Phyllotaxis
the spirals are counted, not admired
Stems and cones

A fifth cluster

A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having.
15 min read 25 figures State the claim, state the testCounted, not admiredTwo routes, one number

Worth reading first: The organ that was taken away · The damage has a period.

When a cut wrecks a stem, two adjacent chains of organs change places, and the size of that exchange has an account. It is one step of the control’s divergence, less about a fifth of the surviving hop’s own angle, and it is right in sign on every row it was fitted over.

The rule’s own file states its weakness plainly. Seventeen rows carry eleven hop values in four clusters, because the hop is nearly constant inside a lag — so a rule fitted on the hop is fitted over four numbers however many rows it has.

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. 1 The hops the correction was fitted over, with the one a search added at the near end.

Why four is a thin denominator

A correction with one free coefficient fitted over four points is a line through four clusters. The coefficient’s value is 0.188 by least squares and is stated as a fifth rather than fitted, because a third digit would be a claim about four measured hops.

Four points also cannot distinguish a linear rule from several others. What they can do is say the sign is right, and the sign is right on every row — the alternatives, the cut stem’s own divergence step and the surviving family’s step, are wrong by up to 82 and 83 per cent. The rival that comes closest is the control’s divergence step with no correction at all, which is wrong by 11.8 per cent on its worst row.

So the rule was in good shape as a description and untested as a prediction. Those are different things and the difference is what a fifth cluster is for.

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. 2 The four candidate accounts of the exchange’s size, and how badly each of them does on the worst row.

Where the fifth came from

A search of the fine end of both branches found one lattice whose cuts keep a lag the census does not: the Lucas branch at a rise of 0.006, on the 7/11 rung, whose offsets 8, 9 and 10 keep a lag of 11.

It was added to the census as one extra row rather than as a replacement for it, so the extended table is the old ten lattices plus one and every existing number is answered from the same cache.

The lattice is one rise below one the census already holds, on the same rung, settling to within a sixth of a degree of its branch angle. Nothing about it is exotic; it had simply never been cut.

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. 3 The fine end of the Lucas branch, where the one available lattice sits.

What the new hop is

12.78 degrees, and it is the smallest in the table by a third. The four clusters the rule was fitted over sit at 19.5, 22.1, 22.8 and 39.1 degrees in magnitude, with the larger negative hops at 21.5 to 37.3.

That matters more than a fifth point usually does. A point between two existing ones is mostly a restatement; a point a third beyond the near end of a fitted range is where a linear rule and a curved one first come apart.

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. 4 The exchange’s shortfall against the surviving hop, with the fitted line through it.

What the rule predicts there

A fifth of 12.78 is 2.56 degrees, so the rule says the exchange should fall about two and a half degrees short of one step of the control’s divergence.

That is a smaller shortfall than anything in the table, which runs from about 4 to 11 degrees, and it is small enough to be swamped by anything sloppy. If the rule were picking up a constant offset dressed as a proportion, this is where it would show.

The correction's coefficient at each of the five lags. How far the exchange falls short of one step of the control's divergence, as a share of the surviving hop's own angle, at every row of the table. The stated rule is a fifth and the least-squares fit over the whole table is 0.184. The new lag's three rows sit at 0.164, 0.163, 0.162, inside the spread the four older lags already covered rather than beyond it.
Fig. 5 The correction’s coefficient at each lag, with the stated fifth and the fitted value drawn as rules.

What is measured

2.09, 2.09 and 2.07 degrees on the three rows, against a predicted 2.56. As a share of the hop that is 0.164, 0.163 and 0.162 against a stated 0.200 and a fitted 0.188.

So the rule is right in sign, right in order of magnitude, and low by about a fifth of itself. The three coefficients land inside the spread the four old clusters already covered — 0.121 to 0.301 — rather than outside it.

That is a confirmation of the useful kind. It is not that the prediction was exact; it is that a point beyond the fitted range did not need a new rule to describe it.

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. 6 The shortfall against the hop, gathered into the clusters that carry the fit, with the new one included.

What it does to the fit

The least-squares coefficient over the whole table moves from 0.188 to 0.184. That is a change in the third digit, on a quantity whose third digit was never claimed.

The spread of per-row coefficients does not move at all: 0.121 to 0.301 before, 0.121 to 0.301 after, because the new rows sit comfortably inside it. So nothing about the rule’s uncertainty changed either.

The correction's coefficient at each of the five lags. How far the exchange falls short of one step of the control's divergence, as a share of the surviving hop's own angle, at every row of the table. The stated rule is a fifth and the least-squares fit over the whole table is 0.184. The new lag's three rows sit at 0.164, 0.163, 0.162, inside the spread the four older lags already covered rather than beyond it.
Fig. 7 Every row’s coefficient against the lag it kept, with the four old clusters and the new one.

And to the two other claims

The exchange’s file makes two claims besides the size. The first is its direction: the chain displaced forwards is one residue below the chain displaced backwards, on every row. That was seventeen rows, quoted as one in a hundred and thirty thousand under a coin.

It is now twenty of twenty, which is one in a million under the same coin. The three new rows are the ones that extend it rather than rows that were already there.

A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in.
Fig. 8 The two chains that change places in a wrecked stem’s profile, which is what the direction is a claim about.

Where the exchange sits

The second claim is about location: the back-displaced chain is the hole’s own chain on ten of seventeen rows, against 2.8 expected by chance. It was reported as a tendency and not a law, because seven rows put it elsewhere.

All three new rows put it at the hole, so the tally is now thirteen of twenty. The share barely moves — 59 per cent to 65 — and the claim stays what it was.

Three rows all falling the same way is what three cuts on one lattice would do whether the tendency is real or not, so this is the weakest of the three extensions. It is reported because leaving it out would be choosing which of a file’s claims to re-score.

Which chain the backward exception sits on, over the census. Chains are numbered from the removed organ, so chain 0 is the chain the hole was on and chain 2 is two organs along it. The exchange is at the hole's own chain on 10 of the 17 rows that carry one, against 2.8 rows for a chain drawn at random from each row's own period. That is far more often than anywhere else and it is not every row, so the position is a tendency rather than a rule — and the file says so rather than rounding it up.
Fig. 9 Which chain carries the backward displacement, across the rows of the census.

The denominator, said properly

Three rows arrived and the honest count went from four to five. Not from seventeen to twenty.

The three sizes are 97.25, 97.25 and 97.28 degrees — one number three times, because the offsets differ and the rise, the seed angle and the history below the hole do not. Any arithmetic quantified over rows would be counting the same measurement three times, which is the exact criticism the exchange’s own file makes of its seventeen.

So the correct sentence is: a rule fitted over four hops is now tested against a fifth, and it survives. Everything else is bookkeeping.

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. 10 The extended table gathered by lag, where five clusters now carry twenty rows.

What a fifth point can and cannot do

It can break a rule and it did not. It can extend a range and it did — the fitted range now runs from 12.78 to 39.14 degrees rather than from 19.5.

What it cannot do is distinguish the rule from its neighbours. A fifth of the hop, a sixth and a quarter all take the worst row from 11.8 per cent to under 6, and one more point at one more hop does not separate them either. That would take several clusters at hops the ladder does not offer.

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 scored on average rather than at their worst, where the three near neighbours are hardest to separate.

Which is a limit of the subject and not of the work

That is worth stating because it is where this thread ends rather than pauses. The hops available are decided by the lags available, the lags are decided by the rungs, and the rungs are what the rule that places organs makes.

Eight rungs across two branches, whose smaller numbers are 1, 2, 3, 3, 4, 5, 7 and 8, and one rung whose larger number can survive. That is the whole supply of hop clusters on this ladder and five of them are now in the table.

Two of the eight are too coarse to wreck at all, so they contribute nothing: nothing behind the front wrecks and at the coarse end the front is short enough that no single removal reaches past it.

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. 12 Every rung on both branches, which is the whole supply of contact numbers this rule offers.

Why the new hop being small is the useful part

A shortfall proportional to the hop and a shortfall that is a constant look the same over a narrow range of hops. Over 19.5 to 39.1 degrees — a factor of two — they are distinguishable in principle and not comfortably.

At 12.78 degrees a constant shortfall of about 4 degrees would give a coefficient of 0.31, at the very top of the observed spread; a proportional one gives 0.16, in the middle of it. The measurement is 0.163.

So the new point does discriminate, mildly, in favour of the proportional reading. That is a modest thing to get from one cluster and it is more than the four could give.

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. 13 The shortfall against the hop without a line drawn through it, which is the reading the new point discriminates on.

The control the extension keeps

Every number the old table reported is unchanged, because the extended census is the old list of lattices with one appended and both are answered from the same cache.

That is the discipline the slot table followed when it went from six lattices to twenty-four. A comparison between a table and a longer table is worth nothing if the two are also two implementations, and the way to guarantee they are not is to make the shorter one a subset of the longer.

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. 14 The census the extension appends to, unchanged by the appending.

What the three rows look like up close

All three are balanced pairs: two adjacent chains displaced in opposite directions by the same amount to within a twentieth. That is what makes them usable — the exchange is only defined on a row with a balanced pair in it.

Their chains are 10 and 0 out of eleven, so the forward-displaced chain is one residue below the backward-displaced one, and the backward-displaced one is the hole’s own. That is the pattern every other row in the table has.

A balanced pair is what a chain slipped by one place would produce, and the alternative — a swap — would give a displacement of nothing in the quantity that measures the slip. That test was made on the seventeen and the three new rows agree with it.

How far every organ moved, 5 places back at a rise of 0.013. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 40 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. two of those levels sit together and three do not.
Fig. 15 A wrecked stem’s displacement profile folded onto the lag it kept, which is what a balanced pair is read from.

And what the other three cuts at that lattice do

The same lattice wrecks at three more offsets, and those keep a lag of 7 rather than 11. None of them is a balanced pair.

So they do not enter the exchange table at all. They go instead to the set the exchange sets aside, where they turn out to be an out-of-sample test of a different rule entirely — one that had been right thirteen times out of thirteen and is wrong on one of them.

The rule tested on a lattice the census does not hold. The three unpaired cuts on the lattice the fifth-lag search turned up, all of them at a lag of 7, which the rule says must close. Two do and one does not. The line the rule was drawn with sits in the gap between the closed and open populations of the original census, and the row that breaks the rule sits well past it — so what failed is the claim rather than the threshold. Out of sample the rule is right 15 of 16 times rather than 13 of 13.
Fig. 16 The three cuts at the same lattice that go to the other thread, and what they do to its rule.

One search, two threads

That is worth a sentence on its own because it was not planned. The search was run to find a lag outside the census’s four and it returned a lattice with six wrecked cuts on it: three that answer the question asked, and three that test something else.

Nobody designed the second half. It is the ordinary return on measuring a whole object rather than the part of it a question needs, and it is the reason the census cuts every offset rather than the ones expected to wreck.

The 13 cuts the exchange sets aside, chain by chain. One row per wrecked cut with no balanced pair, one cell per chain of the lag it kept, and the chains displaced away from the common level filled. The lag is printed at the left and the sum of the displacements at the right, folded into half a turn either way. seven of the 13 sum to within 12 degrees of nothing and six do not, and every row that closes kept a lag of 7 or 8 while every row that does not kept 4 or 5. The rows are stacked by that lag.
Fig. 17 The rows the exchange sets aside, three of which arrived with the fifth cluster.

What would break the rule

It is worth naming, since a rule that cannot be broken is not being tested. A sixth cluster at a hop of, say, 60 degrees with a shortfall of 4 would put the coefficient at 0.07, outside the observed spread, and the proportional reading would be in trouble.

No such hop exists on this ladder. The largest is 39.1 degrees at a lag of 4, and larger hops need coarser lattices, where a single removal wrecks nothing and there is no exchange to measure.

The hops of a 5/8 lattice, shortest first — golden, rise 0.010Every lattice hop at this rise, ordered by how long its step is across the surface of the stem, with the two that a wrecked stem here leaves standing picked out. The two shortest are the contact families the counter returns, 5 and 8, and they differ in length by a factor of 1.076. The lags left standing after a removal are 5 and 8, sitting at rank 2 and 1 in this order, so the family the rule holds is a short step but not always the shortest one.85133161021181122624629lag, in organs — ordered by the length of its stepstep length, in turns of the stemkept: lag 5, lag 8golden, rise 0.010 · pair 5/8 · offsets that wreck: 4, 5, 6, 7, 8generated from a stated rule, not drawn to look right
Fig. 18 How a hop’s angle depends on the lag, which is what bounds the range any test of the rule can cover.

So the rule is as tested as this subject allows

Five clusters spanning 12.8 to 39.1 degrees, a factor of three, with a coefficient that stays inside 0.12 to 0.30 across all of them and a fit that moves by 0.004 when the fifth arrives.

That is the state of it, and it is worth saying that this is the ceiling rather than a stage. The next test would need a lattice this ladder does not carry.

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. 19 The five clusters and their hops, which is the whole population the rule can be scored on.

The prediction that could not be tested

There were two predictions and only one of them had somewhere to be tested. The other was for a golden lattice at a lag of 11, where the exchange was expected to fall about six degrees short of one divergence step.

No golden lattice on this ladder keeps a lag of 11 or 13. At every golden rise on a rung the surviving family ranks second or worse by hop length, and below the finest rung the stem stops settling onto its branch at all. So that prediction is not unconfirmed; it has no object.

The six degrees would not have transferred anyway. It was arithmetic on a golden hop, and the lattice that does keep 11 has a different divergence and therefore a different hop — which is why the rule transfers and the number does not.

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. 20 The golden branch’s fine end, which keeps 8 or 4 at every rise that stays on a rung.

A number and the rule it came from

That distinction is worth keeping. A prediction usually arrives as a number, and the number is the rule evaluated at conditions the predictor had in mind.

When the conditions turn out not to exist, the number is dead and the rule may be perfectly alive. Reading the dead number as a dead rule is the mistake available here, and the way to avoid it is to write down which part of a prediction is the rule before looking.

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. 21 The rules the accounts are, rather than the numbers they produce, which is what a fifth cluster tests.

What it cost

Twenty rises cut at every offset, about eight minutes of stems, and a census extended by one lattice, which is another two. Ten minutes to test a rule that has been in the collection since the exchange was first measured.

That is cheap enough to be worth saying out loud, because the alternative was leaving a fitted coefficient unexamined on the grounds that examining it looked like work. The expensive part was deciding where to look, and that was arithmetic on which rungs carry an eleven.

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. 22 How far past the front a cut can be made, which is what bounds the offsets the search had to try.

What is left open

Two things, and both are small. The rise at which the surviving family changes on the 7/11 rung is located to within a sixth of the rung, because the search stepped from 0.0070 to 0.0060 in one move; nine rises would place it.

And whether the whole lower quarter of that rung keeps 11, which would make the fifth cluster several rises rather than one. Neither would change the rule’s score and both would make the one case a little less lonely.

The ratio is a U across every rung, and its floor is the number that was reported. The ratio of the second comb to the main comb, on five stems at each of 15 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.29, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.
Fig. 23 One rung read across its whole span, of which the search cut four rises.

What a reader should carry

That a rule fitted over four numbers was tested against a fifth from outside its range, and came back right in sign and low by a fifth of itself — inside the spread its own four already had.

And that the fifth is one lattice and one hop, three rows deep, so the denominator went from four to five rather than from seventeen to twenty. Reporting it the other way would have been the same mistake the rule’s own file was written to avoid.

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. 24 The five clusters with their rows, where three of the twenty carry one measurement.

What the picture at the top shows

Five marks, one per surviving lag, placed by the lag and by the angle of the hop that lag keeps. The shaded band is the range of hops the correction was fitted over.

Four of the marks are inside it. The fifth — the lag of 11 — sits at 12.78 degrees, outside the band at its near end, and the number above it says the table holds three rows there. That is what a test looks like as opposed to a restatement.

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. 25 The five clusters once more, with the fitted range shaded and the new one beyond it.

The one line

The exchange’s correction was fitted over four hop clusters and is now tested against a fifth at 12.78 degrees, a third smaller than any of them: it predicts a shortfall of 2.56 degrees and the measurement is 2.09, a coefficient of 0.163 inside the 0.121 to 0.301 the four already spanned.

The fit moves from 0.188 to 0.184, the direction claim goes from seventeen rows to twenty, and the honest denominator goes from four to five.

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. 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. A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in. One way round, seventeen times The two chains that change places in a wrecked stem are adjacent, which is symmetric and says nothing about direction. Label them by lag from the hole and the one displaced forwards is always the lower of the two — on every row of the census, without an exception.

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.

AblationClaim testingContact familyDivergence stepExchangeFalsifiabilityFitted constantHonest limitsHop lengthOut of sampleReplicationSample size