One crossing or two

Walking the ladder’s six rungs at the grid returns five counts of one and a count of five. The five is on the Lucas 3/4, the coarsest rung there is, and a number that sits alone in a table of six is a number that wants an account before it is used for anything.

The account here is not about that lattice. It is about how the quantity a crossing is a zero of is computed, and it makes a prediction with a threshold in it that the other five rungs can refute.

They do not.

What a hop length is arithmetic on #

Every hop length on this ladder is closed form. Given the settled divergence and the rise, the distance across the surface between two organs a fixed number apart follows without growing anything further.

So the whole of a crossing — where the two contact families exchange the shorter hop — is a function of one angle and one rise. The rise is a number anybody can type. The angle is measured.

A divergence read in steps #

The settled divergence is a mean of angles read off a lattice placed on a fixed set of azimuths, 1,536 of them. It is therefore read in steps rather than continuously, and between two of those steps the reading does not move at all.

The consequence is a quantity that ought to be smooth in the rise and is not. Between two steps of the reading the ordering slides with the rise; at a step it jumps.

That is not a defect anybody introduced late. It is the same discreteness a convergence study on a different quantity already priced, measured here on the one that decides where a handover is.

What a handover was defined as #

The two contact steps are equal along a computable curve, and a handover is where the rule’s settled divergence crosses it. That definition was written when the curve was the object and the divergence was a number.

It is still right, and it now has a resolution attached to it. The curve is exact and the divergence is read in steps, so a crossing is the intersection of an exact thing with a staircase — and a staircase can cross a shallow line as many times as it has steps in the neighbourhood.

Five crossings is what that looks like when the line is shallow enough and the steps are large enough. It is not a second definition of a handover; it is the first one, evaluated where it is fragile.

Slide against jump #

Both are measurable per grid step of the rise, and both were measured inside each rung’s tie window so that the six are read over comparable regions.

If a jump is smaller than a slide, the reading nudges a monotone track and the crossing stays where it was. If a jump is larger, the reading can send the ordering back across a zero it has just passed — and every time it does, two more crossings appear.

The prediction is a threshold at one, stated before the counts were looked at.

The threshold is right on all six #

The Lucas 3/4 sits at 12.71: one step of its divergence reading undoes 12.71 grid steps of its own slide. It is the only rung above one and it is the only rung with more than one crossing.

The Lucas 7/11 sits at 0.53 and the golden 8/13 at 0.31, both below one and both crossing once. On the golden 3/5, the Lucas 4/7 and the golden 5/8 the reading does not step inside the window at all, so there is no jump for a slide to be compared with and the track is as smooth as the arithmetic.

Six rungs, one threshold, six correct verdicts, and the separation is not a hair’s breadth: the one rung past the rule is past it by a factor of twelve and no other rung reaches half.

Why that rung and not another #

Two quantities go the wrong way at once there, and either alone would not have done it.

Its slide is the slowest on the ladder — 6.96e-5 per grid step, half the next slowest and a fourteenth of the golden 8/13’s. A slow slide means the ordering takes many rises to travel any distance, so it is near zero for a long stretch.

And its reading step is the largest on the ladder, 0.0391 degrees against the 0.0039 that is the only other step measured anywhere. Ten times the quantum, against a fourteenth of the speed.

The largest single jump #

Not the median but the worst of them: 2.297e-3, which is 33 grid steps of that rung’s own slide.

A jump of that size at a rise where the ordering is within a few parts per million of zero does not perturb the track. It carries it across and back, and the two crossings that produces are a step of the azimuth grid rather than anything the lattice does.

That is the sentence the whole account reduces to. On five rungs a reading step is a wobble; on one it is the dominant term.

Why the coarse end is where it happens #

Both of the quantities that produce it belong to the top of the ladder. A coarse rise carries few organs per turn, so the ordering of two hops moves slowly as the rise falls, and a divergence read off a fixed set of azimuths lands on fewer distinct values there.

That is the same regime in which a grown pattern’s counts change exactly where the static ladder says they should, and for a related reason: at the coarse end the placement rule has very little to be confused by, so everything moves slowly and small quantisations dominate.

It would be neat if that made the effect predictable from the rise alone. It does not: the golden 3/5 is the second coarsest rung on the ladder, is refused over 3.4 per cent of itself, and crosses once.

The comparison was got wrong once #

The first version of it read each rung’s slide and jump over the whole rung rather than over its tie window, and scored 5 of 6 — missing the golden 8/13, the one rung walked whole.

The quantities were right and the regions were not. That rung’s table is 201 rises of rung against a neighbour’s 24 rises of window, so reading them side by side compares a statistic over a rung with a statistic over a crossing.

It is worth recording because the failure is not visible in the answer. Five of six looks like a good account with one exception in it, and it was an account measured on two different things.

Where the ordering cannot be read at all #

The counter declines to order two hop lengths within 1.01 of each other, and that refusal covers a share of every rung. The share is where the same fact appears from the other side.

On the Lucas 3/4 the ordering is refused across 16.6 per cent of the rung — 196 of the 311 rises read. On the golden 3/5, the Lucas 4/7 and the golden 5/8 it is 3.4, 3.3 and 3.6 per cent. On the Lucas 7/11 and the golden 8/13 it is 7.3 and 10.0.

So the rung that crosses five times is refused over a sixth of itself, about five times the share of the three rungs whose reading never steps.

How near the two steps come #

At its closest the Lucas 3/4’s two contact steps differ by a ratio of 1.0000038 — under four parts in a million.

The other five come no nearer than 5, 10, 18, 28 and 7 parts in a hundred thousand. That is one to two orders of magnitude further apart at their tightest than this rung is at its own.

A quantity that depends on which of the two steps is shorter is, at that rise, reading a distinction of four parts in a million off an angle read in steps of four hundredths of a degree. It is not that the reading is wrong there. It is that there is nothing to read.

What the survivor rule is doing when it refuses #

The refusal is not caution about a hard case. The family a wrecked stem keeps is a contact family at twenty-nine of thirty offsets, so naming a survivor means naming one of the two steps the counter ranks — and if the counter cannot rank them, the survivor has no name.

That is why the tolerance is 1.01 rather than something tighter. Two steps closer than a per cent are two steps whose order is decided by where the azimuth grid happened to fall, and a rule that returned an order there would be returning the grid.

The share of a rung it refuses is therefore a measurement of the instrument and not of the lattice, which is exactly what makes 16.6 per cent against 3.3 comparable across six rungs.

The ends are decisive on every rung #

This is the check that keeps the previous section from being a statement about a rung nobody can read. At its coarse and fine ends the Lucas 3/4’s two steps stand at ratios of 1.204 and 1.036, and every other rung’s ends sit between 1.034 and 1.103.

So the ordering is emphatic almost everywhere on all six and marginal only around a crossing. The refused stretch is a feature of the crossing, not a property of the rung.

The same reading counts distinct values of the settled divergence: 47, 73, 72, 52, 31 and 24 across everything read on each rung, against 6, 1, 1, 1, 2 and 2 inside the tie windows. The reading is only ever in question over a stretch this narrow.

So that rung’s handover is not a rise #

It is the honest form of the finding and it does not depend on the count. Over 690 grid steps the two contact steps of that lattice stay within a per cent of each other, and inside that stretch the ordering is decided by a reading rather than by the rise.

A handover was defined as the rise at which the two families change places. On five rungs that definition names one rise. On this one it names an interval, inside which the ordering changes hands at least five times and would change hands somewhere else if the azimuth grid were finer.

Nothing built on that rung’s handover is safe in the way the same thing on another rung is safe.

What this changes about that band #

The Lucas 3/4’s band was already the strange one. Sixteen rises against seventy to a hundred and twenty-six, and every quantity read off it holds still across the whole of it — which is what made it a control rather than an experiment.

The reason it holds still is now visible. Its two contact steps stay within four parts in a thousand of each other across the entire band, because the whole band sits inside a tie window 690 grid steps wide. A band that flat is a band grown inside the region where the ordering cannot be read.

So the control is a good control and a poor experiment for exactly the same reason, and scoring it beside the bands that wreck has to say which of the two it is being counted as.

What was hidden outside the window #

The count on each rung rests partly on an argument about the intervals nobody read. To dip through zero and back inside one unread interval, the ordering must travel its own value at both ends and return, and it cannot go faster than the fastest change measured anywhere on that rung.

The margin is how many times over that fails: 5.64 on the golden 3/5, 6.52 on the Lucas 4/7, 6.07 on the golden 5/8 and 3.35 on the Lucas 7/11. The golden 8/13 was walked whole and has no unread intervals at all.

On the Lucas 3/4 it is 0.05 — a factor of twenty short of excluding a hidden pair.

The instrument said which rung it could not vouch for #

That margin was computed from the walk’s own fastest measured change, and it is the same 0.05 whether the count came back one or five. It is a bound on what was not read rather than a summary of what was.

So the walk named its weakest rung in advance and the weak rung is the anomalous one. That is a much better position than finding an anomaly and then discovering the coverage there was poor.

It is also why five is written as a floor. A margin under one does not merely fail to exclude a hidden pair; it says the region has intervals wide enough to hold one comfortably.

The other instrument says nothing about that rung #

The ablation thread reads the same six objects by taking an organ away at every rise of a band and asking which contact family the wreckage leaves standing. On this rung it returns nothing at all: 96 cut stems, six offsets tried at every one of the band’s sixteen rises, and not one wrecked stem between them.

A cut that does not wreck leaves no rigid hop, so there is no survivor, so there is nothing for any account of the survivor to be right or wrong about. The census that reads a band refuses this one, and the refusal is correct.

The golden 3/5 is the other such band, at 490 cut stems and the same zero.

The rung where both instruments speak #

The Lucas 4/7 is the useful contrast. Its cuts wreck 144 times across 86 rises and keep one family throughout, and its walk returns a single crossing located to one grid step with a margin of 6.52 against a hidden pair.

That is the only rung where a strong statement from the cutting sits beside a strong statement from the walk, and the two are about different rises. The claim they jointly test is that the rise where the steps change places is not the rise where the survivor changes — which the 7/11 rung already separated by 164 grid steps with both numbers located.

Five rungs of six now have a located crossing to make that comparison against. The sixth has five of them and no survivor at all.

Two instruments, opposite verdicts on one pair #

The two quiet rungs are the same object to the cutting and are as far apart as the ladder allows to the walk. It cuts 586 stems across the pair and reports nothing on either; the walk grows 559 rises across the pair and reports one crossing on the golden 3/5 and five on the Lucas 3/4.

So the rung whose handover turns out not to be single is precisely the rung on which no cut can say anything about a handover at all. Nothing on that rung has ever been checked by two instruments and now nothing can be.

Whether that is coincidence is not decidable from six rungs. Both facts follow from the same place on the ladder — the coarse end, where a lattice carries few enough organs per turn that a removal closes and a rise moves the ordering very little — but follow from the same region is not are the same phenomenon.

Where the sweep put it #

None of this moves the rise the ladder recorded on that rung by much. The nearest crossing to 0.04299 is 0.043025, three and a half grid steps away, which is well inside the bound the sampling predicts and the smallest offset of the six.

To the coarsest of the five it is 1.703 sweep steps, and that is the only value on the ladder outside a single step. The two numbers are not in tension: one measures a rounding and the other measures a choice among crossings.

Which is the honest reading of the recorded rise there. It is not the handover rounded; it is the first crossing the sweep happened to arrive after.

What a finer azimuth grid would do #

Move the crossings, and possibly reduce them to one. The account says the extra four are a reading step outrunning a slide, and a finer reading has a smaller step.

That is a test and it has not been run. It would need the whole tie window rewalked at a finer azimuth grid, and the prediction attached to it is specific: the Lucas 3/4’s crossing count falls as the quantum falls, and the other five rungs do not move at all.

An account that predicted nothing about a setting nobody has changed would be worth much less than this one.

What the account does not explain #

Why the slide on that rung is slow. The ratio of jump to slide is measured from two things and the account only says which of them wins; nothing here says why the ordering travels 6.96e-5 per grid step there and 9.68e-4 on the golden 8/13.

Nor does it say why the reading step is ten times larger. The quantum is a property of how a mean of angles lands on a fixed set of azimuths at that lattice, and it was measured rather than derived.

Both are answerable and neither is answered. The account is a threshold that sorts six rungs, not a model of either quantity.

What six rungs can do here #

Refute, and that is all. A threshold at one scored on six points is right six times, and six points would be six chances to be wrong.

They were genuinely chances. Three rungs sit at zero, two below one and one far above, so a single rung with a jump worth two slides and one crossing would have ended it, as would a rung crossing twice with a reading that never steps.

What six points cannot do is establish that the threshold is at one rather than at 0.7 or at 3. Nothing here separates those, and no rung sits between 0.53 and 12.71.

What is claimed #

That every hop length here is closed-form arithmetic on a settled divergence read off 1,536 azimuths, so the quantity a handover is a zero of slides between reading steps and jumps at them.

That inside its tie window one step of that reading on the Lucas 3/4 undoes 12.71 grid steps of slide, against 0.53 and 0.31 on two rungs and no step at all on three, and that a threshold at one accounts for the crossing count on all six.

That the same rung’s ordering is refused over 16.6 per cent of itself against 3.3 to 10.0 elsewhere, and that its two contact steps come within four parts in a million of each other — so its handover is an interval rather than a rise.

And that it is also one of the two rungs on which no cut wrecks anything, so the one rung whose handover is not single is the one rung the cutting instrument has never been able to speak about.