Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
14 min read 23 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.

A band is the stretch of rise around a handover on which a stem’s counted pair holds and its settled divergence stays flat, so that the only thing moving is which of the two contact steps is shorter. Six of them sit on the ladder, and all six were built so that the step ordering could be isolated from everything else the rise controls.

One of them has been cut at every rise it holds. The golden 8/13 band is 126 rises wide, and cutting all of them came back with something the design had not predicted: three of its six wrecking offsets change which family they leave standing, somewhere inside. The changes turned out to be thirteen short islands with uneven gaps rather than a period, at three different rises, while three of the six offsets never change at all.

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. 1 Both wide bands at full resolution. One row per offset, one column per rise, and the family each cut leaves standing.

What the first band left open

That result came with a question attached and the question is about generality. A transition region on one band is a fact about one band until a second one is cut, and the second one is what says whether the shape belongs to the design or to the lattice the design was run on.

The specific form of the question was about where the changes sit. On the golden band they sit 8 to 58 rises below the handover, at three different rises, and none of them is at the crossing itself. If a second band’s changes also scattered below its own handover, the shape would be a property of bands.

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. 2 The first band cut whole, which is the result this one is checked against.

Which band, and why that one

The Lucas 7/11. It is the second widest on the ladder at 124 rises against 126, it sits on the other branch, and its handover sits at 57 per cent of it against 48 per cent on the golden band.

Those three make it the right control. Width, because a band sixteen rises wide has no room for a feature two rises across and could not show a transition region even if bands had them. Branch, because a result that holds on one branch and not the other is a different finding from one that holds on one band. And a handover in a different place, because a shape that follows the handover would move with it.

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.
Fig. 3 Every band on the ladder with its extent. The two long ones are the two cut whole.

The cost

One control and one cut stem per offset at every rise: 1,612 cut stems and 124 controls, thirty-eight minutes. That is the same instrument at the same settings as the first band, run again on a different lattice, which is the only way a replication is worth anything.

Nothing about the sweep was changed for it. The step between consecutive rises is two parts in a thousand, the same ratio, because the ladder is geometric and an absolute step is three different instruments on it.

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. 4 One cut: the removed organ, the organs placed after it, and a control that shares the history below the hole.

What came back

Nothing changes. Not one offset on the Lucas band changes its surviving family anywhere on it — no changes, no islands, no transition region, nothing to fit a period to.

Every one of the five offsets that wrecks keeps the 7 family, at every rise it wrecks at, from the coarse end to the fine one. Where the golden band’s picture is a clean half, a speckled middle and a clean half, this one is a single tone across its whole width.

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. 5 Which offsets wreck at each rise of the Lucas band. The tone is uniform because the family kept is.

So the third answer was the one nobody wrote down

The question had two branches: either the changes cross at separate rises, in which case the shape belongs to bands, or they cross together or at the handover, in which case the golden band is unusual.

The second band has no changes to cross. That is the third answer, it is the strongest of the three, and it points the same way as the second: the golden 8/13 band is the unusual one, and everything said about transition regions is a statement about it.

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 The same picture without the handovers marked, so each band can be read on its own terms.

What survives, and it is the load-bearing claim

The claim the whole thread rests on is that no change of surviving family sits at the handover — that the rise where the two contact steps change places is not the rise where the survivor does.

It survives, and the way it survives on this band is worth stating carefully. On the golden band it is carried by nineteen changes, not one of which is at the crossing. Here it is carried by there being no changes at all, which is a weaker kind of support and is reported as such rather than counted as a second confirmation.

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. 7 How far apart the two contact steps get across each band, which is the quantity a handover is the crossing of.

The width was not the difference

The first thing to check is whether the two bands are the same experiment. They are close: 126 rises against 124, spanning factors of 1.28 and 1.16 in the rise.

So the golden band is a little wider and covers rather more of its rung — 72 per cent against 48. That is a difference and it is not enough of one. A band with 124 rises has room for thirteen islands one to three rises wide; it simply has none.

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.
Fig. 8 Each band’s width against what the divergence’s curvature at its handover predicts, which is how the extents were checked.

Nor was the handover’s position

The second candidate is where the crossing sits. On the golden band it is at 48 per cent of the band and the changes are 8 to 58 rises below it; on the Lucas band it is at 57 per cent.

If the transition region were something that trailed a handover, moving the handover nine per cent along the band would move it too, not remove it. Both bands put their handover near the middle and only one of them has anything on either side of it.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 9 Where the handovers sit inside their own rungs, which is the distribution any band-sampling design assumes something about.

What is different: the families

The difference that is legible is which families the cuts choose between. On the golden band a cut keeps 8 or 4 — the smaller counted number, or a sub-multiple of it. On the Lucas band every cut keeps 7, and never reaches for 4 or for 11.

A band with two available answers can change between them and a band with one cannot. That is not an explanation, because it does not say why one lattice offers two families and the other offers one; it is the difference that is actually measured, stated without a mechanism attached to it.

Which family a cut leaves standing is not the shorter of the two contact steps and is not settled by the offset alone, so a lattice offering one family rather than two is not something the counted pair predicts.

The ordering changes and the survivor does not. Every offset that wrecks, at every rise of the band, with the family left standing written in the cell. The counted pair is 4 and 7 at all 19 rises and the settled divergence is held to a twentieth of a degree, so the one quantity moving across the columns is which of the two contact steps is the shorter — and it changes hands at the marked rise. The cells do not: the 4 family survives at all 31 wrecked cuts, on both sides. Scored on this band, the reading that a wrecked stem keeps its shortest hop is right 6 times out of 31, for an answer that never changed.
Fig. 10 The families the cuts keep across a band, on the design’s earlier and narrower case.

And one rise below this band, the survivor does change

The Lucas band runs from a rise of 0.00922 down to 0.00721. Its rung runs further — to 0.0057 — and one rise below the band’s fine end, at 0.006, three of six wrecking offsets keep a lag of 11 rather than 7.

So the survivor on this rung is not fixed. It is fixed inside the band, which is what a band is for, and it changes outside it. That is a pleasing result for the design and an awkward one for anybody who wanted the band’s uniformity to be a fact about the lattice.

Twenty rises at the fine end of both branches, cut at every offset. One row per rise searched, coarse at the top of each block. The bar names the counted pair the stem shows and the numbers on the right are the lags its wrecking cuts leave standing. A pale row is a rise whose settled divergence has left the branch it was started from by more than 20 degrees, which is what happens below the ladder's finest rung — the pairs there are 2 and 4, 8 and 16, 11 and 22, which are not two consecutive terms of any additive sequence. One rise on the Lucas branch keeps a lag of 11 while still on it.
Fig. 11 The fine end of the Lucas branch, cut at every offset. The rise below the band keeps a different family.

The negative, at full strength

A sweep that finds nothing is worth what its resolution is worth. Nine rises finding nothing is consistent with a feature two rises wide sitting between them, and on the golden band that is exactly what happened: its nine-rise design hit two of thirteen islands.

A hundred and twenty-four rises finding nothing leaves nowhere for a feature to hide. That is the whole return on the thirty-eight minutes, and it is the return a null result usually cannot buy.

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.
Fig. 12 What a nine-rise design finds on each band against what all the rises find.

Which makes the coarse design right here

Run the nine-rise design on the Lucas band and it finds no change of answer. Run the full sweep and it finds no change of answer. The two agree exactly, which they do not on the golden band.

That is worth saying because the sample has been criticised in this thread and the criticism was specific. It was never wrong about whether anything changes — it said yes on the golden band and no here, and both are right. It was wrong about how many and where.

What nine rises could see of 126. Above, offset 7's answer at every rise of the band. Below, the same row with only the rises a 9-cut design visits, which is one every 16. The design was built for a quantity expected to be constant and it reports the ends and the crossing correctly; what it cannot report is where inside the band the answer changes, or that it changes back. Every island here is 2 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 13 One offset’s answer at every rise of the golden band, with only the rises a nine-cut design visits marked.

What that says about when a sample is enough

The honest generalisation is uncomfortable and it is the one the two bands support. A sample is enough when the thing being sampled has no structure narrower than the sample’s own step — and whether it does is exactly what the sample cannot tell.

One step of the nine-rise design is sixteen rises on the golden band and fifteen on the Lucas one. On the first there are features one to three rises wide; on the second there are none. The design cannot distinguish the two cases from inside itself.

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.
Fig. 14 The changes of answer the nine-rise design found across all six bands, which is the reading these sweeps check.

The wrecking set moves here too

One result from the golden band does replicate, and more strongly. Which offsets wreck at all is a function of the rise: offset 4 wrecks at only 27 of the 124 rises, in five separate stretches, and offset 8 wrecks at 88 of them in one run that starts a third of the way down.

So a census taken at one rise of this band and a census taken at another are censuses of different sizes. That half of the earlier finding is about bands rather than about one band, and this is what says so.

It also matters for how the ablation census itself should be read. Every claim of the form at every offset that wrecks is quantified over a set the rise decides, and a single rise per rung is a sample of that set as much as of anything else.

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. 15 The same reading on the golden band, where the wrecking set also moves but never breaks into five stretches.

The control the design already had

There is a third band worth naming here even though it was not cut whole. On the Lucas 3/4 rung the two contact steps stay within four parts in a thousand of each other across the entire band, so the ordering changes hands three times inside it and neither end is separated enough for an ordering to mean anything.

It is in the table as a control rather than as an experiment, and it is the reason the six bands are not six trials of one thing. Two of them are wide and were cut whole; one is a control; the other three are narrow and were sampled.

Where a handover sits inside its rung varies as much as the bands do, and the band around one is grown outwards from it rather than placed, so no two of the six are the same size by construction.

The two steps changing places inside the 4/7 band. Measured at every rise of a band on the Lucas branch, where a counter returns 4 and 7 spirals throughout. The settled divergence moves by 0.0195 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 1.0010 and the ordering changes hands at a rise of 0.0225, so above that rise the shorter step belongs to the 4 family and below it to the 7 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. 16 The gap between the two contact steps across a band, which is the quantity that has to open for a band to be an experiment.

The step is still a ratio

Two parts in a thousand between consecutive rises, on both bands, because a fixed step in the rise is one per cent at one handover and half a per cent at another.

That matters more here than it did on the first band. The two handovers sit at 0.00605 and 0.008, a third apart, so an absolute step would have given the two bands different resolutions and the comparison would have been between a fine sweep and a coarse one rather than between two lattices.

The flat band, re-measured on a finer grid. A quantity that comes out constant is the first thing an azimuth grid should be suspected of, so the whole band is grown again on a grid of 6144 steps against the 1536 the site uses. The finer grid does resolve structure the coarse one flattened: a shallow minimum 0.0820 degrees deep, with its floor at a rise of 0.0158. What it does not do is separate the ends, which still agree to 0.0000 degrees while carrying opposite step orderings. The matched pair the band is for survives the check that would have broken it.
Fig. 17 The same band on an absolute grid and on a proportional one, which is the correction both sweeps inherit.

What a second case is for

The value of this sweep is not the finding but the shape of the finding. A single striking result on a single lattice is a hypothesis; the same instrument run again somewhere else is what turns it into either a law or a curiosity, and here it produced a curiosity.

That is the more useful of the two outcomes, because it puts a boundary on a claim rather than extending it. What can be said about transition regions is now: one band on this ladder has one, and the other wide band does not.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 18 The ladder both bands sit on, with the rungs the two of them belong to.

What is not settled

Three differences between the two bands are available and none of them is established as the cause. The golden band is a little wider, its counted pair is larger, and its offsets run to 9 rather than 8.

Separating those would need a third wide band and the ladder does not have one. The next widest is 112 rises — the golden 5/8 — and cutting it whole is another half-hour and the obvious next thing to do, though it shares a branch with the band that has the speckle and so tests the branch account rather than the pair account.

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. 19 The band that would be cut next, and the two quantities it holds while the ordering reverses.

The two branches are not two samples of one thing

It is tempting to read the golden and Lucas branches as replicates, and they are not. The Lucas branch is started from a different seed angle, its rungs carry different counted pairs, and the angles its rungs settle to name it as surely as the golden ones do.

So a result that appears on one branch and not the other is under-determined in a particular way: it could be about the branch, or about the pair, or about the one lattice. Two bands cannot separate those three, and saying so is more useful than picking one.

What can be said is that the two branches have not disagreed about anything else in this thread. The handover claim holds on both, the wrecking set moves on both, and the ordering reverses exactly once inside five of the six bands regardless of which branch they sit on.

One rise, two seeds — the response of each. How far the next organ moves when the organ a given number of places back is removed, at a rise of 0.013, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice. The shading is the size of the displacement and the vertical line is where the run of felt offsets ends: 8 on the golden stem, whose lattice is 5/8, and 7 on the Lucas stem, whose lattice is 4/7. Nothing else differs between the two runs. Everything that changes with the rise — the spacing, the heights, the size of the neighbourhood the rule sums over — is the same in both rows.
Fig. 20 A quantity measured on both branches, which is the comparison this sweep is a second instance of.

What it would take to settle it

Three accounts are live and each has an experiment. If the transition region belongs to the counted pair, then a golden 5/8 band should not have one and a Lucas band with a large pair should. If it belongs to the branch, the golden 5/8 band should have one. If it belongs to that one lattice, neither should.

The golden 5/8 band is 112 rises and about half an hour, so the branch account is the cheapest to test and would be the next thing to cut. It is the weakest of the three tests as well, since a positive there is consistent with both the branch account and the pair account.

The expensive one is the honest one: a band on the Lucas branch whose counted pair is larger than 7/11. The ladder does not carry one above the finest rung, which is where the stem stops settling onto its branch at all — so that experiment is not available and the pair account may not be separable on this ladder.

Both bands, on two branches and two pairs. One row per band. Each runs from its coarse end on the left to its fine end on the right, with the rise at which the two contact steps change places marked, and the family that survives every wrecked cut written at the end. The 5/8 band on the golden branch keeps the 5 at all 24 of them and the 4/7 band on the Lucas branch keeps the 4 at all 31. Two branches, two counted pairs, one result: the quantity the band varies is not the quantity that decides the answer.
Fig. 21 Two of the narrower bands side by side, which is the comparison the wide ones extend.

What a reader should carry

That the striking picture from the first band is one band’s picture. Three offsets changing their answer in short islands with uneven gaps is a real measurement and it is not a property of bands, of handovers or of the design.

And that a null result at full resolution is a different object from a null result at nine rises. The first says there is nothing there; the second says nothing was found. Only one of them is worth thirty-eight minutes.

Offset 8 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 123 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark ten islands — runs of 1 and 1 and 2 and 1 and 2 and 1 and 2 and 1 and 3 and 1 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.
Fig. 22 The offset that carries thirteen of the golden band’s nineteen changes. Nothing on the other band looks like this.

What the picture at the top shows

Two blocks. The upper one is the golden 8/13 band, six rows and 126 columns, and its lower rows have a clean left half, a speckled middle and a clean right half with the speckle in a different place on each.

The lower block is the Lucas 7/11 band, five rows and 124 columns, and it is one tone all the way across wherever a cut wrecks at all. The pale cells in it are offsets that recover at that rise, which is the other thing the rise controls and is the subject of its own reading.

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. 23 The two bands once more, with both handovers marked. Neither vertical rule is near anything.

The one line

The second widest band on the ladder, cut at all 124 of its rises and 1,612 cut stems: not one offset changes its surviving family anywhere on it, where the widest band changes at three offsets and nineteen times.

So the transition region is a property of the golden 8/13 band and not of bands, the handover claim survives on both, and the half of the earlier finding that replicates is the one about which offsets wreck.

Where this goes next

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. 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. 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.

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.

  • Six lattices were not enough — both name ablation, claim testing, control, honest limits, lattice offset, matched design, negative result, parastichy pair, rise, rung, sampling
  • The exception was already labelled — both name ablation, claim testing, control, handover, honest limits, lattice offset, matched design, negative result, parastichy pair, rise, rung
  • The side the census sat on — both name ablation, claim testing, control, handover, honest limits, lattice offset, negative result, parastichy pair, rise, rung, sampling
  • When nine rises are enough — both name ablation, claim testing, handover, honest limits, matched design, negative result, resolution, rise, rung, sampling
  • When the second wall is free — both name ablation, claim testing, control, honest limits, lattice offset, matched design, negative result, parastichy pair, rise, rung
  • The front deepens down a rung — both name ablation, claim testing, control, honest limits, lattice offset, negative result, parastichy pair, rise, rung

Named objects

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

AblationClaim testingControlGeneralisationHandoverHonest limitsLattice offsetMatched designNegative resultParastichy pairReplicationResolutionRiseRungSampling