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.
A wrecked stem keeps one lattice hop rigid, and
the hop it keeps is one of the two contact
families at twenty-nine of thirty
offsets. That is as far as the census had been pushed, and it leaves a question
with only two possible answers, which is
the kind of question that usually gets settled in a sentence.
It is false. Across the census it is right at twelve of thirty offsets and wrong
at the rest.
Fig. 1 The whole census, with every cell shaded where the family left standing is the second-shortest step on its lattice rather than the shortest.
What “shortest” means here, and how close the two are #
A hop’s step is not measured up the stem. It is measured across the surface: the
azimuthal difference and the height difference taken together, on a cylinder whose
circumference is one turn. That is the distance the placement rule itself uses,
which is why it is the one to rank by.
Fig. 2 The ranking, at one rise. Each lag’s step across the surface, with the two shortest sitting well below everything else and very close to each other.
Ranked that way, the first two entries are always the contact
pair. The gap
between them is the number this essay turns on, and it is small. Across the ten
lattices that produce a wrecked stem it runs from
1.0024 to 1.1072 — that
is, the second-shortest step is between a quarter of a per cent and eleven per
cent longer than the shortest.
Fig. 3 One lattice’s hops in order, with the families a wrecked stem here leaves standing marked. The slider walks every lattice in the census.
So the two candidates are nearly the same length, and the question of which one
the rule holds is not the question of which one is nearer by any comfortable
margin. On a lattice where the difference is a quarter of a per cent it is not
really a question about length at all.
Set aside the single offset whose survivor is not a contact family — the long
four-hop at 8/13, which is neither the shortest nor the second — and twenty-nine
offsets remain with a survivor that is one of the pair.
At twelve of them the survivor is the shortest step. At seventeen it is the
second-shortest.
Fig. 4 One of the seventeen. This lattice’s shortest step belongs to its eight-family, and the lag standing rigid after the cut is five.
Some individual rows are worth reading, because the effect is not a scatter of
near-ties. At the 5/8 lattice with a rise of 0.013 the shortest step belongs to
the eight-family, and both wrecked offsets keep the five — whose step is
1.074 times as long. At 8/13 with a rise of 0.005 the shortest step is the
thirteen and four of the five wrecked offsets keep the eight, at 1.088 times
the length. On the Lucas branch at 4/7 and a rise of 0.020, the shortest is the
seven and both wrecked offsets keep the four, at 1.082.
Fig. 5 The 8/13 lattice, where the shortest step is the thirteen and the family the cuts leave standing is the eight.Fig. 6 And the Lucas branch at 4/7, where the shortest is the seven and both wrecked offsets keep the four.
Those are three different lattices, on two different branches, all keeping the
family with the longer of the two contact steps. And the reverse happens too:
at 3/5 both wrecked offsets keep the five, which there is the shortest, and at
the Lucas 4/7 with a rise of 0.013 four of five offsets keep the seven, which is
the shortest there.
Fig. 7 The intervention that produces every row of the table: one organ out of a settled stem, with the heights prescribed and only the azimuths of what follows left free.
Neither reading survives. “The shortest hop is kept” holds at twelve of
twenty-nine; “the longest of the two is kept” holds at seventeen. Both are
descriptions of a subset dressed as accounts — which leaves the offset as the only
ingredient not yet tried, and it does not finish the job
either.
One clarification before the counting, because the ranking and the counted pair
are two different measurements that agree, and the agreement is doing work.
Fig. 8 The counting, done the way it is always done here: chains of near neighbours followed through the positions, with no angle anywhere in the computation.
The ranking comes from the angle. Given the settled divergence and the rise, the
step of the lag-k family is a two-line calculation, and sorting those gives an
order over every lag out to forty.
Fig. 9 The ranking, computed from the divergence and the rise alone. It knows nothing about where any particular organ is.
The two agree: at every lattice in the census, the first two entries of the
ranking are the pair the counter returns. That is what “contact family” means, and
having it come out of two independent computations is why the phrase can be used
in a claim rather than as a definition dressed up as one.
Before the cut there is no asymmetry to appeal to #
It is worth ruling out the possibility that one family was already the weaker of
the two, since that would make the whole question a matter of which was more
fragile to begin with.
On an undisturbed stem both contact hops are exactly rigid. Measured over the
same hundred and twenty organs the census uses, the divergence of a settled stem
holds to a hundredth of a degree or better, and every hop derived from it holds
to the same. There is no wobble in one family and steadiness in the other; there
is a lattice, in which every lag is as fixed as every other.
Fig. 10 The state the cut is made in. A stem placed organ by organ by the rule, settling into a lattice in which every hop is as steady as every other.Fig. 11 The same thing read as an angle, on a stem climbing its ladder: wherever it stops, the divergence is flat to a hundredth of a degree, which is what makes a shift of forty-five degrees after a cut a signal rather than noise.
So the asymmetry that decides which family survives is created by the removal. It
is not a pre-existing weakness that the removal reveals, which is what a
length-based account would have to be.
The lattice where the words stop meaning anything #
One row deserves separating out, because it is the case that shows why a claim
about length was never going to work.
The Lucas lattice at a rise of 0.008 carries 7/11, and its two contact steps
differ by a factor of 1.0024 — a quarter of one per cent. All four of its
wrecked offsets keep the seven, which is the second-shortest. But on a lattice
where the first and second differ by two parts in a thousand, saying that the
rule “chose the longer one” is reporting a rounding.
Fig. 12 The 7/11 lattice at a rise of 0.008, where the two shortest steps are within a quarter of a per cent of each other and the ordering between them carries no information.
The machinery flags rows like that rather than dropping them: a separation of one
per cent is the threshold, two of the ten lattices are under it, and their rows
are counted in both totals with the flag attached. Dropping them would have moved
the score and hidden the reason.
Fig. 13 The general form of the same caution: a threshold applied to a quantity whose two sides nearly touch is a threshold that decides the answer rather than testing it.
Both families are inside the neighbourhood, and that is measurable #
The reason “shortest wins” fails is not mysterious once the rule’s own profile is
looked at directly.
At the growing tip, the rule minimises a sum of terms, one per organ already
placed, each falling off as a power of the distance. Rank those terms by size and
read off which organs they belong to. The five largest belong to lags 13, 8, 5,
21 and 26 — and they are the same five at every falloff exponent from 1.5 to 6.
Fig. 14 Where that comes from: the rule’s neighbourhood, counted four different ways as its falloff exponent is swept. The organs at the top of the profile do not change.
Both members of the contact pair are near the top of that list, and by amounts
that are comparable rather than separated. The rule is not holding one family
while barely noticing the other; it is holding both, and a cut has to decide
between two things it is doing at once.
Fig. 15 The same point from the geometry rather than from the sum: at a lattice both contact families sit at nearly the same distance from the tip, and the pair is the pair because of that.Fig. 16 What the rule actually sees at the moment of placing an organ: a handful of near neighbours at very similar distances, and a long tail that contributes almost nothing.Fig. 17 And the same fact in the packing: a cell in a spiral lattice touches members of both families, which is what makes both of them contact families in the first place.
So the shape of the answer is: the rule holds two families, a single removal
breaks the symmetry between them, and the length of the step is not what decides
which way it breaks. A quantity that varies by four per cent across the census
cannot be what settles a binary outcome that changes within a single lattice as
the offset moves by one organ.
Fig. 18 The scale of the thing being decided: at one lattice, moving the removal by one organ changes whether the stem repairs at all, let alone which family it keeps.
The gap between the two contact steps is the quantity every version of the guess
depends on, so it is worth having the whole distribution rather than the range.
Across the ten lattices that produce a wrecked stem, the second-shortest step is
longer than the shortest by 1.0024, 1.0131, 1.0738, 1.0750, 1.0760, 1.0817,
1.0867, 1.0883, 1.0917 and 1.1072. Eight of the ten lie between seven and eleven
per cent; two lie under one and a half.
Fig. 19 One of the two narrow ones. The 5/8 lattice at a rise of 0.016, where the two contact steps differ by one and three tenths per cent.
Those numbers have a reason and it is the ladder. A rung is the range of rises
over which one pair is the shortest two, and the boundaries of a rung are exactly
the rises at which the second-shortest overtakes the shortest — so the gap
necessarily passes through zero at every transition and is largest in the middle.
The two narrow lattices are the two nearest a boundary.
Fig. 20 Where the gap goes to zero: at each transition the two shortest steps exchange places, so the ordering between them is undefined there and small on either side.Fig. 21 The same structure drawn as the classical diagram, where the rungs are branches and the transitions are the points at which two families are equally short.
So “the shorter family survives” is a claim whose subject matter shrinks to nothing
twice on every rung, and whose margin is eleven per cent at best. Even where it is
right, it is right by an amount that a placement rule reading a hundred and eighty
organs has no obvious way to notice.
The choice is not cosmetic, and the size of what it decides is worth stating,
because it is the reason a binary outcome is worth an essay.
A wrecked stem’s divergence lands at the one it was cut from plus a whole number
of turns of the surviving family, divided by that family’s period. Keep the
eight and the available destinations are 45° apart. Keep the five and they
are 72° apart. The measured slips bear that out exactly: at the offsets that
keep an eight the divergence moves by 44.97° to 45.03°, and at the offsets that
keep a five it moves by 71.95° to 72.00°.
Fig. 22 The consequence, on the axis it lands on. Which family a cut leaves standing sets the spacing of the places the stem can end up, and the places are what a reader of the finished stem would measure.
So a difference of four per cent in step length — or a quarter of a per cent, on
the lattice above — is deciding a difference of twenty-seven degrees in where the
stem ends up. That ratio is what makes “the shorter step wins” an unsatisfying
account even where it happens to be right: the quantity it appeals to is far too
small and far too smooth to be producing an outcome that large and that discrete.
Fig. 23 For scale, the largest thing a cut can do to a stem at any rung: reverse it. The choice this essay is about sits between that and no change at all.
Everything above is stated over a census that mixes two branches, and it is worth
separating them, because a result that held on one and not the other would be a
result about seeds rather than about rules.
The golden branch contributes nineteen wrecked offsets and the Lucas branch
eleven. On the golden branch the survivor is the second-shortest step at eleven of
nineteen; on the Lucas branch at six of eleven. Both branches keep the shortest
sometimes and the second-shortest more often, in about the same proportion, and
neither is close to doing one or the other consistently.
Fig. 24 The two branches at one rise. The same rule from different seeds, settling on lattices with different counted pairs and the same structure.Fig. 25 And their fronts across seven rises, which is what a removal is made inside. Nothing about the response distinguishes the branches in a way that would separate this result.
That matters because the Lucas branch is where the two clearest individual cases
sit. The 4/7 lattice at a rise of 0.020 keeps its four-family at both wrecked
offsets, and four is the longer of its two contact steps by eight per cent. The
7/11 lattice at 0.008 keeps its seven at all four, and seven is longer than eleven
by a quarter of a per cent. If the census had been golden-only, the second of
those — the case that shows the ordering can be meaningless — would not be in it
at all.
Fig. 26 The Lucas branch responding to a removal at the rise where its longer contact step is the one that survives.
Two things, and they are the reason this is an essay rather than a footnote.
The first is that it removes the account that would have stopped the search. “The
rule keeps its nearest neighbours” is satisfying, sounds like mechanism, and would
have been repeated. It is worth knowing that it is wrong before it is repeated,
and the way to know is to count the cases rather than to check the case that
suggested it.
Fig. 27 The blocks that gave rise to the guess, at four lattices. They are contact numbers, which is what makes “the nearest family survives” sound right until the two candidates are ranked.
The second is that it points at what the answer has to depend on. If neither
member of the pair is favoured by length, then what distinguishes them must be
something about the cut rather than about the lattice — where the removal lands
relative to each family, rather than how long each family’s step is.
Fig. 28 The outcome the choice decides. Two lattices, two wrecked stems, two different repeating motifs, and the period of each is the family that was kept.Fig. 29 The next question, marked on the same census: the cells a rule stated over the offset alone gets wrong.
That is the direction the next essay takes, and it produces the best rule
available over those two numbers — together with the pair of runs that shows no
such rule can be the whole account.
Essays that name at least two of the same things, and that neither author linked.
A wreck has a short list— both name ablation, falsifiability, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rigid hop
One turn per survivor— both name ablation, falsifiability, honest limits, lattice, measurement, parastichy pair, the placement rule, rigid hop, rise
The shallower front turns over— both name ablation, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung, tolerance
Three organs and no mirror— both name ablation, falsifiability, honest limits, lattice, measurement, parastichy pair, the placement rule, rise, rung
Two accounts of one number— both name ablation, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
A cut of two organs— both name ablation, honest limits, lattice, measurement, parastichy pair, the placement rule, rise, rung