The fourth band, cut whole

A band is the stretch of rise around a rung’s handover on which the counted pair holds and the settled divergence does not move. Cutting one at every rise it holds asks a single question: does the contact family a wrecking cut leaves standing change anywhere inside it?

Three bands had been cut that way, and the answer separated cleanly along the branch. Both golden bands changed their answer somewhere inside; the one Lucas band did not change it once. Three of the four accounts written down before any of this began are wrong on a band each, and the one left standing says that whether a band’s cuts change what they keep is decided by the branch it sits on.

That account had been right three times and explained nothing. So the essay that scored it named its own executioner: the Lucas 4/7 band, 86 rises, the second band on the Lucas branch and the cheapest thing that could refute the survivor.

The prediction, as it stood #

Written before the sweep and carried unchanged: if it changes, the branch account goes the way of the other three.

That is not hedged and it is not a hypothesis about how much. A single rise anywhere on this band at which a wrecking cut kept something other than what its neighbours kept would have left the ladder with no surviving account at all, and four explanations dead on four bands.

What would have refuted it #

A change is a rise at which one offset’s wrecked cut keeps a different family from the same offset’s wrecked cut at the rise before. Nothing else counts. An offset that stops wrecking is not a change, an offset that starts wrecking is not a change, and a family that appears on one offset and not another is not a change either.

That definition is narrow on purpose. It is the same one the widest band’s nineteen changes were counted under, and loosening it here to catch something would have been fitting the test to the band.

A test named before the result #

The value of this band is almost entirely in the order the two things happened in. The prediction was written when the band was uncut, in an essay that could not have known the answer, and the sweep was run against it rather than the other way about.

An account rewritten after the band that refuted it is not an account, and the four predicates scored across this ladder are carried unchanged for that reason. It matters here because the branch account is the one with no mechanism behind it: an explanation that cannot say why has only its record to stand on, and a record assembled after the fact is not a record.

Why this band was the one #

Because it is the second Lucas band, and the surviving account had been carried by exactly one.

Two golden bands agreeing with each other is a replication; one Lucas band disagreeing with both is a single observation wearing the weight of half the argument. Everything a reader could reasonably doubt about the branch account sat on the 7/11, and the honest way to doubt it was to grow another band on the same branch and cut it.

The two bands that could not have been the test #

There are two cheaper bands on the ladder and neither would have answered anything. Nothing behind the front wrecks at the coarse end of the ladder, and a band with no wrecked cut has no surviving family to change. What they are worth is a separate measurement and it is worth more than it looks.

So the Lucas 4/7 was not the cheapest band available. It was the cheapest band that could have said no.

The sweep #

Eighty-six rises, from 0.02366 at the coarse end to 0.01997 at the fine, every offset the front reaches at each of them, and 774 cut stems.

The band is built the way every band here is built: outwards from the handover while the counted pair holds at 4/7 and the settled divergence stays within a twentieth of a degree of its value there. Its rises are spaced by a ratio rather than by a fixed step, which is the same argument the ladder itself is swept with — a fixed step is a finer sample at one end of a rung than at the other.

Each rise is cut beside a control: the same stem grown to the same length with no organ removed. That is what makes a displacement a displacement rather than a divergence, and it is what lets the counted pair and the settled divergence be read at every rise rather than assumed from the ends.

The result #

Nine offsets are tried at every rise. Two of them wreck: offsets 4 and 5, at 144 cells of the 774. Every one of those wrecked cuts that leaves a family standing leaves the same one, 4, at every rise of the band.

Zero changes. The band the branch account most needed did what the branch account predicted, and the prediction was made in the open with the band uncut.

One hundred and forty-four and one hundred and thirty-eight #

The two counts in the paragraph above do not agree, and the disagreement is the honest part.

One hundred and forty-four cells are recorded as wrecked. Following the two offsets down the band gives 83 for offset 4 and 55 for offset 5, which is 138. The missing six are wrecked entries whose survivor is null: cuts the sweep reached and which then recovered, leaving no rigid hop and nothing to name. A recovery is not a wreck and a table that counts it as one is counting a stem that healed.

Six of 144 is four per cent, and on a band whose whole result is nothing changes that four per cent is exactly where a change could have hidden. It did not: the six sit together.

The distinction is worth keeping rather than tidying away, because the two numbers answer different questions. One hundred and forty-four is how many cuts the sweep found something at; 138 is how many left a family to name. A reading that quotes the first as though it were the second overstates the evidence by six cells, and this band’s finding is a claim about every cell that holds a family.

Three rises where nothing wrecks #

All six fall at three consecutive rises near the fine end — 0.02017, 0.02013 and 0.02009 — and both offsets recover at all three of them.

So this band has a three-rise window in which not one of its nine offsets wrecks anything, and three rises further down the wrecking resumes on both offsets with the same family standing. Whatever closes the hole there closes it for both cuts at once and then stops.

That is the only structure in the table, and it is a structure in which cuts wreck rather than in what they keep — which is the distinction the ladder keeps returning to.

Two offsets of nine #

Offset 4 wrecks at 83 of the 86 rises and offset 5 at 55.

Offset 5 does not wreck at all until 28 rises into the band, and then wrecks continuously until the three-rise window. So the two rows of this table are not two versions of the same row: one is nearly solid and one has a long empty coarse end, and they agree in every cell where both hold something.

Neither of them wrecks everywhere #

That makes this the second band on the ladder with no offset that wrecks at every one of its rises. The golden 5/8 is the other; the widest golden band has one such offset and the Lucas 7/11 has three.

Which offsets wreck being a function of the rise rather than of the lattice has now replicated on every band cut whole, and with four bands it has stopped being a replication and become a range with two ends to it.

The family kept is the smaller counted number #

4 is the smaller member of this band’s own counted pair. Nothing here keeps 7, and nothing here keeps anything off the pair.

The Lucas 7/11 band behaves the same way: five offsets, 487 wrecked cuts, and the 7 at every one of them. Two Lucas bands, two smaller counted numbers, and no exceptions on either.

What the golden bands keep instead #

The widest golden band chooses between 8 and 4, and 4 is half of 8. The golden 5/8 chooses between 5 and 20, and 20 is four times 5. In both cases the default is the smaller counted number and the change is to something off the pair.

So a golden band’s changes are departures from the value a Lucas band never leaves, and the rule that seemed to govern which departure it makes survived one case and died on the second.

Which makes the default the interesting part #

Every band cut whole keeps the smaller of its counted pair almost everywhere. What separates the branches is not what they keep but whether they ever stop keeping it.

That is a smaller finding than the branch decides the survivor and it is the one four bands support. The survivor is the same kind of thing on all four; what varies is whether the band holds it at every rise.

The handover, and what does not happen at it #

This band’s handover sits at 0.02251, 29 per cent of the way along it, and the ordering of the two contact steps crosses there exactly once: the 4-hop is shorter above it and the 7-hop below.

Nothing in the table marks the crossing. The same offsets wreck either side of it and the same family stands. That is the claim the whole thread rests on holding for a fourth time, and holding here in the weakest way available, because a band that changes nowhere cannot fail a claim about where changes are not.

Saying so is not a formality. Two of the four bands cut whole have no change anywhere in them, so two of the four cannot test the claim at all, and a reader counting four confirmations is counting two measurements and two silences. The claim has been tested where there was something to test it with — nineteen located changes on one band and two on another — and this band adds a rise at which the ordering reverses under a table that does not move.

What a control at every rise is for #

The control is not there to be compared with the cut stem’s angle. It is there to establish that the lattice being cut at rise 40 is the same lattice that was cut at rise 1.

Its counted pair is read at every rise and comes back 4/7 at every rise; its settled divergence is read at every rise and moves less than the band’s own tolerance. Without that the sweep would be a comparison of 86 slightly different lattices, and a uniform answer across them would say something much weaker than a uniform answer across one.

The offset that arrives three rises below it #

Offset 5 begins wrecking at the twenty-eighth rise, which is three rises below the handover at index twenty-five.

That is close enough to be worth writing down and far too close to a coincidence to be worth anything else. Three rises is 3.5 per cent of this band; an arrival placed at random lands within three rises of the handover about one time in twelve. It is recorded because a later band with the same near-coincidence would make it two, and nobody would go back to look.

The band held still #

The counted pair is 4/7 at every one of the 86 rises. The settled divergence moves by 0.039 degrees across the whole band, which is inside the twentieth of a degree the band was grown to.

That is what makes this a sweep of one lattice rather than a comparison of two. A band whose divergence slid would be a band whose ends are different objects, and the result would be about the sliding.

What nine rises would have said #

The coarse design visits ten rises of this band at a step of eleven, and reports no change, which is right.

It is right for the reason a coarse design is usually right — there was nothing to miss. Its record across the ladder is a separate measurement with one failure in it, and this band adds a correct negative rather than a test.

What a fourth band buys #

One thing, and it is the thing that was asked for: the surviving account is no longer carried by a single band on one side.

Two golden bands change and two Lucas bands do not. Every account that ordered the bands by the counted pair, by the number of wrecking offsets or by the span puts at least one of these four on the wrong side, and each of those accounts was refuted on a band cut at every rise it holds rather than on a sample.

An account wrong on a whole band is wrong for good. That is the one property of this instrument worth more than its resolution: a finer sweep can add changes to a band, and it cannot remove the ones already located, so an account that predicted the wrong side of a band stays refuted whatever anybody measures next.

And what it does not #

It does not make the branch account true, and the reason is not modesty.

A branch is a seed angle and a sequence and every quantity that follows from them, including the divergence its stems settle to and the counted pairs its rungs carry. Four bands sorted by branch are four bands sorted by all of that at once. Nothing here separates the branch from what a branch decides, and nothing here proposes a mechanism by which a seed angle would reach a cut organ.

The shape of the confound is worth stating plainly, because it is the reason a fifth band on either branch would buy so little. Adding golden bands raises the count on one side of a comparison whose two sides differ in everything. What would break it is a lattice that carries one branch’s divergence and the other’s counted pair, and no rung of this ladder offers one.

Two Lucas bands is two #

The uniform result now covers 210 Lucas rises and 625 wrecked cuts with a family standing, and every one of them keeps the smaller counted number of its own band.

That is a much stronger negative than the same statement made on one band, and it is still a negative over two bands of one branch. A third would be worth more than a fifth golden one, and there is not a third.

It is also a negative whose strength comes from resolution rather than from volume. Ten sampled rises finding nothing on this band would have been worth very little, because the feature it was looking for is one rise wide on the band where such features exist. Eighty-six rises finding nothing is what makes the golden bands’ speckle a property of the golden bands.

There is no fifth band #

Six rungs of this ladder carry a handover and all six now have a band grown on them and cut at every rise it holds. Two of the six wreck nothing. Four can answer, and all four have answered.

The next thing that would say more is a rung the ladder does not currently hold, which is a different measurement with a different cost. What the six together say is as far as this instrument reaches.

What the picture at the top shows #

Two rows and 86 columns, coarse on the left. The upper row is offset 4 and is filled almost end to end; the lower is offset 5 and is empty for its first third. Both have the same three-cell gap near the right.

Every filled cell is the same tone, and that is the finding. On the widest golden band the same picture has two tones arranged in islands.

The one line #

Eighty-six rises, 774 cut stems, 144 wrecked cuts of which six recovered, and one family kept in every cell that holds one: the band that would have refuted the branch account did not refute it.

Which leaves an account that is right on every band that can test it and explains nothing at all — the same place the third band left it, with twice as much behind it and no more reason to believe it.