If reversing a stem means rearranging its whole front, then a stem with a shallow front should reverse more often. Measured across three rungs and four hundred and seventy-three cuts: 6.8 per cent at a front of three, 4.7 at five, and none at all at eight — where the nearest approach is two tenths of a degree away and stays there.
A stem that reverses its handedness after a cut is the most dramatic thing a
removal does. Every organ ends up on the
other side; the divergence lands at 360° minus what it
was; the counted pair and the order of the
shortest steps come back unchanged and
mirrored. It happens at the 3/5 rung and it does
not happen at 5/8, and the proposed reason is that a reflection has to be reached.
Fig. 1 What is being counted. A stem cut of two organs, settling on the reflection of the divergence it came from, with every family carried across at once.
The prediction that follows is about the front
— the run of most recent organs at which a removal is felt at all, which
this collection has measured and found to be
the larger of the two spiral counts. A front of three has fewer arrangements to
pass through than a front of eight. So a stem with a shallow front should reverse
more often.
This essay measures it at three fronts, and the ordering holds — though the larger
thing the coarse rung turns out to do is not a reversal at
all.
The comparison, and what makes the rows comparable #
The intervention is the same at every rung: two organs removed from the recent
history, named by an offset and a gap, with the heights prescribed by the rise and
only the azimuths of what follows left free. Every arrangement of the two within
the span is tried, and every wrecked stem is compared with a control that shares
its history to the last digit.
Fig. 2 The sweep at one rung, drawn as it is made: a cell per arrangement, an offset along one axis and the gap to the second organ along the other.Fig. 3 And what the second axis is for: moving the second removal through the profile the first one deformed swings the outcome over most of a turn with no trend in it.
The statistic is the share of all cuts that reverse the stem, not the share of
wrecked ones. That choice matters and is the conservative one: the coarse rung
wrecks more readily than the fine, so a share of wrecked stems would flatter it,
and a share of all cuts asks the question the prediction actually makes — how
often does an intervention of this size, applied to a stem with this front,
produce a reversal.
Fig. 4 The quantity the prediction is stated in, measured rather than assumed: the depth of the front at each rung, which is the larger of the two counts.
The rises are the ones the survey of the coarse rung
admitted. The coarse rung contributes nine
of them — every rise from 0.050 to 0.085 and the one at 0.120 — and the five in the
middle where the stem locks on three eighths of a turn are excluded, because a cut
made on a stem that never settled is answering a different question.
Fig. 5 The rises admitted and the rises refused, with the scatter each was judged on.
Three rungs is three different stems, so it is worth being explicit about what the
comparison controls and what it cannot.
Held fixed: the rule, in every particular — the falloff exponent, the loop bound,
the azimuth grid, the number of organs grown before the cut and the number placed
after. The intervention, which is two organs named by an offset and a gap. And the
comparison, which is always against a control sharing the cut stem’s history to
the last digit.
Fig. 6 The habit that makes the controls trustworthy: where two pieces of machinery compute the same thing, they are made to agree to the last digit rather than to a tolerance.
Not held fixed: everything geometric. Going from the coarse rung to the fine one,
the rise falls by a factor of five, the local spacing shrinks with it, the settled
divergence moves, the two contact steps change length and swap ordering twice, and
the number of organs in a turn of the stem goes from about three to about eight.
Fig. 7 Some of what moves: the divergence and its steadiness across the coarse ladder, both of which change while the pair does not.Fig. 8 And the step lengths at the coarse end, which are a different set of numbers from the fine end’s in every respect except the ordering.
So the front is not isolated. It is the quantity the account names, it moves
monotonically across the three rungs, and so does the outcome — which is what a
three-point test can establish and is not the same as an experiment that varies
one thing.
Fig. 9 The coarse rung broken out by rise. Each bar is the cuts that never repaired there, split by where they ended up.
Four hundred and seventy-three arrangements across three rungs, and the share
falls at every step. That is the ordering the reachability account predicts, and
it is not a small effect at the ends: a coarse stem is turned over by about one
cut in fifteen, and a fine one by none of a hundred and twenty-one.
Fig. 10 The background the shares sit against: how a single removal fares at each rung, which is the same axis with the dose set to one.
Three points make a trend and not a law, and the middle point is the one that
keeps it honest — 4.69 per cent is nearer to 6.79 than to zero, so the fall is not
a straight line in the front and nothing here claims it is. What the three rows
support is an ordering.
What a reversal is, and why it is not a small change made larger #
A reflection is worth describing precisely, because the reachability account
depends on how large a rearrangement it is.
Take a stem with divergence d. Its mirror has divergence 360° − d, and the two
have the same counted pair and the same ordering of shortest steps — a 3/5 lattice
at 139.69° and one at 220.31° are geometrically identical patterns of opposite
handedness. Nothing about the packing is different. What is different is which way
the spirals wind.
Fig. 11 A mirrored stem beside the lattice it was cut from. The same arrangement, wound the other way, with every family carried across at once.
For a stem to get from one to the other, every organ in the front has to move. There
is no path along which one family reverses and the others follow later: the
families are determined by the divergence, so they all change at once or none of
them does. That is what makes a reversal a whole-front rearrangement rather than a
large version of a small displacement.
Fig. 12 The size of the displacements a cut produces at this rung, offset by offset. A reversal is not at the top of this scale; it is off it.Fig. 13 And the intervention that has to accomplish it: two organs out of a front of five, with everything below untouched.
That is the sense in which a shallower front should help. Three organs is three
organs to rearrange; eight is eight, and the rearrangement has to happen while the
rule is placing organs one at a time against what is left.
The zero deserves more than a zero, because “none of a hundred and twenty-one” is
consistent with two very different situations: a rung where nothing goes near the
mirror, and a rung where things go near it and miss.
It is the second. Thirty-two of the hundred and twenty-one arrangements at 5/8
never repair, and the nearest any of them comes to the mirror is 0.199°.
Fig. 14 Where the fine rung’s wrecked stems actually go. A short list of destinations, one of which sits two tenths of a degree from the reflection without being it.
Two tenths of a degree is four times the tolerance at which a mirrored stem is
recognised as one. The mirrored cells at the coarse rungs agree with the
reflection to 0.0000° — four decimal places, at every one of the
twenty-eight — so the threshold is not separating two clouds that overlap. It is
separating a set of exact hits from a near miss, and the near miss is reported as
near rather than as on.
Fig. 15 The near miss in context. The fine rung’s destinations, one of which is a slip that happens to land close to where a reflection would be.Fig. 16 And what that near-miss destination actually is: a slip of the old lattice with a family left standing, which arrives at its divergence for reasons that have nothing to do with reflection.
That distinction is the reason the tolerance in the machinery is a twentieth of a
degree rather than something comfortable. A tolerance of half a degree would have
turned this rung’s zero into a one, and the resulting table would have shown the
prediction failing for a reason that was entirely an artefact of a threshold.
Why a share of all cuts rather than of wrecked ones #
The choice of denominator decides the table, so it deserves its own section.
Three denominators were available. All cuts tried is the one used here.
Cuts that never repair is the obvious alternative. Cuts that are felt at
all is a third.
On the share of wrecked stems the ordering does not hold. The coarse rung
reverses 22 of its 71 wrecked cells — 31 per cent — and the 3/5 rung reverses 6 of
8, which is 75. That table would have said the middle front reverses most
readily, and it would have been reporting the fact that the 3/5 rung barely
wrecks at all: eight wrecked cells out of a hundred and twenty-eight.
Neither table is dishonest and they answer different questions. “Given that a stem
was wrecked, how often did it reverse” is a question about destinations. “Given
that two organs were removed, how often did the stem end up reversed” is a
question about outcomes, and it is the one the reachability account makes a
prediction about — the account is about whether the rearrangement can be reached,
not about how the wrecked stems divide once it has been.
Fig. 17 The other question, at the finest of the three rungs: given a wrecked stem, which of the available destinations it reached.
Both are reported here. The share of all cuts falls 6.79, 4.69, 0.00 with the
front; the share of wrecked cells runs 31, 75, 0 and is not monotone in anything.
Saying which one the prediction is about, before computing either, is the whole of
the discipline available on a three-point comparison.
Two other things move with the front and are worth reporting beside the shares,
because a single statistic moving in the predicted direction is weaker evidence
than a picture that hangs together.
The rate of wrecking rises as the stem gets finer. At the coarse rung, 71 of
324 arrangements never repair — about one in five. At 3/5, 8 of 128. At 5/8, 32
of 121, which is more than one in four. So the coarse rung is not simply more
fragile in every respect; it is more likely to reverse and less likely to be
wrecked at all than the finest of the three.
Fig. 18 The rate as the intervention grows, at one rung. More organs removed wrecks more arrangements, which is the ordinary half of the picture.
The number of destinations rises too. The coarse rung’s wrecked stems reach
two places and one of them is the mirror; the 5/8 rung’s reach seven. A rung with
more places to go is a rung where any particular place is reached less often, and
the mirror is one place among however many there are.
Fig. 19 The coarse rung’s two destinations, rise by rise. There is not a spread of outcomes there; there are two.Fig. 20 A wrecked stem at each of two rungs, read as a sequence. Both settle; what differs is how many places there are to settle at.
The coarse rung’s 6.79 per cent is an average over nine rises, and the nine do not
behave alike. Four of them reverse nothing at all; three reverse every wrecked cell
they have; one splits; and one has two wrecked cells and reverses both.
rise
cuts that never repaired
reversed
0.050
2
2
0.055
14
0
0.060
13
0
0.070
7
0
0.075
7
4
0.080
7
7
0.085
7
7
0.120
2
2
Fig. 21 The rung broken out, so that the average is visible as an average. Four rises contribute nothing to it and three contribute everything they have.
That is a much lumpier picture than a single share suggests, and the honest
reading is that the coarse rung has two things going on rather than one rate. The
rises at either end of the rung reverse; the rises in the middle do something
else. Reporting the rung’s share without the breakdown would be reporting a
number that no rise on the rung actually produces.
Fig. 22 The rung the shares are taken across, for the record: nine rises, each a settled lattice, each contributing thirty-six cells.
It also means the three-point comparison is weaker than three points look. The
coarse rung’s number is an average over a rung whose behaviour changes across it;
the other two rungs contribute one and two rises. Nothing here separates “a
shallower front reverses more often” from “the ends of a rung reverse and the
middles do not, and the coarse rung has proportionally more end”.
One more comparison is available and worth stating because it costs nothing. The
mirrored cells at both coarse rungs agree with the reflection to four decimal
places, and they do so from arrangements that differ in where both organs sat. A
destination reached to that precision from several different interventions is a
state of the rule rather than a residue of the damage, which is the same thing the
four-cycle turns out to be.
What the account gets right and what it does not #
Right: the ordering, at three fronts, on the conservative statistic, with the
extreme case at exactly zero and a near miss reported as a near miss.
Fig. 23 The response a reflection has to rearrange, at the coarse rung. Every organ in the run is felt, which is why the reversal is not a small change made larger.
Not right, or at least not established: that the front’s width is the operative
quantity rather than something that varies with it. Everything about a stem
changes as the rise does — the divergence, the step lengths, the number of
destinations, the wrecking rate — and three points cannot separate the front from
its correlates. The account survives a test it could have failed, which is worth
having and is not the same as being shown to be the mechanism.
There is also a result the prediction did not anticipate at all, and it is the
larger of the two findings on this rung. At four of the nine coarse rises, not one
wrecked cut reverses — and what those stems do instead is not settle anywhere.