The shallower front turns over

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.

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.

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.

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.

What is held fixed and what is not #

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.

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.

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.

The measurement #

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.

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.

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.

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 fine rung comes close and does not arrive #

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°.

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.

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.

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.

The share is not the only thing that changes #

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.

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.

Nine rises, and what averaging them hides #

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.

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.

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.

Which step of the ordering is actually established #

Three shares falling is an ordering, and the two steps in it are not equally supported. The arithmetic takes a paragraph and it changes what the essay should be quoted as showing.

The step to the fine rung is decisive. If the fine rung reversed at the middle rung’s rate of 4.69%, a hundred and twenty-one cuts would return no reversal about three times in a thousand; at the coarse rung’s 6.79% it would happen two times in ten thousand. Nothing about that reading depends on a model — it is the chance of a run of a hundred and twenty-one misses at a stated rate, and the observation is a hundred and twenty-one misses.

The step between the two coarse rungs is not established at all. 22 of 324 against 6 of 128 is a difference of 2.1 percentage points with a standard error of 2.3, which is under one standard error. Two shares that close on those denominators are what one rate looks like sampled twice.

So the reportable result is narrower than three falling numbers suggest, and it is worth stating in the form a reader can use. A front of eight reverses at under 2.5% and probably at zero, that being the upper bound a run of a hundred and twenty-one misses supports; a front of three and a front of five both reverse at something around five to seven per cent, and this measurement does not separate them. The reachability account predicted a fall with front depth and got one; what it did not get is a fall at every step, and two of the three points are a single point measured twice.

The nine rises are not a rate at all #

The lumpiness recorded below is not merely a caution about averaging — it can be priced, and priced it stops being a caution and becomes the finding.

Over the nine rises, fifty-nine cuts never repaired and twenty-two reversed, so a single common rate would be 37% of wrecked cells. Test the two rises that reverse nothing and have the most cells to do it with. Fourteen wrecked cells at a rate of 37% give no reversals about fifteen times in ten thousand; thirteen cells give twenty-four times in ten thousand. Those two rises alone, taken together, are a chance in a quarter of a million.

The other end is as strong. A rise with seven wrecked cells reversing all seven happens one time in a thousand at that rate, and two rises do it.

So there is no common rate to estimate. A rise on this rung either reverses its wrecked cells or does not, and the 6.79% is the fraction of rises that are of the first kind, dressed as a fraction of cuts. The number no rise produces is the number the table reports. That is the same defect this collection has recorded elsewhere as a mean over a bimodal ensemble, and it is worth naming as the same one rather than as a local untidiness.

Which changes the question the account has to answer. It is not what rate does a front of three reverse at but which rises reverse, and the answer visible here is the ends of the rung and not its middle — 0.050 and 0.120 at one extreme reversing everything they have, and 0.055 to 0.070 in the interior reversing nothing.

What the gap in the table is #

One feature of that breakdown deserves a sentence because it is easy to read past. The rises run 0.050 to 0.085 and then jump to 0.120, with nothing between.

That gap is not an omission. It is the band where the stem locks onto three eighths of a turn and never settles, and those rises are excluded before any cut is made, because cutting a stem that is not on a lattice answers a different question.

The consequence for the reading above is worth being explicit about. The claim that the ends of the rung reverse and the middle does not is drawn across a hole in the middle-to-upper part of the rung, so the 0.120 row is an end of the sampled range rather than a demonstrated end of the rung’s behaviour. Whether the pattern is genuinely a U across the rung, or a rise in reversal rate that the locked band interrupts, cannot be told from eight rises with four of them contributing two cells each.

That is a concrete next measurement rather than a limitation to live with: the rung below, which is coarser still and wider, has room for both a locked band and several rises on either side of it.

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.

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.