Three organs and no mirror

Removing two organs from the coarsest arrangement this collection grows — a stem carrying three spirals one way and five the other — produces something that took some looking at to name. Four of sixty-four pairs of offsets never repair, and three of them settle at 220.3125°, against an undisturbed 139.6875°. That is 360° minus the original, to the last digit.

Exactness of that kind is worth pricing before it is interpreted. A settled divergence in this collection carries a scatter of a couple of tenths of a degree, so agreeing to the fourth decimal is not a measurement landing near a prediction; it is the same number arrived at twice. Under any account in which the wrecked stem merely drifts to somewhere new, the chance of that somewhere being the reflection to within the printed precision is of order one in a million per stem. Three of the four wrecked pairs do it, and the fourth does something else entirely, so the reading is not that a reflection is likely — it is that a reflection is an exact equilibrium of the rule and the stem either reaches it or does not. A counter shown the wrecked stem’s positions returns 3/5, the pair it was cut from, and the three shortest steps are 5, 3 and 2 organs, in that order, exactly as before.

The stem is not degraded. It is the same lattice wound the other way.

That is a prediction about a quantity a real census could record for nothing and does not. Handedness is one of the four fields this collection’s survey specification asks for; no divergence angle and no pair of counts distinguishes it, so it is a property of the individual plant and it is thrown away in every published count.

The finer arrangements do not do this. Cut a 5/8 stem or an 8/13 stem and the wrecked ones settle into repeating orbits — a fixed cycle of angles — and never into a mirror. Two accounts of the difference were available and could not be separated:

• Coarseness. A mirror is what a coarse lattice does when it is wrecked, because there is not much else for it to be.
• The share of the neighbourhood removed. Two organs out of a front of five is forty per cent; two out of thirteen is fifteen. Perhaps a finer stem would mirror too, if enough of its front went.

This essay makes the experiment that separates them.

The intervention, generalised #

The machinery for it is a small extension of what was already there. A cut is named by the set of organs removed, counted back from the growing tip: [2, 5, 6] takes out the organs two, five and six places back. Everything else about the experiment is unchanged — the control and the cut share a history to the last digit, the heights are prescribed by the rise so a removal leaves a vacancy rather than shifting anything, and only the azimuths of organs placed afterwards are free.

One detail matters enough to state. The organs are spliced out furthest first. Removing the nearer one first shifts every later index by one, so every offset in the table would then mean something other than what it says — silently, with every other assertion still passing.

The check that the generalisation is the same instrument is a limit: a cut of one organ made by the general machinery must reproduce the original single-organ intervention to the last digit. It does, at four offsets, which is what says the general cut removes the organs it names.

What the sweep can afford, and what that costs the claim #

A cut of m organs is named by a first offset and m − 1 gaps, so the number of arrangements grows as the number of gaps to the power m − 1. A sweep that kept the same range of gaps at every size would be unaffordable by four organs and absurd by six, and each arrangement costs a continued run of three hundred organs against a control.

So the sweeps are shaped to hold the cost roughly constant: eight first offsets and four gaps at two organs, eight and three at three, five and three at four, four and two at five. Between forty and eighty arrangements each.

What is traded is the range of gaps at the larger cuts, and what is deliberately not traded is the range of first offsets at the small ones, since that is what the front result is about. It is worth being clear about which claims that weakens.

A claim of the form this destination is reached is unaffected: reaching it once is enough, and the coarse mirror is found in a sweep of twenty. A claim of the form this destination is never reached is weakened, because a narrower sweep has looked at fewer places. That is the shape of the negative here, and it is why the tolerance on it is six degrees rather than a twentieth: the assertion is not that nothing lands on the mirror but that nothing lands anywhere near it, which is a much harder thing for a sparse sweep to miss.

A rate — this fraction of arrangements wreck — is the most fragile of the three, since it depends on which arrangements were tried. The rates are reported with their denominators printed on every bar for that reason, and the claim made from them is only that the ordering is monotone.

Three organs at the finer arrangements #

At the 5/8 arrangement, a three-organ cut has seventy-two arrangements in the sweep — eight positions for the nearest organ, three choices of gap to the second, three to the third. Fifty-one of them never repair. That is seventy-one per cent, against twenty-five per cent for a single organ at the same arrangement, so the intervention is doing a great deal.

Not one of the fifty-one reaches the mirror. The mirror of this lattice sits at 223.22°, and the nearest any wrecked stem comes to it is 14.4° — sixty steps of the azimuth grid the runs are computed on, and a hundred times the tolerance that decides whether a stem is at the mirror or not.

At the 8/13 arrangement a three-organ cut wrecks seventy of seventy-two, which is very nearly everything, and again reaches no mirror. Its nearest approach is 5.6°, which is closer and is not a near miss: that stem sits at 227.85°, which is 137.84° plus exactly a quarter turn, and a quarter turn is what a stem gets when the family it keeps has period four. It is a slipped lattice arriving near the mirror by arithmetic rather than a wreck reaching for it.

What the finer stems do instead #

The wrecked stems at the finer arrangements are not disordered. They settle into exactly repeating blocks of divergence angles, and what those blocks are is a question with an answer: each keeps one family of the lattice it was cut from, rigid organ by organ, with everything else rebuilt around it, and gains a whole number of turns per period of that family.

That description does not fit a mirror at all, and the reason is worth being explicit about. A slip preserves one family and moves every other; a reflection moves every family at once, by reversing the sign of the whole azimuth coordinate. They are different operations, and only one of them leaves anything rigid.

So the coarse stem’s mirror and the fine stems’ orbits are not two points on a scale of damage. They are two different things a stem can become, and the question this essay asks is which stems can become which.

The answer is coarseness #

Three of eight organs is thirty-eight per cent of the front — close to the forty per cent that mirrors a 3/5 stem, and not clearly past it. So three organs is not by itself the test. Four and five are.

Five organs removed from a front of eight is sixty-three per cent of the neighbourhood, which is half as much again as the share that reverses a coarse stem’s handedness. Sixty-three of the sixty-four arrangements tried never repair — so the failure is not that the stems came back. None mirrors, and the nearest approach is 11.8°.

The share of the front removed does not order the outcome. The mirror belongs to the coarse lattice.

Why a coarse lattice is the one that can turn over #

The account is not proved here and it is worth stating, because it makes a prediction the measurement can be checked against.

A reflection maps a lattice to a lattice with the same counts. For it to be a state the rule will hold, the reflected arrangement has to satisfy the same placement condition the original did — which it does automatically, since the rule has no handedness in it. So a mirror is always an available equilibrium and the question is only whether a disturbance can reach it.

Reaching it means passing from one to the other, and the two are separated by a rearrangement of the whole pattern rather than of one family. On a front of five that rearrangement involves five organs; on a front of thirteen it involves thirteen, and the intermediate states it would have to pass through are ones the rule can leave for a nearer equilibrium at every step. The coarse lattice is not more fragile — a single organ never wrecks it at all, where a single organ wrecks two of eight offsets at 5/8 — it is closer to its own reflection, in the sense that fewer organs have to be got past.

If that is right, then a cut large enough to displace the entire front of a fine stem at once should be able to mirror it. Five of eight does not; whether eight of eight would is not tried here, and it is not obviously a well-posed intervention, since removing the whole front leaves the rule placing against organs it was never sensitive to.

An aside the sweep produced by accident #

There is one row of the fine sweep worth keeping, because it is the kind of thing a negative result usually buries.

At the 8/13 arrangement, three organs removed at offsets one, three and six leaves a stem that settles at 227.85° — 5.6° from the mirror, and much the closest approach anywhere in two hundred and seventy arrangements. Taken on its own it looks like the fine arrangement nearly doing what the coarse one does.

It is not. Read the same stem by lags and its four-hop is rigid: the angle from each organ to the one four places above it is unchanged from the control, organ by organ, while the divergence swings. A rigid four-hop forces the mean divergence to move by a whole number of turns over four organs, and one turn over four is ninety degrees. The stem is at 137.84° + 90.00°, and the mirror of 137.84° happens to be 222.16°, which is 5.7° away.

So the near miss is a coincidence between two unrelated arithmetics: a quarter turn on one side, a reflection on the other. Nothing about that stem is reaching for the mirror, and the reason the essay can say so is that the machinery for identifying what a wrecked stem kept was built for a different question altogether.

What a census would have to record #

The transferable claim is short. An ablation on a coarsely patterned stem can reverse its handedness permanently; on a finely patterned one, nothing yet tried does.

Handedness is a field that costs nothing to record and is recorded by almost nobody, so this is a prediction against a quantity that is free and absent. It is also a prediction with an unusual property: it is about the comparison between coarse and fine specimens rather than about a number, so it survives a great deal of measurement error. An experimenter would need to know which way each plant’s spirals turned before and after, and how coarse its pattern was, and nothing else.

Two organs at the coarse arrangement, recomputed #

The comparison this essay rests on is the coarse mirror, and it is worth saying that the number it is compared against was recomputed here rather than quoted.

The sweep used for the comparison is the same shape as the fine ones: five first offsets, four gaps, twenty arrangements. Two of them never repair. One settles at 258.75°, which is a 3/4 lattice and neither the original nor its reflection; the other settles at 220.3125° against an undisturbed 139.6875°, which is the mirror to four decimal places.

That is a smaller sweep than the sixty-four-cell table the mirror was first found in, and it finds it once rather than three times. The rate is therefore not comparable between the two and no comparison of rates is made. What is comparable is the fact of arrival, and the point of recomputing it inside the same machinery that produces the fine sweeps is that the mirror is then identified by the same function, at the same tolerance, on the same day, as the mirror the fine sweeps do not reach.

Quoting the earlier number would have been cheaper and would have left the negative resting on a comparison between two measurements made by different functions — which is the failure this collection has recorded elsewhere and has no excuse for repeating.

What this does not say #

It does not say a fine stem cannot mirror. It says that no cut of one, two, three, four or five organs, at two hundred and seventy arrangements over two lattices, reaches the mirror or comes within six degrees of it. A larger or differently shaped intervention is not ruled out and is not tried.

It does not say the coarse mirror is common. It is one of four wrecked cells in a sixty-four cell table, and the smaller sweep used here finds it once in twenty. It is a rare outcome of a rare outcome, and the interest is that it exists at all.

It does not say the share of the front is irrelevant to anything. It orders the rate of wrecking very well — the share that never repairs climbs from a quarter to nearly everything as the cut grows — and it fails only as an account of which destination a wrecked stem reaches.

And it does not say a meristem would survive this. Removing five of the eight most recent primordia from a real apex is not a small intervention, and whether anything recognisable would grow back is a question about tissue rather than about a placement rule.

The check that would refuse it #

Three assertions run while the figures are drawn.

The first is the limit that makes this one instrument rather than two: a cut of one organ made by the general machinery reproduces the single-organ intervention’s displacement to the last digit, at four offsets. It is the check that catches the splice order, and it would fail immediately if the organs removed were not the organs named.

The second is the dose: the share of arrangements that never repair must rise at every step from two organs to four. Without it the sweep could be measuring nothing — an intervention whose size changed no outcome would satisfy every other claim here trivially.

The third is the negative, and its tolerance is deliberately generous. At every size of cut from three organs to five, at least twenty arrangements must be wrecked, and none of them may settle within six degrees of the mirror. Six degrees is more than twenty steps of the azimuth grid and a hundred times the tolerance that decides whether the coarse stem’s mirror is a mirror. A negative asserted at a twentieth of a degree would be satisfied by a stem two degrees away, which would refute the point while passing the check.