The handovers corrected

The ladder finds a handover by stepping down a rung at a ratio of one per cent and reporting the first rise at which the two contact families have changed places. That rise is the one the sweep visited, not the one the crossing sits at, and the crossings are now located to one step of the grid on all six rungs.

So the six recorded handovers can be scored against them. The scoring has a sign and a bound in it that were stated before the walk ran, which makes it a test rather than a summary.

The two predictions #

A sweep that reports the first rise at which the ordering has already changed must report a rise on the fine side of the crossing. It cannot report a coarser one, because at a coarser rise the ordering had not changed yet.

And it must report a rise within one of its own steps of the crossing, because the rise immediately before it was on the other side.

Both follow from how the sweep works rather than from anything about lattices, so both are predictions with six chances to be wrong.

Six of six on the fine side #

Every located crossing is coarser than the rise recorded for it. The offsets are +3.5, +2.5, +5.5, +2.5, +3.5 and +34.5 grid steps, on the Lucas 3/4, the Lucas 4/7, the golden 5/8, the golden 8/13, the Lucas 7/11 and the golden 3/5 respectively.

Not one is negative and not one is zero. A single negative offset would have meant the sweep reported a rise at which the ordering had not yet changed, which is a bug in the sweep rather than a fact about a rung.

Six of six inside one sweep step #

In the unit that matters the six run from 0.082 to 0.835 of a sweep step, mean 0.374.

That is the whole discrepancy: a rounding, always in one direction, never as large as the instrument’s own resolution. The largest of them belongs to the golden 3/5, whose thirty-four grid steps sit inside a sweep step forty-one grid steps long.

The distribution is what a rounding should look like — spread across the interval rather than piled at either end — though six values cannot say much about a distribution and this essay does not try to.

The one value outside the bound #

There is one, and it is measured to a different crossing. The Lucas 3/4 carries five crossings, and the offset from its recorded rise to the coarsest of them is 0.000725 — 1.686 per cent of the rise, or 1.703 sweep steps.

That is the only number on the ladder outside one step, and it is outside for a reason the account already gives: the recorded rise there is a choice among crossings rather than a rounding of one.

Measured to the nearest crossing instead, the same rung gives the smallest offset of the six at 0.082 of a step. Both numbers are true and they are about different things, which is why they are drawn together.

In grid steps rather than sweep steps #

The same six offsets in the unit the rises are named in are 3.5, 2.5, 5.5, 2.5, 3.5 and 34.5, and read that way they look like six different sizes of error.

They are not. A sweep step at the coarse end of the ladder is about forty-two grid steps and at the fine end it is six, so the same fraction of a step is many more grid steps at the top of the ladder than at the bottom.

Converting is not a presentational choice. It is the difference between a table that says the golden 3/5 is fourteen times worse than the golden 8/13 and a table that says it is twice as far through its own sampling interval.

The grid is not the same size everywhere #

Stated as a fraction of the rise it lands on, the five-decimal grid is 0.0233 per cent at the Lucas 3/4’s handover and 0.1653 per cent at the golden 8/13’s — seven times coarser at the fine end of the ladder than at the coarse end.

That is a fixed grid meeting a quantity that varies by a factor of twenty-three across the ladder, and it is the same asymmetry the ladder sweep’s ratio step exists to avoid in the other direction.

It also sets a floor nobody can go under here. A rise is rounded to five decimals before anything is grown from it, so the finest a claim on this site can be located is one step of a grid that is seven times blunter at one end of the ladder than the other.

Nothing about the rung explains the offset #

If the discrepancy were a property of the lattice rather than of the sampling, something about the rung would predict it. Six properties were written down before the walk ran and the offset was scored against every one of them.

The strongest is the branch, at r = 0.65. Then how steeply the divergence runs at the handover at −0.54, how much of its rung the band spans at +0.20, how wide the rung is at −0.15, how near the two steps come at +0.11, and how coarse the grid is there at +0.09.

A correlation of 0.65 on six points is nothing #

It is worth being blunt about this rather than reporting the number and moving on. With six points, a correlation of 0.65 is what a coin flip produces often enough to be unremarkable, and the branch is a two-valued property, so the correlation is a comparison of two means of three.

The list was fixed in advance, which is what keeps 0.65 from being the best of a search. But a list of six scored on six points is six chances to refute and no chances to establish, and none of the six refutes.

So the honest reading is the null one: the offset is the sampling, and there is nothing else in it.

What the branch correlation would have meant #

Of the six properties, the branch is the one with a story attached. It is the account left standing on which bands change the family their cuts keep, after three rivals were eliminated a band at a time — so a branch effect here would have been the second appearance of a term that already explains something.

That is exactly why it is the one to be careful about. A property that has been right once is the property a reader will accept at 0.65 on six points, and a comparison of two means of three is not evidence about anything.

The offset is a sampling artefact and the branch decides a seed angle and a sequence. There is no route from the second to the first, and none is proposed.

What a refutation would have looked like #

A strong correlation with the grid’s coarseness would have meant the offset is where the rounding lands rather than where the sweep does. A strong correlation with how steeply the divergence runs would have meant the crossing is being smeared by the reading rather than sampled past.

Either would have been informative and each would have pointed somewhere specific. The one at +0.09 and the one at −0.54 are the two that would have carried them.

That is why six properties were scored rather than one: a null result is only worth anything when the alternatives were named and given room to appear.

Two routes to one crossing #

The walk reads the crossing two ways at once and both are in the table. The ordering flips between two adjacent grid rises, and the ratio between the two shortest lengths dips to its minimum across the same pair.

They agree on all ten crossings the ladder holds, which is what this collection asks of any number it reports twice. The ratio is the more useful of the two because it is continuous: it says how sharp a crossing is as well as where it is.

That agreement is also what makes the relocation meaningful. A crossing is defined as an intersection of two computable curves, and a bracket found by a sign change is only the same object as that intersection if the continuous quantity turns over there too.

Nothing built on a recorded handover moves #

Measured against the band grown around it, each offset is 0.41, 4.12, 0.56, 1.76, 2.19 and 2.06 of that band’s own steps, against bands of sixteen to a hundred and twenty-six rises. As a fraction of a band’s whole extent the largest is 6.0 per cent, on the golden 3/5.

So every result read off those bands stands. That includes the width each band should have, the flatness of the divergence across it, and every count of surviving families the ablation thread has made.

Which raises the obvious question of why bother, and the answer is not that something moved. It is that a rise used in a comparison should be a rise somebody has located, and until now the two quantities being compared on these rungs were known to different resolutions.

Where the handovers sit inside their rungs #

The position reader asserts that every handover falls in the coarse half of its own rung, and that assertion had only ever been checked on rises known to a sweep step. Remade on located rises it holds, and every one of the six moves a little in the making.

They sit at 40.31, 10.72, 5.38, 14.22, 32.37 and 33.33 per cent of the way down their rungs, against 40.40, 11.59, 5.50, 14.59, 33.23 and 34.52 recorded. Every one moves coarser and none moves by more than 1.2 points.

The located range is 5.38 to 40.31 per cent against 5.50 to 40.40. The four handovers that were sampled but never located are among them, and none of them was anywhere but where it was thought to be.

The two-pass design against a whole walk #

One rung had been walked whole before any of this — 378 consecutive grid rises on the Lucas 7/11, every rise it holds. The same rung was walked again at four grid steps, with the ordering refined at the grid where it changes and the tie window walked rise by rise.

The two agree completely: one crossing against one, bracketed between the same two adjacent rises at 0.00804 and 0.00803. The cheaper design used 120 rises against 378 — 258 grown stems fewer, at 32 per cent of the cost.

Why the rung that needed no walking was walked #

Because a sampling design compared only with itself has nothing to be wrong about. The other five rungs have no whole walk to check against and never will, so the only evidence that the design measures what a whole walk measures has to come from the one rung where both exist.

It cost 120 rises, which is under a tenth of the walk’s total. That is the cheapest insurance in the design and it is the reason the other five counts can be read as counts.

The check is narrow, and saying so matters: it shows the design agrees with a whole walk on a rung that crosses once. It shows nothing about how the design behaves on a rung that crosses five times, because the rung that does was never walked whole.

The band offsets are the number that matters #

A reader with one number to carry away should carry this one rather than the sweep steps. An offset of 0.835 of a sweep step sounds like a large fraction of something; the same offset is 4.12 steps of the band grown around that handover, and the band is seventy rises long.

A band is grown outwards while the divergence holds still, which is what makes rises either side of a handover a matched pair. Moving its centre by four of its own steps moves nothing that pairing depends on.

The one exception is the band that holds sixteen rises, where 0.41 steps of the band is a smaller fraction still — and where the problem is not the offset at all but that the rung crosses five times.

A cost claim is a claim like any other #

The essay that walked the first rung priced the extension: about three minutes a rung, and fifteen minutes for all five. That number is wrong, and it is wrong by a factor of about eight.

Cost claims are easy to leave unchecked because nothing depends on them the way a measurement does. But a price is what decides whether a check gets run, and a price off by a factor of eight is a check that gets scheduled as an afternoon’s rounding error and then does not finish.

So it is corrected here rather than quietly superseded, and the essay that carried it now carries the corrected arithmetic instead.

What the arithmetic assumed #

That every rung holds about the same number of rises as the one that had been walked. The Lucas 7/11 holds 378 at the grid, and three minutes for 378 rises is about half a second a rise, which is very close to right.

The rungs hold 201 to 4,011. The five unwalked ones hold 9,572 between them, against the 1,890 that five rungs of 378 would be — a factor of 5.1 before anything else is counted.

A rung is a stretch over which one counted pair holds, and the ladder’s transitions are spaced geometrically, so a rung near the top of it covers many more five-decimal rises than a rung near the bottom. Nothing about that is subtle; it simply had not been multiplied out.

Where the correction lands #

The essay that priced the walk carried the arithmetic in two places — a per-rung figure and a total for the five — and both have been rewritten there rather than left standing with a correction filed elsewhere.

A collection that keeps its errors in follow-ups is a collection where the wrong number is the one a reader meets first. The rule this follows is the one the collection already applies to a measurement: the claim is repaired where it lives, and the essay that repairs it says what it repaired.

What survives unchanged is the comparison the price was made for. Geometry is still two orders of magnitude cheaper per rise than ablation, and that ratio is what makes a rung walkable and a band only samplable.

What a rise actually costs #

Between 0.44 and 1.8 seconds, and the cost rises as the rise falls. A finer rise carries more organs per turn, so the stem grown to find its settled divergence is longer.

The mean across the 1,224 rises actually grown is 0.55 seconds, which is close to the coarse end because most of the rises grown are on coarse rungs. A whole walk of every rung would be weighted the other way.

So the correct figure is about two hours to walk the five unwalked rungs whole, and over two hours for all six — a factor of five from the rise counts and about half as much again from the cost of a rise at the fine end.

Which is what the two-pass design is for #

Two hours is not prohibitive and the walk did not spend it. It read 1,224 of the 9,950 rises the six rungs hold, or 12.3 per cent, in 675 seconds.

The saving is not the point on its own; a check that cost two hours would have been worth running. The point is that the design was scored against a whole walk before it was trusted, so the 12.3 per cent is a sampling with a measured agreement behind it rather than a budget somebody accepted.

And what it does not buy is coverage on the one rung that needed it. That rung’s tie window is 690 grid steps and 181 were read, which is the one place the design’s economy shows up as a qualification in an answer.

What the coarse designs have in common #

Two samplings on this ladder now have a record. The nine-rise design that reads a band is right about whether a band changes on five of six and wrong in a way it cannot see from inside itself; the one-per-cent sweep that finds a handover is right about where one is to within a step of its own, on six of six.

The difference between those records is not diligence. The sweep is sampling a smooth quantity that crosses zero, so its error is bounded by its own step and its direction is forced. The band design is sampling a discrete quantity that changes at single rises, so a feature one rise wide is invisible to it about thirteen times in fourteen.

A sampling of a smooth thing can be argued about. A sampling of a discrete one has to be looked at.

What this does not support #

It does not support reading the six offsets as an error model. Six numbers with a common sign and a common bound are consistent with a rounding, and they are equally consistent with a rounding plus a small systematic term nobody can see at this sample size.

Nor does it support any claim about rungs the ladder finds no handover on. Every rung here was selected by the sweep having found something, so the sample is exactly the set of cases where the sweep succeeded.

And it does not establish that the sweep would stay inside one step at a different ratio. One per cent is the only step ever used, so the bound has been tested at one setting.

What would refute it #

A recorded handover on the coarse side of its crossing, on any rung. That would mean the sweep reports a rise at which the ordering has not changed, and no reading of the sampling survives it.

An offset larger than one sweep step, measured to the nearest crossing on a rung that carries one. That would mean the sweep skipped a rise it should have visited.

Both are cheap to look for and both were looked for on all six rungs. Neither is there.

What is claimed #

That the six handovers the ladder records sit at 0.04299, 0.04172, 0.02251, 0.01558, 0.00800 and 0.00605, and the crossings they name sit at 0.043025, 0.042065, 0.022535, 0.015635, 0.008035 and 0.006075.

That all six of those are on the fine side of a crossing and all six are inside a single step of the sweep that recorded them, at 0.082 to 0.835 of a step with a mean of 0.374 — and that neither the sign nor the bound was fitted afterwards.

That nothing about the rung explains the size, the strongest of six pre-declared correlations being 0.65 on six points.

And that walking five rungs whole costs about two hours rather than the fifteen minutes it was priced at, because the rungs hold 201 to 4,011 rises each and the arithmetic assumed 378 apiece.