Rung interior — the series

Which offsets give short hops, at a rise of 0.013. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.

Where a handover sits

Inside every rung there is a rise at which the two contact steps change places, so that the shorter hop belongs to the other family below it. Six of the eight rungs on this ladder have one, each has exactly one, and every one of them sits in the coarse half.

part 1 · cylinder

Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose.

Two lines that cross once

The divergence at which a lattice's two contact steps are exactly equal is a curve across each rung, computable from the geometry with nothing grown. The rule's own settled divergence is a second, shallower curve, and where they cross is where the step ordering changes hands.

part 2 · cylinder

$A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.$

A band that holds the angle still

Around every handover the settled divergence has a shallow floor, so a run of rises either side of it share a divergence to a twentieth of a degree while their two contact steps change places. That is a matched pair with one quantity varying, and it is the design the ablation thread had no way to state.

part 3 · cylinder

Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.

Four crossings nobody visited

Six rungs of the ladder carry a handover and two of them had a band built on them. The other four are here: 70, 126, 16 and 124 rises wide, found by sweeping at a ratio rather than at a fixed step in the rise, which is why the fine ones had been stepped over.

part 4 · cylinder

What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.

A band that moves nothing

One of the six bands holds the counted pair, holds the divergence, and does not move the ordering: its two contact steps stay within four parts in a thousand of each other across the whole of it, so the ordering changes hands three times and neither end has one worth the name.

part 5 · cylinder

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.

How wide a band should be

A band ends where the divergence has slid a twentieth of a degree, so its width should follow from how fast the divergence slides. Predicted from the rung's slope that is wrong by factors of 0.20 to 5.92; predicted from a stationary point it is 0.41 to 1.08, and the outlier is the rung that has no stationary point.

part 6 · cylinder

The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism.

The ordering on six bands

A hundred and fourteen wrecked cuts across four bands, and at every offset of every one of them the family left standing is the same immediately above the handover and immediately below it. Where the answer does change — on the widest band, at three offsets — it changes somewhere else.

part 7 · cylinder

Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it.

Every rise of a band

A band is cut at nine rises because the quantity it was built to test is a constant, and a constant is checked at the ends and at the crossing. On the widest band that quantity turned out not to be constant, which makes nine the wrong number. This is all hundred and twenty-six.

part 8 · cylinder

The alternation is not a period

Nine sampled rises gave 8, 4, 8, 4 at one offset of one band, and a period was the obvious thing to look for. At full resolution it is thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 — and a fitted period buys exactly nothing.

part 9 · cylinder

Offset 7 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 126 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark one islands — runs of 2 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.

Three offsets, three crossings

The claim the band design rests on is that the survivor does not change where the two contact steps change places. It holds at full resolution: nineteen changes and not one at the handover. Where they are is three different rises, eight, nineteen and twenty-nine below it.

part 10 · cylinder

Offset 4 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 98 of them and keeps the 8-family throughout. The ticks below mark no island: the answer changes once and stays changed. The handover is the taller rule and the change of answer is nowhere near it.

The offsets that never change

Three of the six offsets that wreck anywhere on the band keep the same family at every rise they wreck at — 98, 22 and 81 rises of the 126. And which offsets wreck at all is a function of the rise, which no reading of a band had drawn.

part 11 · cylinder

Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.

A band with nothing inside it

Five offsets wreck on the Lucas 7/11 band and every one of them keeps the same family at every rise it wrecks at. There are no islands, no transition region and no period to look for, which is what makes the picture from the other band a picture of that band.

part 12 · cylinder

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.

The second band, cut whole

One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it.

part 13 · cylinder

The wrecking set moves again

Which offsets wreck a stem was assumed to be a property of the lattice. On one band it turned out to be a property of the lattice and the rise, changing on nearly a fifth of that band's steps. On the second band it changes more, and one offset's wrecking is broken into five separate stretches.

part 14 · cylinder

What nine rises find on each band, against what all of them find. Two bars per band: the changes of surviving family a nine-rise design finds, and the changes the full sweep finds. On the Lucas band the two agree exactly, at none and none. On the golden band they agree that something changes and disagree about how much — 5 against 19 — because one step of that design is 16 rises and the band carries features one to three rises wide. A sample was never wrong about whether; it was wrong about how many.

When nine rises are enough

A coarse design was shown to be misleading on one band and it has been criticised on that ground ever since. On the second band it is exactly right, and the difference between the two cases is a property of the band rather than of the design — which is the awkward part.

part 15 · cylinder

All three bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all.

The third band, cut whole

Two bands cut at every rise disagreed about whether the family a cut keeps ever changes, and three explanations were available for a difference between two things. The cheapest third band settles which of them survives, and it settles it against the account nobody was betting on.

part 16 · cylinder

The 17 rows of the exchange table, gathered by the lag they kept. One bar per surviving lag, its length the number of rows the table holds at that lag, with the hop that lag's stems keep written beside it. The hop is nearly constant inside a lag, so a correction fitted over 17 rows is fitted over four hops — which is the denominator that matters and is much smaller than the row count suggests. Adding a lag to the table is worth more than adding rows at a lag already in it.

A family that is a multiple

When a wrecked cut keeps a family that is not one of the lattice's counted pair, the first case looked like a rule: it was half of one of them. The second case is four times the other, which makes the rule a coincidence and leaves a weaker statement that is probably the true one.

part 17 · cylinder

The Lucas 4/7 band cut at every one of its 86 rises. One row per wrecking offset on each band, one cell per rise, coarse on the left, with each band's width in proportion to the rises it holds. A pale cell is a rise at which that offset's cut recovers and has no survivor to report; a dark cell keeps one of the band's own counted pair and a warm cell keeps a family off it, so a change of answer is where the tone changes. The Lucas 4/7 is the test the branch account most needed: 86 rises, 774 stems, 144 wrecks across two offsets and not one change of the surviving family. Across the one bands drawn, 144 cuts wreck of 774 grown. The dashed rule on each row is that band's own handover, where its two contact steps change places.

The fourth band, cut whole

Three bands cut at every rise left one account of which bands change their answer standing, and the account was the one nobody had a reason to prefer. The band that would have killed it has now been cut, and it did not kill it.

part 17 · cylinder

Four accounts of which bands speckle, scored on the three cut whole. Each candidate explanation of why one band's cuts change the family they keep and another's do not, against what the three bands cut whole actually do. A tick is an account that puts that band on the side the sweep does. The branch the band sits on is right on all three; the size of the counted pair, the number of wrecking offsets and how much of its rung the band spans are each wrong on two. Three bands can eliminate and cannot confirm, and this eliminates three of the four.

The branch is what is left

Four accounts of why one band's cuts change what they keep and another's do not were written down before a third band was cut. Three of them are now wrong on a band each, and the survivor is the one with no mechanism behind it.

part 18 · cylinder

The two bands that answer nothing, 586 stems and no wreck anywhere. The g35 and L34 bands cut at every rise, with one row per offset tried and one cell per rise, coarse on the left. A pale cell is a cut whose stem recovers and a mid cell is one that neither recovers nor leaves an orbit to count; the warm tone means a cut that wrecks, and it appears on the reference row below, which is offset 4 of the Lucas 4/7 wrecking at 83 of its 86 rises. g35 grew 490 stems across 70 rises at seven offsets and L34 grew 96 stems across 16 rises at six offsets, and wrecked none of them, so this is a zero that was measured rather than a band nobody cut. The dashed rule on each block is that band's own handover, where its two contact steps change places.

Two bands that wreck nothing

The census that reads a band refuses two of the six, and the refusal is correct: a band with no wrecked cut has no surviving family, so it has no answer to change. Cutting them anyway turns a refusal into a measurement, and the measurement has a third tone in it that the census cannot see.

part 18 · cylinder

What nine rises find on each band, against what the whole band holds. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, and asks whether the family a cut keeps changes anywhere. On the golden 8/13 it finds five of nineteen changes and on the Lucas 7/11 it finds none of none, so it had never been wrong about whether anything changes. On the golden 5/8 there are two changes and it finds neither, both of them at single rises with the offset recovering on either side. Its record on that question is now 2 of 3.

Nine rises were not enough

A coarse sample of a band had never been wrong about whether anything changes inside it, and that record was the argument for trusting a negative from it. The third band cut whole makes it two of three, and the missed feature is one rise wide.

part 19 · cylinder

Four accounts scored on the four bands of six that can score them. One column per band of the ladder and one row per candidate account of why a band's wrecking cuts change the family they leave standing. Each cell is what that account predicts of that band, in plain type where the band agrees with it and pale where it does not, and the header of each column is what the band actually does. The two bands with no wrecked cut are shown silent, because a band with no surviving family has no answer to change; over the four that can answer, the branch the band sits on is right 4 times of 4 and the other three are wrong twice each. A warm block marks each cell where the account and the band disagree.

Six bands, one table

Every rung of this ladder that carries a handover now has a band grown on it and cut at every rise it holds, and four accounts of which bands change their answer are scored on all six at once. The survivor is right on every band that can test it, and the same table read one cell differently kills it.

part 19 · cylinder

A change with nowhere to be

The claim this whole thread rests on is that a survivor does not change where the two contact steps change places. Nineteen located changes never put one there. The twentieth is flagged at a handover, and it is flagged because the offset stops wrecking for thirty-four rises.

part 20 · cylinder

What nine rises find on six bands, against what those bands hold. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, plus both ends and the handover, and asks whether the family a cut keeps changes anywhere on it. Scored on six bands it is right about whether on five and wrong on the 5/8, where it finds none of 2. On the one band where it finds anything it finds 5 changes of 19, so its record is a record about whether and never about how many. The figure under each band's name is that design's step on it, in rises.

The coarse design scored

Every claim this thread has made about an uncut band rests on a sample of nine rises, and its record was the argument for trusting it. Six whole bands close that record, and two of its six correct verdicts are correct only because there was nothing on those bands to find.

part 20 · cylinder

How broken the wrecking is on four bands, from 2.43 rises a stretch to 126. Every offset that wrecks on every band that wrecks, placed on a logarithmic axis of rises wrecked per separate stretch — the smaller the number, the more broken the wrecking. A filled mark is an offset that wrecks at every rise of its band and an open one comes and goes. The range runs from 2.43 on g58 offset 3 to 126 on g813, a factor of 52, and two of the four bands have no offset that wrecks everywhere at all. A band with no offset that wrecks at every rise is said so at the right.

A wrecking set with a range

Which offsets wreck a stem was taken to be a property of the lattice, and every band cut whole has found it to be a property of the rise instead. Six bands turn that replication into a measured range, and the range is a factor of fifty-one.

part 21 · cylinder