Phyllotaxis
the spirals are counted, not admired
Stems and cones

Two rises far apart

The whole handover thread rests on the claim that where the two contact steps change places is not where the survivor does. On one rung both rises are now located to a single step of the grid, and they sit forty-five per cent of the rung apart.
14 min read 20 figures State the claim, state the testCounted, not admiredTwo routes, one number

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

Two things happen on a rung as the rise falls. The two contact steps change places — the family with the shorter hop across the surface becomes the other one. And somewhere, the family a wrecking cut leaves standing changes.

The claim this thread rests on is that those are not the same rise. It has been tested on bands, where the ordering is known to a sweep step and the changes to a bracket. On one rung it can now be tested with both quantities located to a single step of the grid the rises are named on.

Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.
Fig. 1 The two rises located on the Lucas 7/11 rung, each to one step of the five-decimal grid.

The two numbers

The hops cross between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all.

The survivor changes between 0.00640 and 0.00639, found by cutting twenty rises across two grids — eighty-nine cut stems and seven minutes.

Both are brackets one step of the hundred-thousandth grid wide. Neither is interpolated, neither is a sweep step, and neither is a bracket that had to be widened because something was silent.

The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.
Fig. 2 The geometry crossing, located by walking the whole rung and reading two closed-form lengths.

How far apart they are

One hundred and sixty-four steps of the grid. As a ratio in the rise that is 1.2564; as a share of the rung’s whole span in the logarithm it is 45 per cent.

The rung runs from 0.00947 to 0.00570. The crossing sits 33 per cent of the way down it and the survivor’s change at 78 per cent, so the two are separated by nearly half the rung.

Both are located to one step and the gap is 164 steps, so the separation exceeds the sum of the two uncertainties by a factor of eighty. That is not a comparison that needs care.

Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.
Fig. 3 The separation drawn on the rung, with the two located rises and the recorded handover.

Why this is a better test than any band

A band holds the lattice still, which is what makes it the right object for asking whether the survivor changes with nothing else moving. What it cannot do is locate a change against a handover, because a change is only bracketed between two rises at which its offset wrecks, and on one of the three bands cut whole that bracket is a hundred and eight grid steps wide.

A rung is looser about the lattice and tighter about the location. Across this stretch the settled divergence moves two tenths of a degree, four times a band’s tolerance — so a change anywhere in it could in principle be the lattice changing.

What makes the reading safe is that the divergence is held across the step: the two rises either side of the transition read 99.2578 degrees identically, and the counted pair is 7/11 on both.

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. 4 The cuts on the rung, whose transition is located to one step with the lattice identical across it.

What the bands had said

Nineteen located changes on the golden 8/13 band, none at its handover, sitting seven to fifty-seven rises below it. Zero changes on the Lucas 7/11 band. Two changes on the golden 5/8 band, one of them flagged at its handover and bracketed across twenty rises.

That is a strong negative with a soft edge. The nineteen are located to a rise of a band’s own sweep, which is two parts in a thousand — coarser than the grid, and they are all on one band.

This rung’s reading is a single case with a much harder edge, and it is on a different branch and a different pair from the band that carries the nineteen.

All three bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all.
Fig. 5 The three bands cut whole, whose nineteen located changes are the rest of the evidence for the claim.

What sits at the crossing

The census’s own row. The lattice the census holds on this rung is at 0.008, which is four grid steps below the crossing, and it is the nearest any census row comes to a handover.

Its cuts keep 7 — at offsets 5, 6, 7 and 8 — and 7 is the longer of the two hops there, because the crossing is above it and the 11 has already taken over as shortest.

So at the rise nearest the crossing the cut keeps the longer hop, which is the claim that the shortest hop is not the survivor at its least comfortable point.

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. 6 Which family each census row keeps against which hop is shortest there, including the row nearest a crossing.

And what sits at the transition

Nothing in the geometry. At 0.00640 and 0.00639 the shortest hop is 11 on both sides, the ratio between the two shortest lengths is 1.0695 and 1.0704, and both numbers are where the smooth curve says they should be.

The 11 has been the shortest hop for 164 grid steps by the time any cut keeps it. That is the sentence the round set out to write, and it is a negative: the ordering is not what moves the survivor.

If the ordering were the actor, the survivor would change at the crossing. It changes at about a third of the rung further down, in a stretch where the ordering has been settled and unremarkable throughout.

The two shortest hops across the Lucas 7/11 rung, and where they cross. How much longer the second-shortest hop is than the shortest, at every one of the 378 rises of the rung, coarse on the left. The ratio falls to 1.0003 at the crossing between 0.00804 and 0.00803 and rises away from it on both sides, which is what a crossing of two lengths looks like from above. It never reaches one, because the two lengths are equal only between two rises of the grid.
Fig. 7 The ratio between the two shortest hops, which is smooth and unremarkable through the transition.

The account this eliminates

The rung has a second kind of handover in it. That was the possibility the round was opened to test: that a cut keeps the shorter hop with a lag, and the survivor’s change is the ordering’s change arriving late.

A lag would be a mechanism — a stem whose pattern holds an old ordering for a while — and it would predict the two rises to be near each other. They are not near each other by any measure the rung offers.

So the survivor’s change is not the ordering’s change, delayed or otherwise, and the census’s decisive negative does not have to be read against a second handover.

Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.
Fig. 8 The two events, whose separation rules out the survivor’s change being the ordering’s change arriving late.

What is left unexplained

Everything about why the survivor changes where it does. The transition at 0.00640 has no geometric marker: the divergence is smooth through it, the hop lengths are smooth, the ratio between them is smooth, and the counted pair is constant across the whole rung.

What changes there is which offsets wreck — offset 9 starts wrecking and keeps 11 — and the wrecking set is itself a quantity that moves from rise to rise with nothing continuous behind it.

So one discrete quantity is explained by another discrete quantity and the chain stops there. That is where this thread’s explanations have stopped before, and saying so is more useful than a story.

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. 9 What changes at the transition, which is an offset arriving rather than anything in the geometry.

Two quantities, two costs

The geometry costs one grown stem per rise and closed-form arithmetic: 378 rises in three minutes. The cutting costs two grown stems per offset per rise: 20 rises in seven minutes, for a twentieth of the coverage.

That ratio is why the two are known to the same precision here only because the cutting was aimed at a bracket somebody had already narrowed. A blind sweep of the rung at the grid would be 378 rises of cutting, which is about two hours.

The design that made this affordable is: search coarsely, find a bracket, refine inside it. It is the same shape as the three sweeps that located a slot losing a wall, and it works because the quantity turned out to change once.

Three sweeps, each inside the last, and where the crossing turned out to be. The stretch of the golden 5/8 rung the design walked, drawn coarse to fine. The first sweep took ten positions between 70 and 97 per cent of the rung, because that is where the two free rows had been reported, and found every one of them already free. The second took ten between 61 and 70 per cent and bracketed the change between two of them. The third took nine at the five-decimal grid and put it between 0.00998 and 0.00997. A mark is a position cut; a filled mark is a rise whose slot still has two walls.
Fig. 10 The same design elsewhere: sweep coarsely, bracket, then refine at the grid.

What a second rung would be worth

A great deal, and there is one place to get it. The golden 8/13 rung is the only other rung whose pair carries a number a cut could keep off the smaller member, and no golden lattice on this ladder keeps 13 or 11 — the survivor ranks second or worse by hop length at every golden rise on a rung.

So the second instance is not available. The comparison of two located rises can be made on one rung of one branch, and nowhere else on this ladder.

That bounds the finding rather than weakening it. One case with both quantities located to a grid step is worth more than several with one of them bracketed, and it is what there is.

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. 11 The golden branch’s rungs, none of which can produce the same comparison.

The recorded handover

The ladder records this rung’s handover at 0.00800 rather than at 0.008035, because the ladder sweep steps by a ratio and 0.00800 is the nearest rise it visits.

That is 0.44 per cent of the rise away, and it changes nothing here: 0.00800 is 160 grid steps above the transition where 0.008035 is 164, and both are enormous compared with the one-step brackets.

It is recorded because a rise used in a comparison should be a rise somebody has located, and until the walk nobody had.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 12 Where each rung’s handover is recorded, by a sweep whose step never lands on the grid.

What the claim now rests on

Nineteen located changes on one band, none at that band’s handover, at a resolution of two parts in a thousand. One rung with both rises located to one part in a hundred thousand and 164 steps apart. One band with no changes at all and one band unable to test it.

That is a stronger base than the previous round’s, and the strengthening is in kind rather than in count: the rung’s reading is the first anywhere that does not depend on the sampling of either quantity.

It is still one rung, one branch, one counted pair.

Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it.
Fig. 13 The band that carries nineteen of the twenty tests, all of them on one lattice family.

The shape of the negative

Worth stating plainly because negatives are easy to overread. The claim is not that the ordering never matters to a cut. It is that the rise at which the ordering changes is not the rise at which the survivor does.

Those are different, and the first is a much larger claim. The ordering is one of four accounts of which family survives and it is not the one that wins, but it is not scored at zero either.

What this rung shows is that on it, the two events are 45 per cent of a rung apart. It does not show that the ordering is irrelevant to what a cut keeps.

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. 14 The accounts of which family survives, of which the ordering is one and is not the winner.

Why the two events were ever expected to coincide

Because the two families in question are the same two. A handover is the two contact steps exchanging which is shorter; a survivor is one of those two families being the one the wrecked run holds rigid. It would be odd if the ordering of two things had nothing to do with which of them survives.

The expectation was never that they coincide exactly. It was weaker and harder to refute: that the survivor is decided by the ordering, so that a change in one implies a change in the other somewhere near.

That is the version this rung refutes. The ordering changes at a third of the way down the rung, the survivor at four fifths, and there is no candidate for what carries a dependence across 164 steps of the grid.

The hops of a 3/5 lattice, shortest first — golden, rise 0.026. 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, 3 and 5, and they differ in length by a factor of 1.092. The lags left standing after a removal are 5, sitting at rank 1 in this order, so the family the rule holds is a short step but not always the shortest one.
Fig. 15 The ordering of hop lengths on one lattice, of which the top two are the pair a handover exchanges.

The alternative the census already refused

The census’s own result is stronger and cruder: the family a cut keeps is not the shorter of the two, scored over thirty wrecked cuts on ten lattices. Fourteen of the thirty keep the longer hop.

That refutes the shortest hop survives outright. What it does not refute is the softer reading — that the shortest hop matters and something else overrides it — and a coincidence between the two rises on a rung would have been evidence for exactly that.

There is none. So the census’s negative and this rung’s separation are two independent refusals of the same family of accounts, one over lattices and one along a rung.

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. 16 The census’s own refusal, over thirty wrecked cuts on ten lattices.

What is being compared, exactly

Two rises, and they are rises of different kinds, which is worth naming.

The crossing is a property of the intact stem. It is computed from the divergence a stem settles to and the rise, with no organ removed anywhere, and it exists at every rise of the rung whether or not a cut there would wreck anything.

The transition is a property of the cut stems. It exists only where a removal wrecks, and the set of offsets that wreck moves from rise to rise. So one of the two quantities is defined everywhere and the other is defined on a scatter.

That asymmetry is why the transition took eighty-nine cut stems to locate and the crossing took none, and it is why this comparison is possible on a rung and awkward on a band.

Both edges of the front heal; the middle of it does notThe same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 4 or 8 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.organ removed, counted back from the tiporgans placed before the stem is back on its lattice102243484never51256never7never8never9never105311591242137— the front ends here140150160rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right
Fig. 17 Which offsets’ removals the stem recovers from at one rise, and which wreck it and therefore have a survivor.

The number 45 per cent

A share of the rung’s span in the logarithm of the rise, which is the only measure that compares across rungs — a rung at the coarse end covers a factor of 1.5 in the rise and one at the fine end a factor of 1.7, and a difference in rise means different things at the two.

Forty-five per cent is large by any reading. The six bands on this ladder cover 3 to 72 per cent of their own rungs, so the separation here is wider than four of the six bands are.

Put another way: the transition is outside the band built around the crossing. The band around this handover runs from 0.00922 to 0.00721, and 0.00640 is well below its fine end — so a sweep of the band could never have found the transition at all.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 18 How much of its rung each band covers, against which the separation on this rung is wider than most.

Which is why the band said nothing

The Lucas 7/11 band was cut at every one of its 124 rises and not one offset changes its answer anywhere on it. That was reported as a clean negative and it is one.

What this round adds is where the change actually is: eighty-one grid steps below the band’s fine end. So the band’s silence was not a band with nothing in it; it was a band that stops above the interesting rise.

That is a useful thing to know about band sweeps generally. A band is defined by the divergence staying still, and there is no reason a survivor’s change should respect that definition.

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.
Fig. 19 Two bands cut whole, one of which stops eighty-one grid steps above the transition on its own rung.

What both numbers being located changes

Every earlier form of this comparison had one side known well and the other known badly. On a band the ordering is read at the band’s own sweep step and the changes are bracketed between rises at which an offset wrecks — so the answer was always these are not near each other, qualified by as far as either can be placed.

Here neither qualification is needed. Both rises are bracketed between two consecutive members of the grid the ladder names its rises on, which is the finest resolution anything on this site has, and the gap between them is 164 of those steps.

That is the difference between a negative and a measurement. The claim was never in much doubt; what it lacked was a case where the doubt could be quantified, and this is that case.

And what it does not settle

Whether the ordering matters to a cut at all. This rung says the two events are far apart; it does not say the shorter hop is irrelevant to which family survives.

The census answers the stronger form of that separately and negatively: fourteen of thirty wrecked cuts keep the longer of the two hops, so the shortest hop survives is refused outright. What is left is a softer reading in which the ordering contributes something and is overridden, and neither the census nor this rung can measure a contribution that is always overridden.

Measuring it would need a case where everything else is held and only the ordering moves. A band is exactly that object and there are three of them cut whole, which is why the band design exists alongside this one.

What is claimed

That on the Lucas 7/11 rung the two contact steps change places between 0.00804 and 0.00803 and the family a cut keeps first includes the larger member between 0.00640 and 0.00639; that both are located to one step of the grid the ladder names its rises on; and that they are 164 steps apart, 45 per cent of the rung’s span.

That the separation exceeds the sum of the two locations by a factor of eighty, so the comparison does not depend on how either was measured.

And that nothing in the geometry marks the transition: the divergence, the pair and both hop lengths are smooth through it, and what changes there is which offsets wreck.

Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.
Fig. 20 The whole comparison: two rises, two instruments, one step each, and 164 steps between them.

Where this goes next

The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there. The hops cross once Walking a whole rung at the grid its rises are named on costs a few hundred stems and no cuts at all, and it answers a question nobody had asked: whether a rung has one handover or several. It has one, and the ladder had recorded it in the wrong place. 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. 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. Not the shorter of the two If a damaged stem keeps one contact family standing, the obvious guess is that it keeps the nearer one. Across thirty wrecked offsets that is true twelve times and false seventeen, and on one lattice the two steps differ by a quarter of a per cent — where the words shorter and longer are doing no work at all.

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, handover, honest limits, measurement, negative result, resolution, rigid hop, rung
  • The alternation is not a period — both name ablation, claim testing, handover, honest limits, measurement, negative result, resolution, rigid hop, rung
  • The offsets that never change — both name ablation, claim testing, handover, honest limits, measurement, negative result, resolution, rigid hop, rung
  • Three offsets, three crossings — both name ablation, claim testing, handover, honest limits, measurement, negative result, resolution, rigid hop, rung
  • A family that is a multiple — both name ablation, claim testing, contact family, honest limits, hop length, negative result, resolution, rigid hop
  • The third band, cut whole — both name ablation, claim testing, contact family, handover, honest limits, negative result, resolution, rung

Named objects

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

AblationClaim testingContact familyHandoverHonest limitsHop lengthMeasurementNegative resultResolutionRigid hopRungTransition