The organ that moved furthest

The family whose member was removed is the family left standing. That reading is right on every row it can be asked on, which is nine of thirty, and the reason it cannot be asked on the other twenty-one is a technicality with a consequence.

An organ k places back lies on the p-chain through the growing tip exactly when p divides k. So the reading needs the tip as a reference, and only nine of the thirty wrecked offsets in the census are multiples of a contact number. The other twenty-one remove an organ that is nobody’s direct neighbour, and the honest thing there was to leave them alone.

The obvious repair is to choose a different reference — and better still, to measure one instead of choosing it. The organ a cut disturbed most is a candidate every row has. This essay measures it, and the answer is that the instrument does not exist.

The measurement #

Every wrecked cut here already carries a control: a run with the same seed angles, the same rise, the same grid and the same organs, continued by the same rule, differing only in that one organ was not removed from it. Organs below the cut are shared history and are identical by construction. Organs above it are free.

So the displacement of each organ is available directly — the azimuth in the cut run minus the azimuth of the same-numbered organ in the control, folded to within half a turn. Nothing new needs growing.

The disturbance has no far edge #

Here is what stops the repair. Across the thirty wrecked cuts, the largest displacement in the window above the hole runs from 105° to 180°.

That upper figure is not an outlier; it is the ceiling. An azimuth folds at half a turn, so 180° is the largest displacement there can be. Several rows sit within a few degrees of it.

And it does not decay. Over the last sixty organs of each run — hundreds of organs above the cut, long after any local rearrangement has finished — the largest displacement is still 70° to 179°.

That is the definition of a wrecked stem restated in a new quantity. The stem never repairs; it settles into a different phase from its control and stays there, so every organ above the hole is permanently displaced by something of order half a turn. The displacement field is not a bump with a far side. It is a step that never comes back down.

The window, and why no width saves it #

The obvious objection is that the maximum was taken over too wide a window. If the disturbance is global, look near the hole instead — say over the first few organs — and the maximum will be local by construction.

That objection is right about the construction and wrong about what it buys. The window used here is three times the larger contact number, which on a 5/8 stem is twenty-four organs and on an 8/13 stem thirty-nine. It is the most generous reading of “near the hole” that is still a window, and it was chosen before the profiles were looked at.

Narrowing it does not help, because the saturation is not something that develops over tens of organs. On most rows the second or third organ after the cut is already displaced by more than a hundred degrees. There is no width at which the field is small enough for a maximum to be meaningful and wide enough to contain the organs a reading would want to talk about.

And narrowing has a cost that is easy to miss: the narrower the window, the more the answer is determined by the offset, since the window starts at the removed organ. A window of p organs on a p/q stem can only ever return a lag between the offset and the offset plus p, so the family it names is nearly a function of the offset — which is the quantity the reading was supposed to be independent of.

That is the deeper reason the instrument does not exist. It is not that the window is hard to choose. It is that every choice either returns a saturated plateau or returns the offset back.

Which makes the maximum arbitrary #

A maximum over a field that saturates is a maximum over a plateau. Measured directly: at sixteen of the thirty rows, a fifth or more of the window sits within a tenth of the largest value. On those rows there is no peak to find; there is a broad region of organs all displaced by roughly the ceiling, and which one comes out largest depends on the fourth significant figure of a quantity that folds.

Scored anyway, because a negative result should be scored rather than asserted: the most-disturbed organ names a contact family on eight of the thirty rows — one fewer than the tip manages, and a different eight — and where it does name one, it names the survivor on three.

The tip’s reading is right on nine of nine. The measured reference is right on three of eight. The repair does not merely fail to extend the reading; it does not reproduce it.

Why the instrument was worth trying anyway #

Because the alternative was to keep quoting a reading that covers three tenths of its table, and because the failure is informative rather than empty.

What the measurement establishes is that the disturbance is global. Before this, the working picture was of a local rearrangement — a hole, some organs shuffling around it, and a stem that settles back into a lattice at a different phase. The first two are right and the third is doing all the work: the phase difference is permanent and it is of order half a turn, so every organ above the cut carries it.

That has a consequence for any future reading of this kind. Any quantity defined as “the organ where something is largest” will fail here for the same reason, and so will anything defined over a decay length, a reach, or a far edge. Those words describe a disturbance that goes away, and this one does not.

The measurements that do work on a wrecked stem are the ones stated over differences rather than positions — the rigid hop, the block, the slip — and that is not a coincidence. A permanent phase difference cancels out of a hop and does not cancel out of a position. That is why this thread’s positive results are all expressed in lags: not because lags are elegant, but because they are the quantities a wrecked stem leaves measurable.

What the eight rows say, since they are there #

The eight rows where the measured reference does name a family are worth a glance, because a reading that is right on three of eight is not obviously different from one that is right on four of eight, and it would be easy to overstate the failure.

All eight are on 5/8 or 4/7 stems, and on all eight the reference organ sits at a lag between eight and twenty-four — well above the offset. Three of them name the survivor and five name the other family. A coin would give four, so the reading is below chance on its own eight rows, which is a way of saying it carries nothing rather than that it carries the opposite.

That is the outcome to expect when a maximum is being taken over a plateau: which lag comes out is close to arbitrary, so whether it happens to be a multiple of five or of eight is close to arbitrary too, and a family named that way is a family named by rounding.

The generalisation that does work #

There is a version of the lost-member reading that reaches every row and needs no reference organ at all.

The two contact families leave the growing tip on opposite sides in azimuth. That is not an assumption; it is what makes them the two nearest neighbours of a lattice point, and it is visible in the arrangement. So every removed organ, whatever its offset, sits on one side or the other.

Read the rule as the removed organ was on the surviving family’s side rather than the removed organ was a member of the surviving family’s chain, and it becomes answerable on all thirty rows. Two things then have to be checked, and both are.

It reproduces the published reading exactly: on all nine rows the tip’s chains can answer, the sidedness reading gives the same family. So it is a generalisation rather than a rival.

And it scores twenty-two of thirty over the whole census — against eighteen for naming the commoner family outright, and against twenty-five for the offset rule.

What twenty-two of thirty is worth #

Less than twenty-five and more than nothing, and it is worth being careful about which.

It beats the null. Naming the commoner family every time gets eighteen; the sidedness reading gets twenty-two, so it is carrying four rows of information beyond the base rate. That is not much on a table of thirty and it is not nothing.

How much information four rows is can be said exactly. Against a null that names the commoner family every time — eighteen of thirty, a rate of three in five — a census of thirty rows has a standard deviation of 2.7, so twenty-two sits one and a half deviations above it. About one census in fifteen would produce twenty-two or better with the sidedness reading carrying nothing at all. That is not a result and it is not a coincidence either; it is the size of thing thirty rows cannot decide.

The comparison worth making is therefore not sidedness against the base rate but sidedness against the offset rule, and thirty rows cannot separate twenty-two from twenty-five either. Both readings would need the same thing to be told apart, and it is not more argument.

It loses to the offset rule, which gets twenty-five and is a statement about the sequence rather than about the arrangement. That is the uncomfortable part, and it has been the uncomfortable part of this thread from the beginning: the mechanism-shaped readings keep scoring below the arithmetic-shaped one.

And it is being scored on a table that sits nearly all on one side of a quantity nobody recorded, which caps how much any score on it can settle. Four rows of advantage over a base rate, on a census with a known confound, is a result to hold loosely.

What the sidedness reading would need to be believed #

An out-of-sample test, and the design for one is available.

The census’s thirty rows are what both readings were built on. A band around a handover grows stems at rises the census never visited, and cutting across one produces fifty-five more wrecked cuts. Neither the sidedness reading nor the offset rule has been scored on those, and both make predictions there.

That is an afternoon’s work and it is deliberately not done here, because the band’s cells were computed to answer a different question and scoring a reading on data gathered for something else is how a table gets read twice. The right version is to state the prediction first — the sidedness reading says the five on every golden-band cell and the four on every Lucas-band cell — and then check.

Written that way it is a strong prediction and an easy one, because both bands have a single answer throughout. The interesting test would be a band where the survivor changes, and no such band has been found.

The pattern this belongs to #

This is the second reading in two rounds where the proposed repair was an instrument and the instrument turned out not to exist. The first was a definition of neighbourhood depth that was supposed to reorder a set of exponents and could not, because all four candidate definitions ordered them alike.

Both have the same structure. A reading fails; somebody proposes that the failure is in how a quantity was defined; the quantity is redefined several ways; and the redefinitions turn out to be either equivalent to each other or incapable of carrying the reading at all.

The lesson is not that redefinition never helps. It is that a redefinition proposed to rescue a reading should be measured before it is adopted, because the cheap outcome — it makes no difference — is more common than either the reading surviving or the reading failing again. Measuring it costs a morning and settles the question; not measuring it leaves a plausible repair sitting in a plan document for a round, which is where this one sat.

What is left #

The profiles themselves are worth more than this essay uses them for. They are the first organ-by-organ picture of what a cut does that this collection has made, and the whole of what has been taken from them is a maximum and a first value. Their shape — which organs stay put, which move by a whole turn, whether the pattern of movement repeats with the block’s period — has not been read at all.

One thing visible in them without any analysis is that a few organs are completely unmoved — displaced by under a degree — while their neighbours move by a hundred and forty. Which lags those sit at, and whether they are the same lags as the rigid hop, is a question with an answer in data already computed.

It is also a question with a prediction attached, and the prediction is two lines of arithmetic on results this thread has already published. A wrecked stem keeps its p-hop rigid and its mean divergence lands one whole turn per p organs away from the control’s. So the organ n places above the cut sits at n times the new divergence where the control puts it at n times the old one, and the difference between them is n · 360° ÷ p, reduced to a turn.

That says three things the profiles can be read against directly. The displacement should return to zero every p organs and nowhere else, so the unmoved organs are spaced at exactly the block’s period. It should take only p distinct values, evenly spaced around the circle at 360° ÷ p. And its largest value should be 180° when p is even and 180° · (p − 1) ÷ p when p is odd — 144° on a stem keeping a five-family, 154° keeping a seven, and the full ceiling keeping a four or an eight.

The last of those already has partial support in the range quoted above, since the ceiling rows are the ones a folding azimuth cannot exceed and the smaller maxima have to come from somewhere. What it does not yet have is the check, which is a single pass over profiles that are already on disk: count the zeros, measure their spacing, and ask whether it is the block. A profile whose zeros were spaced at anything else would refute the picture that a wreck is a lattice at a different phase, which is the picture every positive result in this thread rests on. It is also the question most likely to turn the profiles from a diagnostic into a result, because an unmoved organ beside a displaced one is a much sharper object than a maximum.

And it is sharper in the way that matters for an experiment as well as for a model. A maximum over a plateau is a quantity nobody could measure on a plant; the spacing of the organs that did not move is a count, which is the kind of reading this collection has repeatedly found to be the only kind a photograph settles.