Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
14 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: Where a handover sits · The organ that was taken away · Counting the spirals.

The ablation census is a list of lattices, each cut at every offset out to the front and two past it, and the wrecked cuts are the rows every result in this thread is quantified over. The set of wrecked cuts is therefore the denominator of a great deal of arithmetic.

It was taken to be a property of the lattice. It is not: it is a property of the lattice and the rise, and both bands cut whole say so.

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.
Fig. 1 Which offsets wreck at each rise of the Lucas band, with a mark wherever a single removal wrecks the stem.

What a wrecked cut is

Remove one organ from a grown stem, continue the run, and compare it against a control that shares the history below the hole. Either the run recovers — the organs placed after the hole return to the arrangement the control has — or it does not, and the stem carries a permanent displacement.

The second is a wreck. Which offsets produce one is the first thing the census measures and the thing everything else is conditioned on — the surviving family, the period the damage falls into, the exchange and its size are all read off wrecked cuts and off nothing else.

Take away the organ twelve places back, and the next one goes into the hole. The last 26 organs of a stem at a rise of 0.008, unrolled. The open circle is the organ removed — twelve places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 139.0° apart, against a local spacing of 32°, and the vacancy itself is 146.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 2 One cut: the removed organ, the organs placed after it, and the control it is compared against.

Where the assumption came from

It came from the census’s own shape. Ten lattices, each cut at every offset, each producing a list of wrecking offsets — and a list attached to a lattice reads like a property of it. The same reading gave the collection its rule for which family a cut leaves standing, which is stated over the offset and the counted pair and scored on the same rows.

Nothing in the design ever said the list would be stable in the rise, because the design never varied the rise while holding everything else. Each of the census’s ten lattices sits at its own rise on its own rung, so a difference between two of them is a difference in several things at once.

Every stem that never repaired, and the lag it kept. The 19 offsets across six lattices at which a single removal leaves a stem that never returns to its divergence. For each one: which organ was removed, the period of the block of angles the stem settles into, the lag whose hop survived the cut unchanged, and how many whole turns the stem gains over one period of that lag. The block and the surviving lag are the same number in every row. The marked row is the one whose survivor is not a parastichy number of the lattice that was cut — a hop 6.8 times the length of a contact hop, which no census would report and which the rule held rigid all the same.
Fig. 3 The census as a grid: ten lattices down, offsets across, and which cuts wreck.

What a band gives that the census does not

A band is a hundred-odd lattices that differ in the rise and in nothing else that has been left free. The counted pair is held, the settled divergence is held to a twentieth of a degree, and the rise moves by two parts in a thousand a step.

So a band is the experiment the census could not run. Anything that changes across it changes because of the rise, and the wrecking set changes across both of them.

That is the same argument the band around a handover was built on, applied to a quantity the band was not built to test. A design that holds two things and moves a third can be read for any fourth thing that happens to be recorded.

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.
Fig. 4 The two quantities a band holds while the rise moves, which is what makes a change across it attributable.

The first band’s reading

On the golden 8/13 band, six offsets wreck somewhere and none of them wrecks everywhere in the same way. Offset 7 wrecks at all 126 rises, offset 8 at 123, offset 6 at 110, offset 4 at 98, offset 9 at 81 and offset 5 at only 22 of them.

The set changes on 23 of that band’s 125 steps. So two censuses taken at two rises of one band, with the same pair and the same divergence, are censuses of different sizes about one in five times.

Offset 5 is the extreme case there: it has no answer at 104 of the band’s 126 rises and a clean one at the other 22. A census that happened to sit at one of those 22 would report a five-offset lattice where a census two rises away reports four.

Which offsets wreck across the golden 8/13 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 22 of its 126 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.
Fig. 5 The same reading on the golden band, where the set changes on nearly a fifth of the steps.

The second band’s, which is stronger

On the Lucas 7/11 band, five offsets wreck. Offsets 5, 6 and 7 wreck at all 124 rises. Offset 8 wrecks at 88 of them in one run that begins 36 rises down from the coarse end. And offset 4 wrecks at 27 of the 124, in five separate stretches of 11, 2, 1, 6 and 7 rises, with gaps of 2, 52, 20, 2 and 21 between them.

Five stretches is more than anything on the golden band, where no offset’s wrecking broke into more than three. So the effect replicates and is larger on the second case than on the first, which is the direction a replication least often goes.

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.
Fig. 6 Both bands at full resolution. The pale cells are cuts that recover, and their pattern is the subject here.

An offset that comes and goes

Offset 4 is worth looking at on its own. It wrecks for eleven consecutive rises, then recovers for two, then wrecks for two, then recovers for fifty-two, then wrecks once, then recovers for twenty, then wrecks for six, recovers for two, wrecks for seven, and recovers for the last twenty-one.

Read as a list of numbers that is not obviously anything. Read as a picture it is a row that is mostly empty with four clumps in it, and the clumps are not evenly spaced and do not fit a period any more than the other band’s islands do.

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.
Fig. 7 An offset’s whole track across a band, drawn as runs rather than as a list.

Which is the same shape as the other finding

That is worth pausing on. The golden band’s changes of surviving family are short islands with uneven gaps that fit no period, and the Lucas band’s wrecking of offset 4 is short stretches with uneven gaps that fit no period.

The two are different quantities on different bands and they have the same texture. Whether that is a fact or a coincidence is not something two rows can say, and it is the kind of thing worth writing down so that a third band can be asked about it.

A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 76 per cent against 76 for no period at all, a gain of 0 points over 123 rises, so the alternation the coarse design reported is not a period being sampled badly.
Fig. 8 Every candidate period scored against the other band’s speckle, with the score for no period drawn as a rule.

Why this matters to the arithmetic

Nearly every claim in the ablation thread is of the form at every offset that wrecks, and the set that quantifies over is decided by the rise. So two statements made at two rises are statements about different sets.

That is not a defect in any particular claim. It is a caveat that has to travel with all of them, and the reason to state it here is that nothing had drawn the set moving before the bands were cut whole.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 5 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.
Fig. 9 Which offsets recover and which wreck at one lattice, which is the reading a census row is.

What it does to the census’s size

The census holds thirty wrecked cuts across ten lattices. Had those ten lattices been placed a few parts in a thousand away in the rise, it would hold a different number.

How different is now measurable rather than guessable. On the golden band the number of wrecking offsets ranges from three to six across its rises; on the Lucas band from three to five. So a census of ten lattices could plausibly have come back with twenty-four rows or with thirty-six.

Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.
Fig. 10 How many offsets wreck at each lattice of the census, which is the number that moves with the rise.

And what it does not do

It does not make the census’s results unstable. The exchange’s orientation is right on every row it has; the correction is right on every row; the periodicity classification splits the rows the same way it did.

Those are claims about the rows that are there, and adding or removing rows of the same kind does not threaten them. What moves is the denominator of any claim that counts rows, and this thread has been careful to state such claims as counts rather than as shares for other reasons.

The one place it bites is the count of rows the exchange is quantified over, which is seventeen and is quoted as a probability under a coin. That probability is right for the rows measured and it is not a probability about the ladder.

Four accounts of one angle, scored on the same 17 rows. Each bar is how far an account of the exchanged pair's size sits from the measurement, as a share of the control's divergence, averaged over the census. A step of the wrecked stem's own divergence is the obvious candidate and the worst of the four. The surviving family's own step is not the unit at all: that angle is a fifth of the exchange. A step of the control's divergence is close, and correcting it by a fifth of the surviving hop takes the worst row from 11.8 per cent to 4.0.
Fig. 11 The accounts of the exchange’s size scored on the rows the census holds.

Where the set changes, against where anything else does

The handover is the one rise a band is built around, and it is not where the wrecking set changes. On the golden band the 23 changes are spread across the whole width; on the Lucas band offset 8 begins wrecking 36 rises from the coarse end and offset 4’s stretches are scattered.

So this is a third quantity that moves across a band without paying any attention to the crossing the band is centred on. The first two were the settled divergence, which is held by construction, and the family kept, which moves on one band and not the other.

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.
Fig. 12 How far apart the two contact steps get across each band, and where each crossing sits.

A recovery is not a missing measurement

The distinction the picture depends on is between a cut that recovers and a cut that was never made. The sweep tries every offset from one out to the front and two past it at every rise, so nothing inside a band is untried.

That matters because a pale cell reads as absence, and absence is the failure mode this collection keeps finding in its own instruments. Here it is a measured state: the cut was made, the run was continued, and it came back to the control.

The block is one of the two numbers, and not always the smaller. Every lattice a single organ was removed from, one row each: golden, rise 0.020 carrying 3/5; golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; golden, rise 0.005 carrying 8/13; Lucas, rise 0.020 carrying 4/7; Lucas, rise 0.013 carrying 4/7. The two open ticks on each line are that lattice's own parastichy numbers; the filled dots are the blocks the stems that never recovered settled into, one per offset that failed. The claim this table was built to test is that the block is the smaller of the two, which held at the two rises it was first measured at. It does not hold here: 3/5 gives 5, 5/8 gives 5 and 8, 4/7 gives 4 and 7. What survives is weaker and still worth something — every filled dot but 1 sits on one of that row's own ticks, so the orbit carries a count of the lattice it was cut from, and which of the two it carries is decided by the offset rather than by the pattern.
Fig. 13 How far past the front a cut can be made before every cut recovers, which is what bounds the offsets swept.

The front is not the boundary either

The obvious account of an offset that comes and goes would be that it sits near the edge of the front — the depth past which nothing wrecks — and that the edge moves with the rise.

Offset 4 is not near that edge. On a 7/11 lattice the front runs deeper than four, and the offsets past the front on this band are 9 and above, which are never tried because nothing behind the front wrecks. So the offset that flickers is an interior one, not a boundary one.

The same rule, the same rise, two lattices, two fronts. How many organs back a single removal is still felt, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice, at seven rises. Everything but the seed is identical at each rise — the rule, the spacing, the heights, the azimuth grid — and the two fronts differ at every one of them. Which branch has the wider front changes hands four times going down the range, so no function of the rise gives the column. The golden branch carries 5/8 at four of these rises, across a factor of two in the rise, and its front is eight at all four.
Fig. 14 How deep the front runs down a rung, which is what bounds the offsets a sweep has to try.

What a flickering offset might be

No account is offered here and it is worth saying why. A cut wrecks when the organs placed after the hole settle into an arrangement the control does not reach, and whether they do depends on the whole geometry of the neighbourhood the next organ is placed against — a neighbourhood whose extent is itself a measurement rather than a setting.

Two rises two parts in a thousand apart give neighbourhoods that differ by a fraction of a degree in every hop. That such a difference can flip a binary outcome is not surprising; that it flips it in clumps rather than at a boundary is the part with no account.

Where a wrecked stem settles, whatever was taken from it. The settled divergences reached by every arrangement that never repairs, at each size of cut, on a stem whose parastichy pair is 5/8, counted rather than assumed. Each point is one destination and its size is how many arrangements reached it. Cuts of one organ and cuts of five land in the same handful of places; the largest cut invents nothing the smallest did not already reach. The dashed line is the mirror of the divergence the stem was cut from — the place a coarser stem goes when two organs are taken from it — and no arrangement at any size comes within 12 degrees of it.
Fig. 15 What happens to the organs above a hole when a cut wrecks, which is the outcome that flips.

The obvious next measurement

The thing to do is to ask whether the flicker is real at a finer step. This sweep steps by two parts in a thousand; a sweep of one of offset 4’s short recoveries at a tenth of that would say whether the boundary between wrecking and recovering is sharp or whether the outcome genuinely alternates.

That is about forty stems and ten minutes for one of the gaps. It has not been done and it is the cheapest open question this reading leaves.

The 13/21 rung, at two azimuth grids. Five stems at each of three disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the earlier work, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 14 of 15 stems; at the fine one it reads all 15. The ceiling was a parameter of the program.
Fig. 16 The same quantity read at a finer step, which is how a flicker is told from a boundary.

What the golden band says about the same question

There is one piece of evidence already. On the golden band offset 5 wrecks at 22 of 126 rises, in a dozen short stretches — the same texture as offset 4 here, at a different offset on a different branch.

So whatever this is, it happens to the offset that wrecks least often on each band. That is a pattern over two cases and it is stated as such.

Offset 5 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 22 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.
Fig. 17 The golden band’s least-wrecking offset, which has no answer at 104 of its 126 rises.

Reading a census row honestly

The practical consequence is a sentence for anybody reading the census. A row that says this lattice wrecks at offsets 4, 6, 7 and 8 is a statement about that lattice at that rise, and a lattice a few parts in a thousand away may wreck at three offsets or at six.

One rise per rung is a sample was written about a different quantity and it applies here unchanged. The census is a sample of the ladder in every column, not only in the ones it was designed to sample.

The same caution has now been earned three times in this collection by three different instruments: a reading window that decided a result, a run length that moved nineteen onsets, and a rise that decides which cuts there are to read at all.

The Lucas ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 6 and 40 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; three of them sit past the mark.
Fig. 18 Where the census’s lattices sit on one branch, which is the sample every claim is quantified over.

What replicates, precisely

The claim that replicates is narrow and worth stating in its narrow form: the set of offsets at which a single removal wrecks a stem is not constant along a band. It changes on 23 of the golden band’s 125 steps and on the Lucas band it breaks one offset’s wrecking into five stretches.

What does not replicate is the family the surviving cuts keep, which changes on one band and not the other. The two were found in the same table and they are not one finding.

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.
Fig. 19 Both bands stripped to their content, where one reading is uniform and the other is not.

Two readings of one table

The band sweep produces one table and this thread has now read it twice. The first reading asks which family a wrecked cut keeps, and the second asks which cuts wreck at all. They use different cells: the first uses only the filled ones, the second uses the pattern of filled against pale.

Those are close to independent questions and they came out differently. The first is uniform on one band and speckled on the other; the second moves on both. Reading one table twice is cheap and it is the thing most likely to be skipped once the table has answered the question it was built for.

There is a third reading available and not taken here: the order in which offsets stop wrecking as the rise falls, which is a statement about the front rather than about any one offset.

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.
Fig. 20 The table both readings come from, with every cell carrying three possible states.

What a stable set would have looked like

It is worth naming the alternative that did not happen, because a negative result needs one. A wrecking set that were a property of the lattice would give a picture with five solid rows and five empty ones — every offset either wrecking at every rise or at none.

Nothing on either band looks like that. Every band has at least one offset that wrecks at some rises and not others, and on the Lucas band three of five rows are solid and two are not.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 4 or 8 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.
Fig. 21 Which offsets wreck at a single lattice, which is the row shape a stable set would repeat at every rise.

Why the effect is bigger on the finer band

One observation with no account attached. The Lucas band sits at rises around 0.008 and the golden band around 0.0058, so the two are not far apart; but the Lucas band’s offsets run to 8 and the golden band’s to 9, and the finer of the two has the smaller front.

What is measured is that the flickering happens at the offset that wrecks least on each band, and that the Lucas band’s flickering offset flickers more. Whether that is about the branch, the pair or the rise is three questions the two bands cannot separate.

The same rule, the same rise, two lattices, two fronts. How many organs back a single removal is still felt, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice, at seven rises. Everything but the seed is identical at each rise — the rule, the spacing, the heights, the azimuth grid — and the two fronts differ at every one of them. Which branch has the wider front changes hands four times going down the range, so no function of the rise gives the column. The golden branch carries 5/8 at four of these rises, across a factor of two in the rise, and its front is eight at all four.
Fig. 22 How deep the front runs down a rung on the other branch, which is the quantity that bounds each band’s offsets.

What a reader should carry

That whether a cut wrecks is decided by the rise as well as by the offset, and that a band is what shows it because a band moves the rise and nothing else.

And that this is the half of the earlier finding that survived being asked twice. The speckle in the surviving family was one band’s; the movement in the wrecking set is both bands’, and larger on the second.

Which offsets wreck across the golden 8/13 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 22 of its 126 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.
Fig. 23 The golden band’s wrecking set, with three of its six offsets absent from long stretches.

What the picture at the top shows

Five rows and 124 columns. Three rows are solid: those offsets wreck at every rise of the band. One row begins a third of the way in and is solid after that.

And one row is mostly empty with four clumps in it, separated by gaps of two, fifty-two, twenty and two rises. That row is offset 4, it wrecks at 27 of the band’s 124 rises, and it is the strongest instance of this effect anywhere in the collection.

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.
Fig. 24 The band once more. The broken row is the measurement; the solid ones are what it is measured against.

The one line

Which offsets wreck a stem is a property of the lattice and the rise: the set changes on 23 of the golden band’s 125 steps, and on the Lucas band one offset’s wrecking is broken into five stretches with gaps of 2, 52, 20, 2 and 21 rises.

So every claim of the form at every offset that wrecks is quantified over a set the rise decides, and two censuses taken a few parts in a thousand apart are censuses of different sizes.

Where this goes next

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. 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. Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it. One rise per rung is a sample Every census on this site takes one rise from each rung, because the question was always which pair. Any rule later scored on those rows inherits a variable that was never varied — and two of this collection's results turn out to be about the sampling as much as about the rule.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AblationCensus designClaim testingContact familyHandoverHonest limitsLattice offsetRecoveryReplicationRiseRungSampling