 114 wrecked cuts rather than fifty-five.

The second is coverage. The two original bands were both on rungs in the middle of the ladder — 5/8 and 4/7 — and the four new ones reach the coarse end and the fine end of both branches. A claim tested only in the middle of a range is a claim about the middle of a range.

What the note underestimated #

“A few minutes of geometry and a few of ablation” was right about the geometry and wrong about the sweep. The four bands were not found by pointing the existing machinery at four new rises; they were found after the step was changed from an absolute amount to a ratio, and before that change two of the four could not be found at all.

The 8/13 band spans a factor of 1.284 around a handover at 0.00605, which is 0.0018 in absolute rise — nine steps of 0.0002, of which the existing loop allowed sixty. That one would have been found, coarsely. The Lucas 3/4 band spans 1.030 around 0.04299, which is 0.0013 — six steps, and its ends would have been located to a sixth of its width.

So the four were not four repetitions of an existing measurement. They were four repetitions of a measurement that had to be repaired first, and the repair is the part worth carrying forward.

Where the handovers came from #

Not from the bands. A handover is found by sweeping the whole ladder at one per cent in the rise and reading, at each rise, which of the two contact steps is shorter; the handover is the rise inside a rung at which that changes.

It is also computable without growing anything. The divergence at which the two steps are exactly equal is a curve, found by bisection on the lattice’s own arithmetic, and the rule’s own settled divergence is a second, shallower curve. They cross once per rung, and the measured handover and the computed crossing agree on all six rungs to within one sweep step.

So the six positions are two independent measurements agreeing, and the bands are built at the place both of them name.

What is held constant across all six #

Every band is grown by the same loop with the same three stopping conditions, at the same step, with the same tolerance on the divergence. Nothing is tuned per band.

That matters because a band is a measurement of a rung rather than a construction placed on one. The widths come out at 1.03×, 1.15×, 1.18×, 1.25×, 1.28× and 1.28× because the six rungs’ divergence curves are shaped differently, and if any of those had been adjusted to make a band usable the spread would be an artefact of the adjusting.

The one place a choice is visible is the tolerance, and it is the same twentieth of a degree the first two bands used, kept rather than re-chosen.

What each band’s ends are bounded by #

Recorded per band and per side, because the two kinds of end mean different things.

Four of the six bands are ended on both sides by the divergence leaving its bound, which makes their widths measurements of how flat the curve is.

Two are ended on their coarse side by the rung’s own transition: the golden 8/13 band and the Lucas 4/7 band. Those two would have run further if the rung had, so their widths are lower bounds rather than measurements — and the 8/13 band, at 72 per cent of its rung, is nearly the whole of one.

Why this could not wait for a finer ladder #

The obvious alternative is to sweep the ladder itself more finely and find more rungs, and it does not help.

The ladder’s range is bounded at both ends by something other than resolution. Above 0.0700 the rise is no longer small and the coarse rungs cannot be wrecked; below 0.0040 a stem does not settle onto a lattice at any run length. Between those two the ladder has eight rungs and six of them have a handover.

A finer sweep would locate the six handovers more precisely and would not produce a seventh. So six bands is the population rather than a sample of it, which is the unusual position the width prediction is also in.

The one line #

Six rungs carry a handover; two had bands and four did not. Grown at a step that is a ratio rather than a fixed amount, the four carry 70, 126, 16 and 124 rises, spanning 1.15, 1.28, 1.03 and 1.28 in the rise and holding their divergence to between 0.039° and 0.070°. Five of the six bands reverse their ordering exactly once and are usable; the sixth does not, and is the subject of the next essay.