Five rungs walked

A rung is the stretch of rise over which one parastichy pair is counted. A handover is a rise inside it at which the two contact families exchange the shorter hop, and six of the ladder’s rungs carry one.

Every one of those six was found by a sweep that steps at a ratio of one per cent, and a ratio step never lands on the five-decimal grid each rise here is rounded to. So a recorded handover is the nearest rise the sweep happened to visit to a crossing nobody had located, and nobody had asked how many crossings there are to locate.

One rung had been walked at the grid and it crosses once. The other five had not been walked at all.

What a second crossing would cost #

A band is grown outwards from a recorded handover while the counted pair holds and the settled divergence does not move, and every result in the ablation thread is a result about a band.

A rung with two crossings has two candidate handovers, so the band grown around the recorded one is a band around whichever of them the ladder sweep reached first. Every claim on that rung that names the handover would then be a claim about an arbitrary member of a set.

That is the worry the walk was written for, and it has an answer per rung rather than one answer.

What it costs to ask #

A hop length is closed-form arithmetic on the settled divergence, so a rise is one grown stem and forty lengths evaluated and sorted. Nothing is cut.

The six walks grew 1,224 rises and no cut stems at all, in 675 seconds on a quiet machine — eleven minutes and a bit, against the thirty-seven minutes two bands of sixteen and seventy rises cost when they were cut at every rise instead. Two earlier runs of the same table came in at 699 and 779 seconds, which is the spread of the machine rather than of the walk.

That ratio is the whole design. Geometry can be walked and ablation has to be sampled.

The three passes #

A coarse walk of the whole rung first, at a step chosen per rung. Then every bracket where the ordering changes refined at the grid, end to end. Then the whole stretch over which the two contact steps stay within a per cent of each other, walked rise by rise.

The third pass is the one that answers the question. Outside it a second crossing cannot sit, because the ordering would have to travel the whole separation and come back inside an interval nobody read.

Why the refinement is not a bisection #

Bisection is the obvious way to find a sign change in a dozen evaluations rather than in a hundred, and it was refused here for one reason: it assumes one crossing, and one crossing is what is being tested.

A bracket where the ordering changes is therefore walked end to end at the grid, every rise of it. That is the more expensive half of the second pass and it is the half that could have come back with an answer nobody expected.

It did, on one rung.

Five rungs cross once #

The golden 3/5, the golden 5/8, the golden 8/13, the Lucas 4/7 and the Lucas 7/11 each carry exactly one crossing, located to one step of the grid, and on each of them the whole stretch a second could sit in has been walked and is closed at both ends.

Those stretches are 135, 54, 21, 67 and 30 grid steps wide, and the walk read 137, 56, 23, 69 and 32 rises across them — two steps past the tie at each end, so that the stretch is bounded by rises where the ordering is decisive rather than by where the budget stopped.

What that is worth to a band #

On those five the band is a band around the crossing, and the offset between the rise it was centred on and the rise the crossing sits at is 0.4 to 4.1 of the band’s own steps against bands of 70 to 126 rises.

So nothing built on any of them moves. The width a band should have, the flatness of the divergence across it and every survivor counted inside it were all read on a band whose centre is now known to be within a few steps of the right rise.

That is a negative result, and it is the negative that lets the other work stand.

The rung that does not #

The Lucas 3/4 carries five crossings, at 0.043715, 0.043495, 0.043375, 0.043225 and 0.043025, each bracketed between two adjacent grid rises. They are spaced 22, 12, 15 and 20 grid steps apart, which is five sign changes inside 181 grid steps.

That rung is the coarsest on the ladder, running from 0.064 to 0.0239 and holding 4,011 rises at the grid. The five crossings occupy less than a twentieth of one per cent of it.

Five is a floor and not a count #

The stretch over which that rung’s two steps stay within a per cent of each other is 690 grid steps wide. The walk covered 181 of them, and both ends of what it covered were still tied when the refinement budget stopped it.

So the region is not closed. There is no argument here that the sixth crossing does not exist — only the observation that five is what 181 grid steps held, and that a further 509 have not been read.

Every other rung’s window is closed, which is why five is the only count on the ladder that carries a qualification.

The threshold the walk borrows #

The stretch walked rise by rise is the stretch over which the two contact steps are within 1.01 of each other, and that number was not chosen here. It is the separation rule the counter already uses to decide when it will decline to order two hops at all.

Reusing it means the walked window is exactly the region where the ordering is a property of the azimuth grid rather than of the lattice. A wider bound would have walked rises whose ordering nobody doubts; a narrower one would have stopped inside the region the question is about.

It is also the one setting in the design that could be accused of having been tuned to the answer, so it is worth saying that it was fixed before any rung was walked and is used unchanged.

The band on that rung is around none of them #

The Lucas 3/4’s band runs from 0.04342 to 0.04214 and holds three of the five crossings. The two it misses are both above its coarse end, and the coarsest sits 30 grid steps above 0.04342.

The recorded handover it was grown around is 0.04299, which is not one of the five. The nearest crossing is 0.043025, three and a half grid steps coarser.

That band is already the odd one on the ladder: sixteen rises against seventy to a hundred and twenty-six, and a control rather than an experiment on every quantity anybody has read off it.

No rung crosses none #

Worth stating separately, because it is the other way the check could have come out. Every one of the six carries at least one crossing, and every crossing is located to one grid step.

A rung with none would have meant that the ladder sweep’s recorded handover there is not a handover at all — that the ordering it saw change was an artefact of visiting two rises far enough apart for the reading to have moved between them.

None of the six is like that. The crossing is what a handover is, and all ten of them are now rises somebody has located.

What a coarse step is per rung #

The rungs are not the same size and were not walked at the same step. The Lucas 3/4 holds 4,011 grid rises and was walked coarsely at 30; the golden 8/13 holds 201 and was walked at one, which is every rise it has.

Between those the steps run 25, 12, 10 and 4. Each is chosen so that the coarse pass costs roughly the same on every rung, which is the same argument the ladder sweep makes for stepping at a ratio applied one level down.

The rung that was walked whole #

The golden 8/13 is short enough to afford at every rise, so it was walked at every rise: 201 consecutive grid rises, one crossing, and nothing unread anywhere on it.

That matters more than its own answer. It is the one rung where the count rests on having looked rather than on an argument about what could fit in the intervals between looks, and it is a check on the design rather than a result the design produced.

The rung that needed no walking #

The Lucas 7/11 had already been walked whole and was walked again here at four grid steps, which is 120 rises against 378.

The two agree exactly: one crossing, bracketed between the same two adjacent rises. That agreement is what makes the other five rungs affordable at all, and it is scored properly rather than asserted.

Walking a rung nobody needed walked is the cheapest evidence available that the cheaper design measures the same thing.

Where the handovers sit inside their rungs #

Relocated to the grid, the six sit at 40.31, 10.72, 5.38, 14.22, 32.37 and 33.33 per cent of the way down their rungs from the coarse end.

Every one of them moved and none moved by more than 1.2 points. All six remain in the coarse half, which is the assertion the position reader makes and the assertion that had only ever been checked on rises known to a sweep step.

So a claim that was made at one resolution survives being remade at a finer one. That is the least exciting outcome available and it is the one that had to be checked.

The golden 3/5 is the largest correction #

Its recorded handover is 0.04172 and its crossing is at 0.042065, which is 34.5 grid steps away — an order of magnitude further than any other rung’s, and still inside a single step of the sweep that recorded it.

That is not a contradiction. The sweep’s step at that rise is 0.00041, which is forty-one grid steps, so a discrepancy of thirty-four steps is 0.835 of one sweep step and nothing more.

It is the clearest case on the ladder that a grid step and a sweep step are different units and that a number is uninformative until it says which.

What the walk cannot say #

Whether any of these crossings means anything to a cut. That is a different instrument answering a different question, and on two of the six rungs it does not answer at all.

The Lucas 4/7 is where the two instruments overlap most usefully: 144 wrecked cuts across 86 rises with not one change of the family they keep, against a single crossing located to one grid step. The claim that the rise where the steps change places is not the rise where the survivor changes is tested there against a located number for the first time.

Two rungs where the walk is the only instrument #

The golden 3/5 and the Lucas 3/4 are the two bands on which no cut wrecks anything: 490 and 96 cut stems between them and not one wrecked stem, which is why the ablation census refuses to read either.

On those two there is no surviving family, so there is nothing for any account of the survivor to be right or wrong about. The walk is the only instrument that returns a number on them, and it returns one crossing on the first and five on the second.

So the two rungs the cutting cannot speak about are the two the walk separates most sharply, which is a coincidence worth stating and not yet worth explaining.

What five crossings do not establish #

That the geometry crosses five times. The account of the extra crossings is an account of the instrument, and it is the settled divergence being read in steps rather than continuously.

A finer reading would move those crossings and might leave one. What is established is narrower and still worth having: at the resolution everything else in this collection is read at, that rung’s contact ordering is not a function of the rise.

The refusals #

Four things are refused before a stem is grown. A rung the ladder finds no handover on. A key naming no rung at all. A step of half a grid step, and a step of one and a half.

The last two are the informative pair. A step below the grid names rises that round onto each other, so the walk would read every rise twice and report a resolution it does not have. A step that is not a whole number of grid steps lands half its rises on the grid and half between them.

Both are the kind of setting that produces a plausible table, which is why they are refused rather than warned about.

What was not read #

Of the 9,950 rises the six rungs hold at the grid, 1,224 were grown — 12.3 per cent. The rest are covered by an argument rather than by having been looked at, and the argument is that a hidden pair of crossings does not fit in the intervals between the rises that were read.

On four of the rungs that argument has a factor of 3.35 to 6.52 in hand, and the fifth was walked whole and has no unread intervals at all. On the Lucas 3/4 it has a factor of 0.05, which is to say none, and that is the rung that came back with five.

The check knew in advance which rung it could not vouch for. Being told so by the instrument before the answer arrived is the strongest thing here.

Why the whole curve is kept #

The crossing could have been reported as a bracket and nothing else, and on five rungs that is all anybody would read. The walk keeps every evaluated rise instead, and two of the results here exist only because it does.

How much of a rung the ordering is refused over is one: it is a count across every rise read, not a property of the crossing. How fast the ordering moves per grid step is the other, and it is what the argument about the unread intervals is built from.

A design that returned only the answer would have had no way to say how far short a hidden crossing falls of fitting, which is the sentence the one qualified count on the ladder rests on.

What would refute it #

A sixth crossing on the Lucas 3/4, found by walking the remaining 509 grid steps of its tie window, would not refute anything — five is already stated as a floor.

What would refute the design is a second crossing on any of the other five, found inside a window this walk reports as closed. That is a bounded claim about a bounded region and it costs one rung’s worth of walking to test.

The cheapest test of it is the golden 8/13, and it has already been made: that rung was walked whole and holds one.

What the count is not a property of #

The branch, the pair, or the coarseness of the rung. Five rungs cross once and they include both branches, pairs from 3/5 to 8/13, and rungs from 201 to 2,847 grid rises wide.

The one that crosses five times is the coarsest, but the golden 3/5 is the second coarsest at 2,847 rises and crosses once. So coarseness alone does not order the two, and what does order them is a quantity neither the branch nor the pair decides.

What is claimed #

That the six rungs the ladder finds a handover on were each walked at the five-decimal grid across the whole stretch their two contact steps are too close to be ordered over, at a cost of 1,224 grown stems and no cut stems, in 675 seconds.

That five of them carry exactly one crossing, each located to one step of the grid, with the stretch a second could sit in closed at both ends.

That the Lucas 3/4 carries at least five, at 0.043715, 0.043495, 0.043375, 0.043225 and 0.043025, that its band holds three of them and its recorded handover is none of them, and that five is a floor because 181 grid steps of a 690-step window were read.

And that no rung carries none, so every recorded handover on the ladder is a real crossing of the two contact steps rather than an artefact of where the sweep looked.