Phyllotaxis
the spirals are counted, not admired
Stems and cones

The lag that never survives

A correction to the exchange's size rests on four hop clusters, and a fifth would be the first real test of it. The prediction was written for a golden lattice at a lag of eleven. No golden lattice on this ladder reaches one, and the reason is a fact about the rule rather than about the search.
13 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: The organ that was taken away · Counting the spirals.

When a cut wrecks a stem, the organs above the hole settle into an arrangement whose displacements repeat at one lag. That lag is the surviving family, and the whole ablation thread is indexed by it.

The census keeps four of them: 4, 5, 7 and 8, across thirty wrecked cuts on ten lattices. That is a small number of distinct answers for a table that size, and it is the reason a fifth would be worth having.

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.
Fig. 1 Which lags the census’s cuts leave standing, gathered by the lag rather than by the lattice.

Why four is the number that matters

The exchange’s size is accounted for as one step of the control’s divergence, less about a fifth of the surviving hop’s own angle. The rule is right in sign on all seventeen rows it was fitted over.

Seventeen rows is not seventeen tests. The hop is nearly constant inside a lag, so seventeen rows carry eleven hop values in four clusters, and inside a cluster the coefficient still runs 0.12 to 0.29. A rule fitted on the hop is fitted over four numbers whatever the row count says.

That was the file’s own reading of itself rather than a criticism from outside, and it is stated beside the direction result, which does rest on seventeen rows because it is a difference between two labels rather than a fit to a number.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 2 The shortfall against the surviving hop’s angle, with the rows gathered into the clusters that carry them.

What a fifth cluster would be worth

A fifth hop is the first observation the rule has not already been fitted to. If it landed near the line the rule would have been tested; if it landed off, the rule would have been broken by one number.

The prediction was written down before the search: at a lag of 11 on a golden lattice the exchange should fall about six degrees short of one divergence step, and at 13 about four degrees long. That is specific enough to be wrong.

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, on its worst row. 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. 3 The candidate accounts of the exchange’s size, scored on the rows the correction was fitted over.

Where such a lag would have to come from

The surviving family is a contact family of the lattice — a survivor has to be a neighbour rather than any lag at all. So a cut keeping 11 needs a lattice with an 11 among its contact numbers, and a cut keeping 13 needs a 13.

On this ladder those are the golden 8/13 rung and the Lucas 7/11 rung, both at the fine end. The census cuts one rise on each. Twenty more rises were cut, ten a branch, running from inside those rungs down past the ladder’s finest.

The golden 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 12 and 35 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; seven of them sit past the mark.
Fig. 4 Where the census’s lattices sit on the golden branch, and which rungs carry a thirteen.

What the golden branch returns

Every golden rise inside the 8/13 rung keeps 8 or 4. Three rises were cut there — 0.006, 0.0055 and 0.005 — and between them their wrecking offsets keep 8 at most of them and 4 at a few.

Not one keeps 13. The counter finds 13 among the contact numbers at all three, and the cut never leaves it standing at any offset at any of them.

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. 5 The fine end of the golden branch, cut at every offset. Every rise that stays on the branch keeps 8 or 4.

And the survivor is never the shortest hop there

There is a sharper way to say the same thing. At every golden rise on a rung, the family left standing ranks second or worse by hop length — never first.

So the 13 is present, it is the shortest step on the surface at those rises, and it is never what survives. That is consistent with the census’s own finding that the survivor is not the shorter of the two contact steps, and it is the reason the golden half of the prediction has nothing to be tested on.

One wrecked stem, lag by lag — golden, rise 0.005, organ 6 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 55 degrees. The lag-4 hop swings by 0.12 degrees and sits 0.01 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 4, which is the surviving lag and not a coincidence.
Fig. 6 Every lag ordered by how long its step is, with the two contact families named.

Below the rung, the stem stops being a lattice

The obvious next move is to keep going down. It does not work, and the way it fails is worth reporting.

Below the golden 8/13 rung’s fine end the settled divergence leaves the branch: the rises searched come back at 185.8°, 225.4°, 229.2°, 228.4°, 183.4° and 178.8°, between 41 and 92 degrees from the golden angle. Every rung of the ladder proper settles within 5.3 degrees of its branch angle, so those are not near-misses.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 7 What happens to a stem below the ladder’s finest rung, where it no longer settles onto the branch it was started from.

What the counter returns down there

The pairs are the other half of the evidence. Below the rung the counted pairs are 2/4, 8/16, 11/22 and 8/11, and the first three are not two consecutive terms of any additive sequence — they are a number and its double.

A counter still returns a pair there, which is exactly why the exclusion has to be made on the divergence rather than on whether a pair came back. A doubled pair is what a counter says about an arrangement with a rotational symmetry in it rather than about a 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. 8 Both branches at the fine end, with the rises that have left their branch drawn pale.

So the golden prediction is not merely unconfirmed

There is a difference between it was looked for and not found and there is nowhere to look, and this is the second. No golden lattice on this ladder keeps a lag of 11 or 13, and below the ladder there are no lattices at all in the sense this collection uses the word.

That is a fact about the rule that places organs rather than about the search. The rule settles a stem onto a branch over a range of rises and stops doing so below it, and the range it stops at is where the pairs with large numbers in them live.

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. 9 The ladder with its finest rung, below which a stem no longer settles onto a branch.

Which is a constraint on the whole thread

Read forwards rather than backwards, this says something about what the ablation census can ever contain. The surviving lag comes from the contact families, the contact families come from the rung, and the rungs the ladder carries are 2/3, 3/5, 5/8 and 8/13 on one branch and 1/3, 3/4, 4/7 and 7/11 on the other.

The smaller number of each pair runs 1, 2, 3, 3, 4, 5, 7, 8. Those are the lags a cut can plausibly keep, and the census keeps four of them.

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. 10 The other branch’s rungs and where the census sits on them.

The larger number of a pair, and whether it ever survives

Stated that way the question becomes sharper than the one that was asked. It is not does anything keep 11 or 13? but does the larger counted number ever survive a cut?

On the golden branch the answer over twenty cut rises is no. On the Lucas branch it is yes, once, one rise below where the census stops — and that single case is what the whole search returned.

The 20 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 20 rows is fitted over five 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.
Fig. 11 The same table with the one lattice the search found added to it.

Why the census stopped where it did

The census’s ten lattices were chosen so that the same counted pair appears at more than one rise, which is what makes its decisive negative available: 4/7 appears twice and 5/8 four times, so the pair and the offset do not decide the survivor is a claim with rows on both sides.

They were not chosen to span the lags. Nobody had asked which lags a cut can keep, because the answer looked like a consequence of which lattices were in the table rather than like a fact about the rule.

It is a fact about the rule, and the way to see that is that the family a cut keeps is read from the stem itself rather than assigned. The census did not choose its lags; it recorded them.

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. 12 The census as a table, whose ten lattices were chosen to repeat pairs rather than to span lags.

What the search cost

Twenty rises, each cut at every offset out to the front and two past it: about eight minutes of stems, and the result is banked so that reading it again is free.

That is cheap for a search and it is worth saying that a search is a different kind of work from a sweep. A sweep visits every point of a stated range; a search goes looking, and its report has to include where it looked as well as what it found.

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 bounds the offsets a search has to try.

What was looked at, exactly

Ten rises on each branch. The golden list starts at 0.0060, inside the 8/13 rung, and runs down through 0.0055, 0.0050, 0.0046, 0.0044, 0.0042, 0.0040, 0.0038, 0.0035 and 0.0032. The Lucas list starts at 0.0090, inside the 7/11 rung, and runs down to 0.0032.

Three of the golden rises are on a rung and seven are below the ladder. Four of the Lucas rises are on a rung and six are below. So the search covered the whole of both fine ends and then some.

The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here.
Fig. 14 The fine end of the ladder at the step the rungs are read at, which is the region the search covers.

The negative is bounded, and the bound is stated

What is not claimed is that no golden lattice anywhere keeps 13. The claim is about this ladder, which is the region a stem grown by this rule settles onto a branch in, at the rises this search cut.

A finer step between 0.006 and 0.00482 could in principle find one. That is 27 rises at one per cent and about ten minutes, and it is the obvious way to make the negative tighter if anybody wants it tighter.

Rises that do not settle, at two resolutions. The count of rises whose divergence never settles, on the rung where the band was found and on the finer rung swept ten times as closely. The coarse rung has six of 19, all of them stuck on three eighths of a turn; the finer rung has none of 23. Sampling is not the explanation: if a band of the same kind sat inside the 3/5 rung it would need to be narrower than a ten-thousandth of rise to have been missed here.
Fig. 15 The ladder read at two steps, which is what a finer search of one rung would look like.

Why the prediction was branch-specific in the first place

The six degrees in the prediction came from arithmetic on a golden lattice: a hop at lag 11 on a golden lattice has a particular angle, and a fifth of it is a particular number.

That arithmetic does not transfer. The Lucas lattice that does keep 11 has a different divergence and therefore a different hop, so the number the prediction names cannot be checked and the rule it came from can. Confusing the two would be reading a prediction’s arithmetic as its content.

The hops of a 4/7 lattice, shortest first — Lucas, rise 0.020. Every lattice hop at this rise, ordered by how long its step is across the surface of the stem, with the one that a wrecked stem here leaves standing picked out. The two shortest are the contact families the counter returns, 4 and 7, and they differ in length by a factor of 1.082. The lags left standing after a removal are 4, sitting at rank 2 in this order, so the family the rule holds is a short step but not always the shortest one.
Fig. 16 How a hop’s angle depends on the lag and the branch, which is why a predicted number does not travel.

What a negative like this is for

It closes a direction. Anybody extending the exchange’s correction now knows that the golden branch cannot supply a fifth cluster, that the Lucas branch supplies exactly one, and that the region below the ladder supplies none because it supplies no lattices.

That is worth more than an unsuccessful search usually is, because it turns nobody has found one into there is one place and here it is.

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. 17 The Lucas branch’s fine end, where the one available lattice sits.

The two halves of the search report differently

The golden half returns a bounded negative. The Lucas half returns a single positive at a rise of 0.006, on the 7/11 rung, whose settled divergence is 0.158 degrees from the Lucas angle.

Both are worth reporting and the second is the one that changed a number. It is the subject of its own reading, and the rule it tests survives it.

The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement.
Fig. 18 The hops the correction is fitted over, with the one the search found at the near end.

The other lattices the search passed over

Four Lucas rises sit on a rung: 0.0090, 0.0080, 0.0070 and 0.0060. The first three keep 7 at every offset that wrecks, which is what the census’s own l008 does, and the fourth keeps 7 at three offsets and 11 at three more.

So the change happens between 0.007 and 0.006, inside the 7/11 rung and below the band that was cut whole across it. The band holds the survivor fixed at 7 across all 124 of its rises and the survivor changes twenty rises below its fine end.

That is a coincidence of design rather than a finding: the band was grown outwards from the handover until the divergence moved a twentieth of a degree, and it happens to stop above the rise where the family changes.

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. 19 The band on the rung where the change happens, which holds one family across every one of its rises.

What is between 0.007 and 0.006

Nobody has looked. The search stepped from 0.0070 to 0.0060 in one move, so the rise at which the family changes on that rung is located to within a sixth of the rung and no better.

Narrowing it is nine rises at the five-decimal grid and about five minutes, and it would say whether the change is a transition at one rise — as the second wall going free turned out to be — or a stretch where both families occur. That question has an answer and it has not been asked.

The 13/21 rung, at two azimuth grids. Five stems at each of five 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 22 of 25 stems; at the fine one it reads all 25. The ceiling was a parameter of the program.
Fig. 20 The same kind of quantity read at a finer step, which is how a transition is told from a stretch.

Why a doubled pair is not a lattice

The counted pairs below the ladder deserve a sentence, because a reader could take 8/16 for an ordinary count. A lattice this collection would call one has a pair of coprime contact numbers, which is what two consecutive terms of an additive sequence always are.

8 and 16 are not coprime and 11 and 22 are not. A counter returns them because it reports the two families whose steps are shortest, and on an arrangement with a rotational symmetry the shortest two can be a family and its double.

That is why the branch test is made on the settled divergence and not on the pair. A pair that looks wrong is suggestive; a divergence 92 degrees from the branch angle is the measurement.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 29 and 47, and neither number is a Fibonacci number.
Fig. 21 What a counter returns across a range of divergences, including the arrangements that are not lattices.

The search’s own refusal

One rise in twenty could not be read at all and the search records it rather than dropping it. A search that stopped at its first unreadable point would be reporting where the reading fails rather than where the lag is.

That distinction has bitten this collection before. The failure mode it keeps producing is absence — a figure that draws nothing, a tick array that comes back empty, a label placer that places nothing — and every one of them passed every check that asks whether what is there is right.

Both vary; only one of them varies enough to find. Each organ's step exponents, divided by its own mean so the two are comparable. The ogive's run over 15 per cent of their mean across 5 rings. The convex head's run over 1.15 per cent across 5 — inside the band a 1 per cent error on each ring position leaves, so no ruler separates it from a flat disc.
Fig. 22 What a measurement can and cannot report about itself, which is why an unreadable point is recorded.

What a reader should carry

That the four lags the census keeps are not an accident of which lattices were in it. They are what the rule that places organs makes available, and the larger counted number of a pair never survives a cut on the branch that was searched.

And that a prediction can be well posed, specific and untestable — not because nobody tried, but because the object it is about does not occur.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 23 The ordered hop lengths at one lattice, with the family that survives never at the top of it.

What the picture at the top shows

Ten rows, one per golden rise searched, coarse at the top. Each row names the counted pair the stem shows, the divergence it settles to, and the lags its wrecking cuts leave standing.

The top three rows are on a rung and keep 8 or 4. The seven below are pale: their divergences are 41 to 92 degrees off the golden angle and their pairs are 2/4, 8/16, 11/22 and 8/11. Nothing anywhere in the block keeps 13.

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. 24 The golden branch’s fine end once more, with the on-branch rises at the top and the rest below.

The one line

Twenty rises cut at the fine end of both branches: no golden lattice on this ladder keeps a lag of 11 or 13, the survivor there ranks second or worse by hop length at every rise, and below the finest rung the stem stops settling onto its branch at all — so the golden half of the prediction has nothing to be tested on.

The Lucas branch supplies exactly one lattice that does, and it is one rise below where the census stops.

Where this goes next

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. One rise below the census Ten lattices were cut at every offset and their surviving lags came back as four numbers. One rise further down a rung the census already sweeps, three cuts keep a lag of eleven — which is a fifth number, on a lattice nothing about was unusual except that nobody had cut it. 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. A survivor has to be a neighbour A stem that never repairs after a removal keeps exactly one lattice hop rigid, and nothing predicted which one. Sweep every offset at twelve lattices and the answer narrows sharply: at twenty-nine of thirty the surviving hop is one of the two families a counter returns, and the one exception is a step six times too long to be one. What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17. A fifth of the hop The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent.

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.

  • Every rise of a band — both name ablation, claim testing, honest limits, lattice offset, negative result, parastichy pair, rise, rung
  • Six lattices were not enough — both name ablation, claim testing, honest limits, lattice offset, negative result, parastichy pair, rise, rung
  • The exception was already labelled — both name ablation, claim testing, honest limits, lattice offset, negative result, parastichy pair, rise, rung
  • The front deepens down a rung — both name ablation, claim testing, honest limits, lattice offset, negative result, parastichy pair, rise, rung
  • The offsets that never change — both name ablation, claim testing, honest limits, lattice offset, negative result, parastichy pair, rise, rung
  • The shortest hop was a coin flip — both name ablation, claim testing, honest limits, lattice offset, negative result, parastichy pair, rise, rung

Named objects

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

AblationClaim testingContact familyHonest limitsHop lengthIdentifiabilityLattice offsetNegative resultParastichy pairRiseRungSearch