Two bands that wreck nothing

Four of this ladder’s six bands have been cut at every rise they hold and have answered the question the design was built for. The other two will not answer it, and the machinery that reads a band says so by refusing to read them.

That refusal is correct. The question is which contact family a wrecking cut leaves standing, and on these two bands no cut wrecks anything at all: 586 stems grown across 86 rises and thirteen distinct offsets, and not one of them leaves a rigid hop behind. A band with no wrecked cut has no surviving family, so it did not change its answer is true of it in the way that a statement about the unicorn in the room is true.

But a refusal is not a measurement, and the two are easy to confuse in a table. So both bands were cut anyway, at every rise and every offset, and the zero was written down as a number rather than left as a blank.

Why the census refuses them #

The reader that assembles a band’s table asserts two things before it will report. That the band is at least sixty rises wide, and that somewhere on it a cut wrecks, so that there is a survivor to follow.

The golden 3/5 band fails the second. The Lucas 3/4 band fails both, and the first failure is independent: sixteen rises is not a stretch anyone could look for an island in, because the shortest feature the other bands hold is one rise wide and the widest is three.

What a wreck is, exactly #

The word carries the whole result, so it is worth pinning down. A cut wrecks when the stem never returns to the lattice it was on and settles instead into a state in which exactly one lattice hop stays rigid — the one the ablation thread has been counting since it was found that the rigid hop is nearly always one of the two the counter reads.

The family that hop belongs to is the survivor, and it is the only quantity any of the four accounts predicts. A cut that produces no rigid hop produces no survivor, and there is nothing for an account to be right about.

Which is the right thing for it to say #

Both assertions protect a claim the file is actually making. A table of surviving families with no surviving families in it is not a sparse table; it is a table of a different thing, and every statistic computed over it — how many offsets change, how many islands, how wide they are — would be a statistic over an empty set reported as a zero.

The failure that produces is not an error message. It is a row in a comparison that looks like a band which was measured and found uneventful.

What was cut #

Every rise of both bands, every offset the front reaches at each of them, with a control beside each cut. 586 stems, thirty-seven minutes.

The golden 3/5 band holds 70 rises from 0.04474 to 0.03898 and tries seven offsets at each of them, which is 490 cut stems. The Lucas 3/4 holds sixteen from 0.04342 to 0.04214 and tries six, which is 96.

Why a coarse stem is so hard to wreck #

The bands here sit at the coarse end of the ladder, where a rung carries few enough organs per turn that a single removal leaves a hole with a great deal of room around it.

That is the same regime in which a grown pattern changes its counts exactly where the static ladder says it should, and for the same reason: the placement rule has little to be confused by. The prediction attached to it — that the coarse end wrecks nothing — is not new here. What is new is that it has been cut at every rise rather than assumed.

The zero #

Not one of the 586 wrecks. There is no survivor anywhere on either band, no island, no change, and nothing for any of the four accounts the ladder was built to separate to be right or wrong about.

Stated that way it sounds like the refusal restated at greater cost. It is not, and the reason is that the cuts did not all do the same thing.

A tried offset has three states, not two #

A cut either wrecks or it does not — that is how every band on this ladder has been read, and on the four that wreck it is enough, because the interesting cells are the wrecking ones.

On these two bands it is not enough. A cut that does not wreck can recover, meaning the stem closes the hole and carries on as though nothing had been taken; or it can hold, meaning it neither heals nor leaves a countable orbit behind. Something happens at a held cut and the something is not a wreck.

What a recovery is #

A recovery is the placement rule doing what it has been measured doing since the correlation that priced it: the next organ lands where the missing one would have pushed it, the one after lands closer to where it belongs, and within a few plastochrons the stem is back on the lattice it was on.

Five hundred and two of the 586 cuts do that. At the coarse end of this ladder a single removal is a small perturbation of a lattice with few enough organs per turn that the hole closes almost immediately.

What a held cut is #

The other 84 do not recover and do not wreck. The stem neither returns to its lattice nor settles into one rigid hop that can be named and counted.

That is a third outcome and it is the one the four-band reading has no cell for. The census that produces a survivor per wrecked cut is looking for the rigid hop; a stem that fails to produce one is, to that machinery, a stem that did not wreck, and it lands in the same undifferentiated nothing happened as a stem that healed perfectly.

Which is what makes the zero a measurement #

Nothing wrecks here is a weak sentence. It is compatible with the cuts having no effect at all, which is what a reader would assume of a band the ladder’s own census will not read.

Eighty-four cuts do something and none of the somethings is a wreck is a much stronger one, and only these two bands can say it. It reports a lattice that responds to a removal without producing the object the whole ablation thread is about, and that is a fact about the coarse end of the ladder rather than an absence of data about it.

The control at every rise #

Each cut sits beside a stem grown to the same length with no organ removed, which is what makes a displacement a displacement.

It also carries the check that a band is one lattice rather than a range of them. The counted pair is read at every rise on both bands and holds at 3/4 and 3/5 respectively; the settled divergence is read at every rise and stays inside the tolerance the band was grown to. Without that, 586 uniform outcomes would be 586 outcomes on 86 slightly different objects, and uniformity across those says much less.

Thirty-two of ninety-six #

On the Lucas 3/4 band the split is completely clean. Offsets 2 and 3 hold at every one of the sixteen rises, and offsets 1, 4, 5 and 6 recover at every one.

Nothing about it varies down the band. Two of the six offsets do one thing everywhere and the other four do the other thing everywhere, which is 32 held cuts and 64 recoveries with no structure inside either.

Fifty-two of four hundred and ninety #

The golden 3/5 band is not clean, and the place where it is not clean is the only structure either band holds.

Six of its seven offsets recover at all 70 rises. Offset 3 recovers for the first eighteen rises from the coarse end and then holds for the remaining 52, in one transition, between 0.04325 and 0.04316.

So the third band’s third offset changes state exactly once, and the state it changes into is the one that has no name in the census.

Which is a boundary and not a wreck #

It is worth being careful about what that transition is. It is not a change of survivor, because there is no survivor on either side of it. It is a change in whether the stem recovers, which is a different quantity measured on a different set of cells.

The distinction matters because the ladder’s whole result is about the first quantity. A reader who saw a tone change in this picture and read it as a change of answer would be reading the one band on the ladder where the two are guaranteed not to be the same thing.

Where that transition sits #

Seventeen rises above the handover, on the coarse side of it, and thirteen above the rise at which the band’s two contact steps change places.

The handover this band was grown around is recorded at 0.04172; inside the band’s own rises the 3-hop is shorter above 0.04214 and the 5-hop below 0.04205. Neither of those is where offset 3 stops recovering, and there is no reason yet to expect them to be.

The reference row is part of the drawing #

A picture of a zero has a problem that a picture of a result does not: an empty row and a row nobody cut look identical, and the reader has no way to tell which is on the page.

So the grid is drawn from the offsets that were tried rather than from the offsets that wrecked — every cell is a stem that was grown — and a row of a band that does wreck is drawn beneath at the same cell size. The tone that is missing from these two bands is then visible on the page rather than merely absent from it.

Two silences that are not the same silence #

Both bands wreck nothing and they are not the same object.

The narrow one is a control in the strict sense: its two contact steps stay within four parts in a thousand of each other across the whole of it, the ordering changes hands three times inside it, and neither end is separated enough for an ordering to mean anything. Its settled divergence moves by 0.070 degrees end to end and sits within 0.039 of its value at the handover.

The wide one is an ordinary band that happens to be quiet: 70 rises, one crossing, and a settled divergence that moves by 0.039 degrees across the whole of it. It is a band the design would have accepted in every respect but the one that matters.

The narrow band overlaps the wide one #

Both sit at the coarse end of the ladder, and the Lucas 3/4’s rises lie entirely inside the golden 3/5’s range.

They are still different lattices, and the size of the difference is worth having: one settles to a divergence near 100.47 degrees and the other near 140.84. Two rungs can be swept at the same rises and share nothing else, which is the reason a rise is not a coordinate a lattice can be named by.

Nine rises on a band of sixteen #

The coarse design visits ten of the Lucas 3/4’s rises at a step of two, and nine of the golden 3/5’s at a step of nine. Both report no change, and both are right.

They are right the way an instrument pointed at a wall is right about the wall. Neither adds anything to the design’s record, because a design cannot be credited with a correct negative on a band where a positive was impossible.

What the two bands cost, against what they buy #

Thirty-seven minutes and 586 stems, which is a tenth of what the whole ladder has cost and the only part of it spent on bands nothing was expected from.

What it buys is one distinction, on 84 cells. That is a poor exchange by volume and a good one by what the distinction does: without it the two rows of the ladder’s table are blanks, and a blank in a comparison of six bands is read as a band that was measured and did nothing. With it they are rows carrying a number and a shape, and the difference between those two readings is what the six-band score turns on.

What this does not support #

It does not support the claim that these bands would never wreck. The offsets tried at each rise are the ones the front reaches, and how deep the front goes is itself a function of the rise — so a band further down either rung would be cut at more offsets, and there is no result here about what those would do.

Nor does it support any reading of the held cuts as a kind of near-wreck. Nothing here measures how close a held stem came to producing a rigid hop, and the two outcomes were separated by whether a survivor could be named rather than by anything continuous.

And it does not support a third state on the four loud bands. The reader those four come through does not carry whether a cut recovered, so on them a cut that holds and a cut that heals are the same cell, and there is no way from here to say how many of either there are. The band cut before this pair has 630 cells that are not wrecks and nothing here divides them.

And what it rules out #

It rules out reading these two rows of the ladder as evidence for or against any account of which bands change their answer. A band that cannot produce a survivor cannot decline to change one, and scoring it as though it had is what turns a silence into a no.

It also rules out treating the ladder’s six rungs as an experiment of size six. Four of them can separate the accounts and two cannot, which is a property of the coarse end of the ladder rather than an accident of how these two were sampled.

What a fourth state would have looked like #

Three outcomes were found and the reading is only as good as the list. A cut that wrecked and then recovered many plastochrons later would sit in the held column here, because the sweep reads a fixed length of stem and asks what the end of it looks like.

Nothing in these 586 cells suggests one. Every held cell on the narrow band holds from the first rise to the last, and every held cell on the wide one belongs to a single offset in a single stretch, which is the pattern of a state a lattice is in rather than of a state it is passing through. But the sweep is not built to see a slow recovery, and a claim that none happened would be a claim about a measurement that was not made.

What replicates from the loud bands #

One thing, in a weakened form. Which offsets do what remains a function of the rise: on the golden 3/5 the set of offsets that hold at one rise differs from the set eighteen rises coarser.

That is the same shape as the result the wrecking set keeps producing on the bands that wreck, measured on a different outcome. Whether it is the same phenomenon is not something these two bands can say, because the quantity is only defined here on cells the other bands do not have.

What the picture at the top shows #

Two grids and a reference row. Every cell of both grids is a stem that was grown and a cut that was made; the pale cells recovered and the mid cells held; the warm tone that would mean a wreck appears only in the row underneath, which belongs to a different band.

The one line #

Five hundred and eighty-six cuts, 84 of which do something and 502 of which do nothing, and not one wreck between them: the two bands the census refuses were refused correctly, and the zero it refused to record is a measurement with a shape.