The share was not the thing

The hypothesis is easy to state and was the obvious one. A coarse stem reverses its handedness when two of its five front organs are removed. A fine stem, cut of one or two, does not — but two out of thirteen is a much smaller intervention than two out of five. Perhaps the quantity that matters is the share of the neighbourhood removed, and a fine stem given the same proportional damage would turn over too.

It is a good hypothesis. It has a number in it, the number is a ratio so it is comparable across arrangements, and it makes a prediction that can be pushed past the point of comfort: take more than forty per cent of a fine stem’s front and it should mirror.

This essay pushes it there and it fails.

What “the share of the front” means, precisely #

The hypothesis names a ratio and a ratio needs a denominator, so it is worth being exact about which one is used and why, since a different choice would make a different claim.

The denominator here is the front: the number 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. At the coarse arrangement that is five; at the fine one, eight; at the finest, thirteen.

That is the right denominator for this hypothesis because it is the neighbourhood the rule is actually sensitive to. Two other candidates were available and are worse. The whole neighbourhood the sum runs over is about a hundred organs at these rises and is dominated by terms that change nothing — so a ratio against it would report every intervention here as one or two per cent and separate nothing. The number of organs within one spacing of the tip is a geometric quantity that changes with the rise for reasons unrelated to the pattern, so a ratio against it would confound the hypothesis with the rise, which is what the front result exists to avoid.

Choosing the front also gives the hypothesis its best chance, which is the point of choosing carefully when the result is going to be negative. It is the denominator that makes forty per cent of a coarse stem and forty per cent of a fine one the most nearly comparable interventions available.

The comparison, set up so it cannot be argued #

The coarse stem carries 3/5, so its front is five organs deep and a two-organ cut is forty per cent of it. Twenty arrangements were tried — five positions for the nearer organ, four gaps to the second. Two never repair. One of those two settles at 220.3125° against an undisturbed 139.6875°, which is the reflection to four decimal places, with the counted pair and the order of the shortest steps unchanged.

The fine stem carries 5/8, so its front is eight organs deep. Five organs is sixty-three per cent of it: half as much again as the share that turns the coarse stem over. Sixty-four arrangements were tried — four positions for the nearest organ, two gaps each for the four that follow.

Sixty-three of the sixty-four never repair. The failure is emphatically not that the stems recovered.

Not one of the sixty-three settles at the mirror. The mirror sits at 223.22° and the nearest approach is 11.8°, which is fifty steps of the azimuth grid and more than two hundred times the tolerance at which the coarse stem’s mirror is recognised as one.

The ratio was measuring something and it was not this #

It is worth separating the two things the share does, because the hypothesis was not stupid and its failure is specific.

The share of the front removed orders the rate of wrecking very well. At the fine arrangement it climbs 13, 25, 38, 50, 63 per cent, and the share of arrangements that never repair climbs with it: 0.25, 0.56, 0.71, 0.83, 0.98. That is monotone at every step and it is what a dose ought to do.

What it fails to order is the destination. Whether a wrecked stem ends up at one place or another is not a function of how much was taken from it, and the strongest form of that statement is the one the sweep makes almost by accident: the destinations reached by a five-organ cut are places a one-organ cut already reached.

So the dose is a knob on one axis and not on the other, and the hypothesis under test conflated them. That is a common enough shape for a wrong idea to have — it took a real quantity that really varies and attached it to the wrong outcome.

The finest arrangement, for a third point #

Two arrangements make a comparison and three make a trend, so the sweep was also run at 8/13.

Three organs out of a front of thirteen is twenty-three per cent — the smallest share of the three points that matter, and well under the forty that turns the coarse stem over. Seventy of seventy-two arrangements never repair, which is the highest wrecking rate anywhere in this work, at the lowest share.

That single row does more damage to the hypothesis than anything else here. If the share of the front were the governing quantity, a twenty-three per cent intervention should be milder than a forty per cent one — and here it wrecks ninety-seven per cent of arrangements where forty per cent of the coarse front wrecks ten. The rate does not follow the share across arrangements at all; it follows the arrangement, and within one arrangement it follows the dose.

So the share is not even a good predictor of the rate once more than one arrangement is in view. It orders the rate within an arrangement, which is the weakest thing a dose can do and is what any monotone measure of intervention size would have done equally well.

And the destination result holds there too: none of the seventy reaches the mirror, and the closest approach is a stem sitting a quarter turn from where it started, which is a slipped lattice rather than a reflected one.

Why the destination is not a matter of degree #

Once the two rungs before this one are in hand, the reason is not mysterious.

A wrecked fine stem keeps one family of the lattice it was cut from — the angle from an organ to the one p places above it is unchanged, organ by organ — and gains a whole number of turns over each period of that family. The destinations available are therefore the original divergence plus 360° divided by p, times a small whole number, for the handful of lags the rule can hold rigid.

That list does not have the mirror on it, and it cannot be given one by removing more organs. A reflection is not a slip: it moves every family of the lattice at once, by reversing the sign of the azimuth, where a slip moves all but one. So enlarging the cut cannot walk a stem towards the mirror by degrees, because there are no intermediate states of the right kind to walk through.

What enlarging the cut does is make it more likely that the stem leaves its own lattice at all. Once it has left, where it goes is decided by which lag survives, which is a small whole number and not a proportion.

What the coarse arrangement has that the fine one lacks #

The claim that survives is that the mirror belongs to the coarse lattice. It is worth saying what property of a coarse lattice that could be, even though this is an account rather than a result.

A reflection of a lattice is a lattice with the same counts, and the rule has no handedness in it, so the reflected arrangement is always an equilibrium the rule would hold. The mirror is available at every arrangement. What differs is whether a disturbance can reach it.

Getting from a lattice to its reflection means rearranging the whole pattern rather than one family of it. On a front of five that is five organs’ worth of rearrangement; on a front of eight it is eight, and on thirteen it is thirteen. At every step of the way there are nearer equilibria available — the slipped lattices, whose whole list this collection now has — and the more organs the rearrangement has to pass through, the more chances there are to fall into one of them instead.

That account predicts that the coarse arrangement should be the only one that mirrors among those tried, which is what is observed, and it predicts that the mirror should be rare even there, which it also is: one arrangement in twenty.

A hypothesis worth having had #

It would be easy to write this essay as though the share hypothesis were obviously wrong, and it is worth resisting, because the reasoning behind it is the reasoning that produces most good experiments in this collection.

It took the one clear qualitative difference between the coarse arrangement and the fine ones — the mirror — and asked what quantity could carry it. It found a quantity that was genuinely different between them, that could be varied independently of the arrangement, and that could be pushed past the value where the effect was known to occur. That is exactly the right shape, and the experiment it prescribed is the one that got done.

What it got wrong is a category error that is hard to see in advance: it treated a destination as though it were on a continuum. Damage is a matter of degree, and it is natural to expect the outcome of damage to be one too. The outcomes here are not — they are a short list of lattices, each specified by two small whole numbers — and no amount of turning a continuous knob produces an outcome that is not on the list.

That is a lesson worth carrying past this thread. Whenever an outcome is discrete — a pair of counts, a handedness, a period — a dose can decide whether and cannot decide which, and a hypothesis of the form “more of it arrives there” needs an account of how the intermediate states are traversed before it is worth testing.

What would test it further #

Two experiments follow directly and neither is done here.

A coarser arrangement still. If reachability falls with the depth of the front, a 2/3 stem should mirror more readily than a 3/5 one. The rise required is large enough that the lattice becomes marginal in other ways, which is why this collection’s coarsest rung is 3/5, so the experiment is not free.

A cut shaped like a rotation rather than like damage. The intervention here removes organs. An intervention that displaced the front’s organs by a prescribed amount, rather than deleting them, could in principle be pointed at the mirror deliberately — and if a fine stem can be driven there by a displacement but not by a deletion, that separates reachable from this experiment from reachable at all.

Both would sharpen a negative that is currently stated over a specific range: no cut of one to five organs, at two hundred and seventy arrangements over two fine lattices, reaches the mirror or comes within six degrees of it.

What this does not say #

It does not say the share of the front is meaningless. It orders the rate of wrecking cleanly at every step, which is a real and useful thing for it to do. What it does not order is which destination a wrecked stem reaches.

It does not say a fine stem can never mirror. It says that nothing in this range of interventions gets there. A larger cut, a differently shaped one, or a displacement rather than a deletion are all untried.

It does not say the coarse mirror is a common outcome. One arrangement in twenty at the coarse rung, and four in sixty-four in the larger table it was first found in. It is rare and it exists, and those are both parts of the claim.

And it does not say what a plant would do. Removing five of a meristem’s eight most recent primordia is a large intervention on real tissue and this collection has no view on whether anything would grow back.

The check that would refuse it #

Three assertions, and the first two exist to stop the third being satisfied trivially.

The first is that the coarse arrangement does mirror in the sweep this comparison uses. That number is recomputed rather than quoted, by the same function and at the same tolerance as the fine sweeps, so the negative is not resting on a comparison between two measurements made by different code on different days.

The second is that the cut it is compared with really does take a larger share of a larger front — sixty-three per cent of eight against forty per cent of five — and that at least twenty of its arrangements are wrecked. A negative asserted over a set of stems that mostly recovered would be a claim about recovery.

The third is the negative itself, at a tolerance of six degrees rather than the twentieth of a degree at which a mirror is recognised. The distinction matters: asserting nothing lands exactly on the mirror would be satisfied by a stem two degrees away, which would refute the point while passing the check. Six degrees is more than twenty steps of the grid, and the nearest approach across every fine sweep is 11.8°.