Phyllotaxis
the spirals are counted, not admired
Stems and cones

An offset that arrives

The rise where a lattice first keeps a new family is located to one step of the grid, and what happens there is not what the question assumed. No cut changes its mind: a cut that was not wrecking starts, and what it keeps is the new family.
15 min read 21 figures Counted, not admiredState the claim, state the testOne parameter, one family

Worth reading first: The organ that was taken away.

The question was where the family a cut keeps changes on the Lucas 7/11 rung. It is answered to one step of the grid: between 0.00640 and 0.00639, with the counted pair, the settled divergence and the ordering of the two hops identical on both sides.

What happens there is not a cut changing its mind. It is a cut appearing.

What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.
Fig. 1 What each offset keeps above and below the transition, and the one that arrives with the new family.

The two sides

At 0.00640 the offsets that wreck are 5, 6 and 7, and all three keep a lag of 7.

At 0.00639 they are 5, 6, 7 and 9. Offsets 5, 6 and 7 still keep 7. Offset 9, which recovers at 0.00640 and at every rise above it in the search, wrecks — and keeps 11.

So the lattice’s set of surviving families goes from {7} to {7, 11} because a new row appeared in the table, not because an existing row moved.

The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere.
Fig. 2 Every rise of the search, offset by offset, with the transition and the offset that arrives at it.

Which changes what the located rise means

The rise located to one grid step is the rise at which the lattice first keeps 11 anywhere. That is the quantity the fine-end search was asking about — it went looking for a lattice whose cuts keep a lag outside the census’s four — and it is the quantity the exchange’s fifth hop cluster depends on.

It is not the rise at which any particular cut changes its answer. Those are different questions and only the first has a sharp answer here.

Nothing was lost by the distinction being noticed late. The number is the same and it means something slightly different from what the question implied.

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. 3 The extended census, whose eleven-keeping rows all come from below this transition.

The offsets that are steady

Five, six and seven wreck at nearly every rise of the search and keep 7 at every one. Across twenty rises spanning a sixth of the rung, with the divergence moving two tenths of a degree, those three cuts do the same thing throughout.

That is worth stating because it is the control. If everything moved from rise to rise, the arrival of offset 9 would be one more piece of noise in a noisy table. It is not: three of the four offsets that matter are constant across the whole search.

So the table has a steady part and a moving part, and the moving part is the membership of the wrecking set.

What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.
Fig. 4 Each offset’s answer above and below the transition, of which three do not change at all.

The one that does change its mind

Offset 8. It wrecks at 0.0068 keeping 7, at 0.0065 keeping 7, and at 0.00637 keeping 11.

And it cannot be located. It does not wreck at 0.0064, 0.00639 or 0.00638, so its change is bracketed across thirteen steps of the grid — a bracket that happens to contain the transition without that meaning anything.

That is the same limit the third band ran into on a different object: an offset’s answer exists only where its cut wrecks, and the wrecking set moves faster than the answer does.

What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.
Fig. 5 The one offset whose own answer changes, bracketed across thirteen grid steps because it is silent between.

At 0.0070 it is {4, 5, 7}. At 0.0069, {5, 6, 7}. At 0.0068, {5, 7, 8}. At 0.0067, {5, 7}. At 0.0066, {5, 6, 7}. At 0.0065, {5, 6, 7, 8}. At 0.0064, {5, 6, 7}.

Seven consecutive rises, seven different sets, each rise one part in a hundred and forty from the next in the rise. Offsets 4, 6, 8 and 9 each appear and disappear; only 5 and 7 are in every one.

The band sweeps found the same instability over stretches of rise. Here it is visible between adjacent readings, which makes it harder to read as a slow drift.

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. 6 The same rung’s band, where the wrecking set’s movement is visible over stretches rather than between neighbours.

Why an arrival is a different kind of event

A cut wrecks when the pattern above the hole never returns to the arrangement it had. Whether it does is a boundary in a continuous problem: the removed organ’s absence propagates, and whether the pattern recovers depends on how much of the front the hole was carrying.

So does this cut wreck is a yes-or-no reading of a continuous situation, and a yes-or-no reading can flip under an arbitrarily small change. That is what an arrival is.

What does the cut keep is different in kind. It is a choice among integers, and there is no continuous quantity underneath it that could be crossing a threshold. Its changes are not the same kind of event as an arrival, and the search found one of each.

The next organ moves for the last 13, and for no othersOne row per organ removed, counted back from the tip of a stem at a rise of 0.005 whose counted pair is 8 and 13. Removing any of the last 13 moves the next organ by 2.6° to 167.6°; removing an older one moves it by at most 0.47°, which is under the azimuth grid. The boundary is at 13, and 13 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.organ removed, counted back from the tiphow far the next organ moves, in degrees1138.0°284.4°353.4°4167.6°529.3°6101.7°7120.7°816.4°9165.2°1056.7°1181.1°12140.6°132.6°— the front ends here140.0°150.0°160.5°rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right
Fig. 7 What each offset’s removal does at one rise of the ladder, of which some wreck and some recover.

What the arriving cut keeps, and why 11

Because 11 is available there and 7 is not, for that offset. The surviving family is the smallest lag whose hop the wrecked run holds rigid, and on offset 9’s wrecked run the 7-hop is not held.

That is a positive reading rather than an absence: the 7-hop is measured on that run and it moves, while the 11-hop does not. So offset 9 is not keeping 11 because 7 was unavailable in some formal sense; it is keeping 11 because that is what its own run holds.

Which makes the two families genuinely coexisting at the same rise, on the same lattice, under different offsets. That is the shape the census’s own rows show elsewhere on the ladder, and it is why the offset is in every rule about survivors.

Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.
Fig. 8 The census’s rows, where one lattice’s cuts at different offsets keep different families.

How the new family fills in

Below the transition the set of offsets keeping 11 grows unevenly. {9} at 0.00639 and 0.00638. {8, 9} at 0.00637. {9, 10} at 0.00636. {8, 9} again for four rises. Then {8, 9, 10} from 0.00631 down to 0.0060.

So there is no second transition at which the thing settles. It fills in over about ten grid steps, with membership moving as it goes, and only at the bottom of the search does it stabilise.

That is the shape a wrecking set has everywhere in this thread, and it means the useful statement is about the first arrival rather than about the final set.

The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere.
Fig. 9 The eleven-keeping offsets filling in below the transition, over about ten steps of the grid.

What the census row at the bottom is

At 0.0060 the lattice wrecks at six offsets: 5, 6 and 7 keeping 7, and 8, 9 and 10 keeping 11. That is the row the exchange’s fifth hop cluster came from, and it is the only lattice on this ladder that produces a lag of 11.

Read against this search, that row is not exotic. It is the state the rung is in over its bottom seventy grid steps, and the census simply had not cut anywhere in that stretch.

The search that found it reported it as one rise below where the census stops, which is true and understates it: it is one rise below the census and eighty steps below the transition.

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. 10 The fine-end search that found the row, whose steps skipped the whole stretch this round cut.

What would have been found by cutting a band

Nothing. The band around this rung’s handover runs from 0.00922 to 0.00721, and the transition is at 0.00640 — eighty-one grid steps below the band’s fine end.

So the Lucas 7/11 band’s clean negative is a negative about the band and not about the rung. Every rise it holds is above the transition, and every cut on it keeps 7 because every cut in that stretch keeps 7.

That reframes the band result usefully rather than undermining it. The band was cut to ask whether the survivor changes with the divergence held still, and the answer there is no. The survivor does change on the rung, where the divergence is not held still.

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. 11 The band on this rung, whose 124 rises all sit above the transition the rung carries.

Which of the two designs is right

Both, for different questions. A band answers does the survivor change when nothing else does. A rung answers where does the survivor change.

The first is the better-controlled question and the second is the one with an answer here. Neither subsumes the other, and running only bands would have missed this transition entirely while running only rungs would leave every change confounded with a moving divergence.

That is an argument for keeping both designs rather than for preferring one, and it is worth recording because the band design is the more expensive and has been the default.

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. 12 A band’s construction, which holds the divergence still and therefore cannot reach the fine end of its rung.

What the offset is, physically

The number of places back from the growing front at which the organ is removed. Offset 5 means the fifth-newest organ; offset 9 means the ninth-newest.

Past the front every cut recovers, because the removed organ is no longer one of the neighbours the next organ is placed against. How far back the front reaches is a measurement and it deepens as the rise falls, which is part of why offsets 9 and 10 become available at the fine end of this search and not at the coarse end.

So the arrival has a candidate explanation of a sort: the front is deepening, and offset 9 comes inside it. What that does not explain is why the new arrivals keep 11 rather than 7.

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 back a removal still wrecks a stem, which decides which offsets are available at all.

The explanation that is not available

That the deeper offsets keep 11 because they are deeper. It is a natural reading and the table refuses it: offset 8 keeps 7 at 0.0068 and 0.0065 and keeps 11 at 0.00637, so depth alone does not decide it.

What the table supports is weaker: at rises below the transition, the offsets that keep 11 are the deep ones and the offsets that keep 7 are the shallow ones, and the boundary sits between 7 and 8.

Above the transition there are no offsets keeping 11 at any depth. So depth sorts the answers where both answers exist and does not produce them.

What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.
Fig. 14 Depth against family below the transition, where the boundary sits between offsets seven and eight.

What the round could not do

Locate offset 8’s own change. That would need it to wreck at adjacent rises across its bracket, and it does not — three rises of silence sit in the middle of it.

Nothing available forces a cut to wreck. Whether it does is decided by the rise and the offset together, and both are already fixed by the question.

So the honest report is one located rise for the lattice and a thirteen-step bracket for the one offset that changed, with the reason for the second stated rather than smoothed.

What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.
Fig. 15 The bracket that could not be narrowed, and the three rises of silence inside it.

What a control is, here

Every cut is grown beside an intact stem with the same rise, the same seed angle and the same history below the hole, and the displacement of every organ is the difference between the two runs. Without that there is no way to tell a stem that was disturbed from a stem that was always going to sit where it sits.

The control is also what makes recovers meaningful. A cut recovers when the displacement falls back to nothing and stays there; it wrecks when it does not. Both are readings against the control rather than properties of the cut run alone.

So an offset arriving in the wrecking set is the displacement on that offset’s run failing to return, at a rise one step finer than the last. Nothing about the control changed: it is the same stem at a rise one hundred-thousandth away.

Take away the organ nine places back, and the next one goes into the hole. The last both organs of a stem at a rise of 0.0064, unrolled. The open circle is the organ removed — nine 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 0.2° apart, against a local spacing of 29°, and the vacancy itself is 161.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 16 A cut and its control, from which every displacement in this thread is a difference.

The reading window, and why it is not the issue here

A surviving family is read over the top hundred and twenty organs of a three-hundred-organ run, and that window is one of the instrument settings this collection has learned to distrust — varying it moved three periodicity verdicts and revealed a drift.

It bites less on this reading. Rigidity is a boolean per lag, and across the census the rigid lags hold their angle to within a fraction of a degree while the others miss by tens, so the verdict is a separation rather than a line.

Where it would bite is on a survivor large enough that a hundred and twenty organs gives few repeats. At 11 that is eleven samples a class, which is comfortable; the readings that have to be declined are the ones at much larger lags.

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step.
Fig. 17 The profile a surviving lag is read from, which needs several repeats of that lag inside the window.

Two ways the table could have come out

A cut changes its mind. Some offset keeps 7 above a rise and 11 below it, at adjacent rises, with everything else held. That would have been the cleanest possible answer and would have made the transition a property of a single run.

A cut arrives. Which is what happened, and it makes the transition a property of the wrecking set — a quantity that moves everywhere in this thread and has no account attached to it anywhere.

The second is the less satisfying and it is what the table says. Writing down which of the two was expected, before the cutting, is what makes it possible to say that afterwards.

What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.
Fig. 18 The outcome the table produced, against the cleaner one it could have produced.

What this does to the exchange’s fifth cluster

Nothing, and the check is worth making. The exchange gained three rows from the lattice at 0.006, at a hop of 12.78 degrees, which is smaller than any it was fitted over — and that is what made it a test rather than a restatement.

Those three rows are offsets 8, 9 and 10, which are exactly the arrivals. So the fifth cluster comes from cuts that do not exist at rises eighty steps higher, and the row is a row of the rung’s fine end rather than of the ladder generally.

That is a limit on how far it generalises, not a doubt about the numbers. The three rows are measured the same way as the seventeen before them and they are the only rows of their kind available.

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 hop clusters the exchange’s correction is fitted over, whose fifth comes from these arrivals.

Where the arrivals sit relative to the front

Offsets 9 and 10 are the ninth- and tenth-newest organs, which on this lattice is near the back of the front. A removal there is a removal of an organ that is only just still one of the neighbours the next placement is computed against.

That is where wrecking is marginal by construction: at offset 11 and beyond nothing wrecks at all, so 9 and 10 sit at the boundary between wrecking and recovering.

An arrival at the boundary is therefore the expected shape. What is not expected is that a cut that only just wrecks holds a longer hop rigid than the cuts that wreck comfortably.

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. 20 Where wrecking stops, against which the arriving offsets sit at the boundary.

Why the arrival is the awkward answer

Because it moves the question one step and leaves it there. Where does the survivor change has a sharp answer; why does offset 9 start wrecking at that rise does not, and the wrecking set is a quantity nothing in this thread accounts for.

It moves from rise to rise on every band cut whole, in stretches on some offsets and single rises on others, with the counted pair and the settled divergence held. Three rounds have measured it and none has predicted it.

So the chain of explanation here is: the lattice first keeps eleven because an offset arrives; the offset arrives because it starts wrecking; and it starts wrecking for no reason anybody has written down. That is where it stops, and saying so is better than the alternative, which is a story about the front deepening that the table refuses.

The reason it refuses that story

Depth would be the natural account — offsets 9 and 10 are near the back of the front, the front deepens as the rise falls, and so the deep offsets come inside it and bring the new family with them.

Offset 8 kills it. It keeps 7 at 0.0068 and at 0.0065 and keeps 11 at 0.00637, at a depth that does not change. So depth cannot be what decides which family a cut keeps, because one offset at one depth gives both answers at different rises.

What the table does support is weaker and worth keeping: below the transition, the offsets that keep 11 are the deep ones and the offsets that keep 7 are the shallow ones, with the boundary between 7 and 8. Depth sorts the answers where both exist. It does not produce them.

What is claimed

That at the located transition on the Lucas 7/11 rung, the offsets already wrecking keep the same family on both sides and a new offset begins to wreck; that what the new offset keeps is the larger member of the counted pair; and that the located rise is therefore the rise at which the lattice first keeps that family anywhere.

That one offset does change its own answer somewhere in the search and cannot be located, because it does not wreck at three consecutive rises inside its own bracket.

And that the set of offsets keeping the new family fills in over about ten further steps of the grid rather than arriving at once.

The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere.
Fig. 21 The whole search read as an arrival: twenty rises, one new offset, and one located rise.

Where this goes next

The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere. One step of the grid, again A search of the fine end stepped from one rise to another a hundred grid steps away, and the family a cut keeps changed somewhere between. Cutting the rises in between puts the change inside one step, with the lattice identical on both sides. Offset 5 on the 5/8 band, and the change that cannot be placed. Every rise of the band, coarse on the left, with offset 5's cut drawn at each: pale where it recovers, dark where it wrecks and keeps 5, warm where it wrecks and keeps 20. It keeps the off-pair family at two rises above the handover and then does not wreck again for 34 rises, so its return is bracketed across a stretch that contains the handover. The flag that says a change sits at a handover fires here for the first time, and it is a statement about where the offset stops wrecking rather than about the handover. A change with nowhere to be The claim this whole thread rests on is that a survivor does not change where the two contact steps change places. Nineteen located changes never put one there. The twentieth is flagged at a handover, and it is flagged because the offset stops wrecking for thirty-four rises. 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. One offset, two answers Which contact family a wrecked stem keeps is decided by where the cut landed, at twenty-five of thirty offsets, by the simplest rule anybody would write down. It is refuted by two runs: the same counted pair, the same offset, two different rises, and two different surviving families.

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.

  • Which chains changed places — both name ablation, claim testing, discretisation, honest limits, lattice offset, measurement, resolution, rigid hop
  • A period the grid invented — both name ablation, claim testing, discretisation, honest limits, measurement, refusal, resolution
  • A step of one organ — both name ablation, claim testing, honest limits, lattice offset, measurement, resolution, rigid hop
  • Every rise of a band — both name ablation, claim testing, honest limits, lattice offset, measurement, resolution, rigid hop
  • One way round, seventeen times — both name ablation, claim testing, honest limits, lattice offset, measurement, resolution, rigid hop
  • The alternation is not a period — both name ablation, claim testing, honest limits, lattice offset, measurement, resolution, rigid hop

Named objects

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

AblationCensus designClaim testingContact familyDiscretisationHonest limitsLattice offsetMeasurementRefusalResolutionRigid hopTransition