Removing a neighbour costs least

Every ablation table in this collection carries a number nobody has read: the displacement of the first organ placed after the cut. It is how the front is measured — an organ that moves is inside the front and one that does not is outside — and beyond that threshold nothing has been done with it.

Split the thirty wrecked cuts by which organ was taken away, and the number separates into two groups with a gap between them.

Remove a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — and the next organ moves by 8.9° to 30.7°. Remove anything else inside the front and it moves by 62.8° to 167.6°. Nine rows against twenty-one, a factor of two across the gap, and nothing lands in it.

That is the opposite of what the words suggest, which is the reason to write it down. It is also a fact about the placement rule rather than about plants, and this collection is careful about the difference: what follows is an account of what a stated rule does when an organ is taken out of the arrangement it is placing against, and any claim about a meristem would need the rule to be the mechanism, which nothing here establishes.

Why it sounds backwards #

The placement rule puts each new organ where the sum of inverse powers of distance to the organs already there is least. The nearest organs contribute the largest terms. So removing one of the nearest organs removes one of the largest terms in the sum, and the obvious expectation is that this is the most disruptive thing a removal can do.

The measurement says the opposite. Removing a chain-neighbour is the cheap removal, by a factor of two, with no overlap between the groups at all.

The account #

It is a statement about where the removed organ sits in azimuth rather than about how large its term was.

The new organ goes into the gap the existing organs leave. On a settled lattice that gap is bounded on two sides, and the organs bounding it are exactly the two contact families’ nearest members — at lags p and q, on opposite sides of the new organ’s slot. That opposition is what makes them the contact pair.

Now remove one of them. The slot’s two walls become one wall and a much more distant one, so the minimum of the sum slides toward the missing wall — but only until the next member of the same family starts to bound it, which on a lattice is one step further along the same chain and in nearly the same direction. The gap widens; it does not open.

Remove an organ that is on neither chain, and the geometry is different. Such an organ sits between the two walls in azimuth — not bounding the slot but sitting across it, several rows down. Removing it opens the interior of the region the new organ is choosing within, and the minimum can move a long way before anything stops it.

So the rule of thumb is: a chain-neighbour is a wall and everything else is furniture. Taking away a wall moves the answer to the next wall along, which is close. Taking away furniture lets the answer roam.

The measurement, stated exactly #

The number is the azimuth of the first organ placed after the cut, minus the azimuth of the same-numbered organ in a control that shares the stem’s entire history, folded to within half a turn. It is not an average over anything and it has no window in it, which is worth saying because the same runs’ later organs are useless for this kind of question: a wrecked stem is permanently displaced everywhere above the hole, so any quantity read further up is measuring a phase difference rather than a response.

The first organ is the exception. It is placed by the rule against a configuration that differs from the control’s in exactly one organ, so its displacement is a clean response to a single change with nothing accumulated in it.

That is also why the result is available at all. Every other quantity in this thread is read off a settled stem hundreds of organs later, where the cut’s signature has been overwritten by the new phase. This one is read immediately, and immediacy is what makes it about the geometry of the hole.

The gap, and why it is a gap rather than a line #

Nine measurements between 8.9° and 30.7°; twenty-one between 62.8° and 167.6°. The ratio across the empty region is 2.05.

That matters because a threshold chosen to make a claim true is how a claim gets made true. Nothing here is fitted: any cut-off between about thirty-five and sixty degrees gives the same two groups, and the tolerance stated in the machinery — forty-five degrees — is the middle of that range rather than the edge of it.

It also matters that the grouping is decided by arithmetic on the offset — the same arithmetic the lost-member reading uses to decide which rows it can be asked on — and not by the displacement. Whether a removed organ is a chain-neighbour is settled by whether its offset divides by five or by eight, with no reference to how far anything moved. So the two variables are independent, and their agreement across thirty rows is a result rather than a definition.

What it predicts #

Three things, and the first two are already visible in tables here.

That the front’s edge should be ragged rather than sharp. The front is the run of offsets at which a removal is felt at all, and it has been reported as a step: large out to the larger counted number and nothing past it. If chain-neighbours are cheap, then the offsets equal to p, q and their multiples should sit lower than their neighbours inside that step — a comb of notches rather than a plateau.

That the effect should not depend on the rise except through the pair. The account is about which organs bound the slot, and that is fixed by the counted pair — which is also why a band that holds the pair and moves the rise would be expected to leave it alone. Across the census the two groups hold at every rise from 0.032 to 0.005, which spans a factor of six and four different pairs.

That a two-organ cut removing both walls should be much more expensive than either alone. That is testable with machinery already here and it has not been done: the multi-organ work measures whether a stem repairs, not how far the next organ moves.

What it does not explain #

It does not explain which family survives. That is the question this thread has been chasing, and the cheap-removal result is about the first organ rather than about the eventual arrangement.

The two are not obviously connected, and the measurement says they are not connected simply. Among the nine cheap removals the survivor is the removed organ’s own family every time — that is the lost-member reading — and among the twenty-one expensive ones the survivor is whatever the offset rule says it is, with the same five exceptions as ever.

So a large first displacement and a small one both lead to a wrecked stem with a rigid hop in it, and how far the first organ moved does not say which hop. If anything that is the more interesting half: the amount of damage and the kind of damage are independent here. It is also a warning against the tempting shortcut of treating displacement as a proxy for severity — the family a stem keeps is not ordered by it, and neither is whether the stem repairs at all, since offsets outside the front move nothing and heal while offsets deep inside it move everything and heal too.

The count of nine, and what it rests on #

Nine cheap removals is not many, and it is worth saying which nine so the reader can see they are not one lattice’s worth.

They are spread across six lattices and both branches: 5/8 stems at three different rises, an 8/13 stem, two 4/7 stems and a 7/11 stem. Five of them remove a member of the smaller family and four the larger. No lattice contributes more than two.

That spread matters because the alternative explanation for a clean split is that one lattice happens to behave differently from the rest and its rows have been gathered into a group. Six lattices, two branches and four counted pairs is not that.

What nine rows cannot support is any statement about the shape of the cheap group. Whether removing a member of the smaller family costs more or less than removing one of the larger, whether the cost grows with the offset, whether it depends on the rise — all of those are questions about the internal structure of nine numbers spanning 8.9° to 30.7°, and nine numbers will answer them however they are asked.

What it does say about the mechanism #

Something narrow and worth having: the rule’s response to a removal is geometric rather than energetic.

An energetic account would predict the response from the size of the term removed, which is largest for the nearest organ. A geometric account predicts it from where the removed organ sat relative to the gap being filled. The two accounts differ in sign here, and the measurement picks the geometric one.

That is a mild result and it is stated mildly. It does not follow that energy is irrelevant — the rule is an energy — only that the response to a perturbation of it is dominated by the shape of the constraint set rather than by the magnitude of the perturbation. In a system where the minimum sits in a narrow valley, removing a wall moves the minimum; removing a distant contributor deepens the valley without moving it much. Here it is nearly the reverse, and it is the reverse because the “distant” contributors are the ones sitting inside the region the minimum lives in.

Where the number was hiding #

This collection has had the thirty numbers since the census was built, and every one of them has been printed in a figure. The reason nobody read them this way is worth a paragraph, because it is the ordinary reason.

The first displacement was introduced as an instrument: a way of deciding whether an organ is inside the front. For that purpose all it needs is a threshold, and the threshold is generous — two and a half degrees, chosen because inside the front the displacements run from a few degrees to a hundred and sixty and outside it they are under half a degree. Once a quantity is being used as a detector, its value stops being interesting; what matters is which side of the line it falls on.

So the number was drawn thirty times, read as a boolean thirty times, and its range — from 8.9° to 167.6°, a factor of nineteen — was never the subject of a sentence. The general form is that a quantity used to make a decision stops being read as a measurement, and the fix is not vigilance but occasionally asking what else a detector’s own output would say if it were a reading.

What would refute it #

A lattice where a chain-neighbour removal is expensive. The account predicts there is none, because the geometry it rests on — two families bounding a slot on opposite sides — is what a contact pair is, on every lattice.

The place to look is the coarse end, where the counted numbers are small and the distinction between “on a chain” and “not on a chain” nearly disappears. On a 2/3 stem, offsets one, two, three, four and six are all multiples of a counted number, so almost every removal is a chain removal and the two groups have nothing to separate. Whether the split survives there is a question about a rung this census does not reach.

The other place is a whorled stem, where organs arrive several at a time and the notion of a chain through the tip is different. A whorled arrangement’s response to a cut has been measured for whether it repairs and not for how far the next organ moved, so the number is one column away.

What is left #

The notches. If chain-neighbour removals are cheap, the front’s displacement profile has structure in it that every table here has drawn as a step, and reading that structure is arithmetic on data already computed — the offset table exists at three rises and has been read only for where it falls to zero.

And the two-organ test, which is the sharpest thing the account predicts and the only one that would need new runs. Remove both walls and the new organ should move much further than either removal alone; remove two pieces of furniture and it should move about as far as one. That is a two-by-two comparison with an interaction in it, which is more than any single measurement here has asked for.