Below the 3/5 rung is a 2/3 rung, and it runs from a rise of 0.050 to 0.120. It is not a lattice across all of it: from 0.090 to 0.115 the divergence stops settling and sticks on exactly three eighths of a turn, wobbling by a degree and a half — while a counter goes on reporting 2/3 as though nothing had happened.
A two-organ cut wrecks a stem at the 3/5
rung, where every single removal heals, and what
the wrecked stem settles into is the lattice it was cut from wound the other way:
the same divergence subtracted from a full turn, the same counted pair, the same
order of shortest steps, mirrored. At the 5/8 rung nothing of the kind
happens.
The account offered for that difference was reachability. A reflection is not
a small change — every organ of the front has to end up somewhere it was not — so
a stem falls into its mirror only if the
rearrangement is short enough to be carried there. A front of five has fewer arrangements to pass through than a front
of eight, so the coarse rung reaches its mirror and the fine one does not.
Fig. 1 The outcome the account is about. A stem cut of two organs, settling into the reflection of the lattice it came from.
That predicts something testable: a 2/3 stem should reverse more readily than a
3/5 one. The rung below 3/5 is coarser than anything this collection has grown,
and the obvious hazard was stated when the prediction was: a stem that coarse may
not be a lattice at all, and cutting a thing that is not a lattice answers a
different question.
Fig. 2 The ladder the rungs sit on. Sweeping the rise, the counted pair holds constant over a range and then steps, and the coarse end of it is where this essay works.
Swept from a rise of 0.040 up to 0.130 in steps of 0.005, the counted pair is 3/5
at the first two, 2/3 from 0.050 to 0.120, and 1/2 from 0.125 upwards. So
there is a 2/3 rung, it is nine rises wide at this resolution, and there is a
coarser rung below it again.
Fig. 3 The whole coarse ladder. Each bar is how much the divergence wanders over the last sixty organs at that rise, with the pair a counter returns written underneath.Fig. 4 What a stem on the 2/3 rung looks like, unrolled. Three organs to a turn and a bit, which is as coarse as an arrangement gets before it stops having two families at all.
The settled divergence slides across the rung the way it does across every rung —
and what that slide does to a front is its own
question, as is what a coarse enough
rung does to the mirror:
140.47° at the coarse end, down through 139.45°, 138.75°, 137.93°, 137.11° and
136.17° to 135.35°, with the counted pair constant across the whole slide. That is
the ordinary picture and there is nothing wrong with it.
Fig. 5 What a rung member does, read as a run rather than as a number: a transient, and then a divergence that holds flat to the resolution of the grid.
At those five rises the mean divergence is 135.0000° — to four decimal places, at
every one of them — and it does not hold still. The scatter over the last sixty
organs runs from 0.79° to 1.60°, where every other rise on the rung scatters
between 0.0000° and 0.3356°.
rise
mean divergence
scatter
0.080
136.1719°
0.0000°
0.085
135.3516°
0.1172°
0.090
135.0000°
1.5982°
0.100
135.0000°
1.5099°
0.115
135.0000°
0.7948°
0.120
135.4688°
0.0000°
Fig. 6 The band on its own, at the resolution that shows it. Five bars an order of magnitude taller than their neighbours, and the line is the threshold a rise has to pass before a cut is made on it.
135° is three eighths of a turn. The stem has not settled on a lattice
divergence; it has locked on a rational one, and it wobbles about it by more than a
degree.
Fig. 7 Three eighths of a turn, drawn exactly. Eight files, radial gaps between them, and none of the packing that makes a spiral arrangement worth studying.Fig. 8 Where 135 degrees sits among the fractions: a denominator of eight, which is small enough to make files and is nothing like the golden angle’s position.
Before the band is worth calling a hazard, the rung it sits in has to be shown to
be an ordinary rung, or the whole ladder is suspect rather than one part of it.
Four things say it is. The counted pair agrees with the contact geometry at every
rise — the counter, shown the positions and told nothing, returns the pair that
the two shortest steps say it should. The rung has boundaries in the right places:
3/5 above it and 1/2 below, each replacing the pair one step along the ladder that
every other rung on this site follows. The settled divergence slides
monotonically across it. And the front, measured by removing one organ at a time
and asking which offsets are felt, is three organs deep, which is the larger of
the two counts, exactly as at every finer rung.
Fig. 9 The front at the coarse rung beside two finer ones. Three organs, which is the larger of its two counts, and the same rule holds at all three.Fig. 10 And the same measurement made a second way across seven finer rises, on stems grown from a stated seed, so that the front is not being read off a single run.
So the coarse rung is a rung. What sits in the middle of it is not a failure of
the ladder; it is a state the rule can occupy that the ladder has no row for.
This is the part that makes the band a hazard rather than a curiosity.
A counter shown a locked stem returns 2/3 — the same pair as every other rise
on the rung. It is not confused and it is not wrong. Three eighths of a turn is
close enough to the rung’s own divergences that the two shortest chains of near
neighbours are still the same two, and the wobble is large enough that the eight
files three eighths would make never quite form.
Fig. 11 Why the counter reports what it does: the pair it returns as the divergence is swept changes at boundaries, and 135 degrees is not near one of them at this rise.Fig. 12 The counter run across the band itself, reporting the rung’s pair at every rise in it. Nothing in the counted output marks these rises as different.
So every instrument that decides what a stem is by counting it says the band is
ordinary 2/3. The only measurement that separates it is the one nobody thinks to
make: how much the divergence moves once it has stopped moving.
Fig. 13 The measurement in question, on a stem that does settle. The transient dies and the divergence goes flat, which is what “settled” is being asked to mean.Fig. 14 And the general form of the caution: a tolerance stated loosely enough will admit a stem that is still moving, and the admitted rows then look like ordinary rows.
It is worth recording what does not break in the band, because “not a lattice” is
a strong phrase and the object is more specific than that.
A locked stem still packs. Its organs are placed by the same rule at the same
prescribed heights, they do not collide, and a counter finds two families in
them. It is not disordered, and none of the disorder statistics this collection
uses reports anything unusual about it.
Fig. 15 A locked stem in the round. Regular, non-colliding, two families, and no divergence.
And it still repeats, after a fashion. Three eighths of a turn brings the
pattern back to itself every eight organs, so the neighbourhood an organ is placed
into is nearly periodic. That is what makes the state stable at all — the rule is
sitting in a landscape that repeats, rather than being pushed around at random.
Fig. 16 The statistic that would have caught genuine disorder, swept across a stretch of the divergence where the rule is well behaved. It reports nothing unusual about the locked band either: a stem stuck on a rational is not disordered.
What it does not do is hold a single divergence, which is the one property every
claim in this thread is stated over. A cut is made against a control, the control’s
settled value is the reference, and a stem with no settled value has no reference
to be compared against.
Fig. 17 Why the reference matters: every claim about what a cut did is a difference from what the undisturbed stem was doing, and that requires the undisturbed stem to be doing one thing.
A rise is admitted to the sweep if its divergence scatters by no more than half a
degree and its counted pair agrees with its contact geometry.
Half a degree is not a delicate number here. The rises that settle scatter by at
most 0.3356° and the rises that do not scatter by at least 0.7948°, so any
threshold between those two sorts the ladder identically. The gap is four tenths
of a degree wide and the threshold sits in the middle of it.
Fig. 18 The gap, drawn. Nine bars below the line and six well above it, with nothing in between.
That matters because the whole point of the test is that it rejects something. A
test written after the rows were chosen would have admitted everything; this one
throws out five of the nine rises in the middle of the rung, which is a third of
the ladder and includes the rises a sweep by round numbers would have picked
first.
Fig. 19 Why a scatter measurement is the right instrument for this: it is the quantity that separates a lattice from a stem still being pushed around, and it is sensitive well below the resolution of a count.
The band is invisible to four of the five things anybody would check, and going
through them is the point of the essay.
The counted pair says 2/3 across the whole rung, band included. The front,
measured by removing one organ and asking which offsets are felt, is three organs
deep across the whole rung. The mean divergence at 135.0000° sits between the
values at 0.085 and 0.120 and is exactly where a smooth slide would put it — it is
not an outlier on the curve. And a drawing of a locked stem looks like a
drawing of a coarse stem, because a wobble of a degree and a half at a spacing of
a hundred and twenty degrees is not something an eye picks up.
Fig. 20 A locked stem, unrolled. It looks like an ordinary coarse arrangement, because at this spacing a wobble of a degree and a half is not visible.Fig. 21 And a settled one at a neighbouring rise, for the comparison. The difference between these two is a number rather than a picture.
The fifth thing does see it, and it is the one this collection has been in the
habit of measuring since the collection began: how much the divergence moves once
the transient is over. That is not a check anybody runs to decide whether a stem
is a lattice; it is a check run to decide whether a run is long enough. It happens
to be the only instrument here that separates the band from the rung.
Fig. 22 The one measurement that sees it, on the rises that matter: how much the divergence of each stem is still moving once its transient is over.Fig. 23 And what settling looks like when it happens, which is the shape the band never reaches.
That is the shape of a defect worth recording. Not a wrong number produced by a
broken instrument, but a state that every instrument reports correctly and that
nobody would think to ask about.
Not the subject of this essay, and worth a paragraph because leaving it unnamed
would suggest it is unexplained.
Three eighths of a turn is a rational divergence with a small denominator, and a
rational divergence is a fixed point of a different kind: the pattern repeats
exactly every eight organs, so an organ placed at the argmin of the profile has a
perfectly periodic landscape to sit in. The rule can hold that, and at rises where
the true lattice divergence passes near it the rule apparently prefers it — but
holding it exactly would require the wobble to vanish, and it does not.
Fig. 24 The general fact behind that: rationals with small denominators are the places a pattern can repeat exactly, and they are sparse and strong.Fig. 25 And the general behaviour: which state a rule settles into is a matter of basins, and a state can be reachable, stable and not a lattice all at once.
Five rises of nine is a third of the rung, and the ones lost are in the middle,
which is where a sweep would naturally have been made.
A sweep at three round rises — 0.05, 0.10 and 0.15 — would have taken one member of
the rung, one member of the band and one member of the rung below. Two of those
three rows would have been about something other than a 2/3 lattice, and the table
would have shown it as a 2/3 lattice, because the counter says so at all three.
Fig. 26 The three rises a round-number sweep would have chosen, and what is actually at each of them.
That is not hypothetical caution. The result the coarse rung is being cut to test
is a share — how often a two-organ cut reverses a stem — and a share computed over
thirty-six arrangements at each of three rises is a share over a hundred and eight
cells, two thirds of which would have been cells of a stem that had no divergence
to reverse.
Fig. 27 The shape of the table such a sweep produces, drawn at a rise that is a lattice. Run on a stem from the locked band instead, every cell of it would be answering a different question.Fig. 28 The band at the resolution that finds it, which is a step of 0.005 rather than 0.05. A coarser sweep of the rise cannot see this at all.
What the test costs, then, is a factor of ten in the resolution of the rise sweep
and a stated criterion. What it buys is a table whose rows are all about the same
object.
Nine rises of the 2/3 rung, from 0.050 to 0.085 and at 0.120, each a settled
lattice by a stated test, each counted at the pair its contact geometry gives, and
each therefore a legitimate thing to remove two organs from.
Fig. 29 And what the second axis of that table means, drawn at a finer rung where the front is deep enough for the gap to have somewhere to go: moving the second removal through the profile the first one deformed, which is why a two-organ cut is not two one-organ cuts.
That is a smaller ladder than the rung, and the difference is the essay. The
prediction under test — that a shallower front reverses more readily — is worth
nothing measured on stems that were never lattices, and the rows it would have
been measured on are the ones a sweep at 0.09, 0.10 and 0.11 would have chosen.