The end of a wrecked run

The slot design measures the first organ placed after a cut. That is what the whole two-by-two is built on, it is what the free rows are free in, and it is what falls by a hundred and fifty-four degrees at one rise of the golden 5/8 rung.

The natural next question asks about the other end. If the first organ goes somewhere different, does the stem also finish somewhere different?

The question is well posed #

It is not a fussy one either. A transition in a placement that washed out over the next few hundred organs would be a transient, and one that persisted to the end of the run would be a change in the arrangement the stem settles into.

Those are different things and the collection has the number that would separate them. Every cut run records the divergence it finishes at, and it has done since the design was written.

And it has no answer here #

Across the twenty-nine swept positions, the doubled cut’s run ends at 65.25 degrees, somewhere between 137.25 and 137.82, at 182.84, or at 209.56. Four values, and it moves between them at rises one part in a thousand apart.

Inside the nine-rise fine sweep alone it changes three times, over a stretch where the walls, the block, the intact stem’s settled divergence and the rigid lags are all held.

That is the same stretch on which five quantities are identical either side of the transition. A sixth quantity that changes three times inside it is not evidence about the transition; it is evidence about itself.

Four values, and they are not four arbitrary values #

Before the endpoint is called a coin it is worth subtracting the intact divergence from each of the four, because the differences are not a scatter.

The stem’s own settled divergence is about 137.5°. Against that, the four endpoints sit at

• 137.25 to 137.82 — no move at all;
• 209.56, which is 137.5 + 72.1;
• 65.25, which is 137.5 − 72.3;
• 182.84, which is 137.5 + 45.3.

Seventy-two degrees is 360 ÷ 5. Forty-five degrees is 360 ÷ 8. This is a stem counted 5 and 8, and the four endpoints are the intact divergence, the intact divergence plus or minus one turn over the smaller counted number, and the intact divergence plus one turn over the larger — every one of them within a third of a degree, which is one or two steps of the azimuth grid.

Those are exactly the destinations the slip law predicts. A wrecked stem keeps one family rigid and its divergence moves by a whole number of turns over that family’s period, so a 5/8 stem can end at d, at d ± 72°, at d ± 45°, and nowhere else. All four observed values are on that list and no observed value is off it.

So the endpoint is not a coin. It is a choice among a short list, and the list is one this collection has already derived from a different measurement on different stems. What is unpredictable is which member of the list a given rise picks; what is entirely predictable is the list.

That distinction is the whole of it and it changes what the quantity is worth. A number that lands anywhere would be evidence of nothing. A number that lands on one of four computable values, moving between them as the rise moves by a thousandth, is the short-list result confirmed on a design built for something else — and it says the run’s destination is quantised even where its choice of destination is not stable.

Which also sharpens what the instability is about. It is not that a wrecked run forgets where it is going; it is that two rises a thousandth apart send it to different members of one small set, which is a statement about how finely the basins of those four destinations are interleaved.

Which is not a defect #

A run that has been wrecked has no reason to settle anywhere in particular. The doubled cut wrecks the stem at twenty-five of the twenty-nine positions — the two that recover both sit above 87 per cent of the rung, far below the transition — so the number being read is a wrecked run’s endpoint.

A stem that does not settle onto its branch is a known state in this collection and it is measured rather than excluded. This is the same state arrived at by damage rather than by the rise, and the floor it sits under is a wall rather than a budget — a longer run does not rescue it.

The two halves of one run behave differently #

That is the finding worth keeping. The first organ’s displacement is reproducible, sits exactly on the azimuth grid, is flat across half a rung, and moves once in twenty-nine positions.

The end of the run is none of those things. Same runs, same code, same tolerances, and one quantity is a measurement while the other is a coin.

Nothing about the code distinguishes them. Both are read off the same object at the same moment, and the difference is entirely in how many placements stand between the cut and the number.

Why the first organ is stable #

Because it is decided by one thing. The organ after a cut is placed at the minimum of a sum over its neighbours across 1,536 candidate azimuths, and the neighbourhood it is placed against is the stem with two organs missing.

Two rises a thousandth apart give neighbourhoods that differ by a fraction of a degree in every hop, so the minimum is in the same place and the placement is the same grid step. That is why 163.59 degrees repeats exactly across four positions spanning half a rung.

And why the end is not #

Because it is decided by three hundred of those placements in sequence, each against a neighbourhood the previous ones made. A difference of a grid step early on is a different neighbourhood for the next organ, and so on.

That is the ordinary behaviour of a sequential rule and it is why the ladder itself is measured on the intact stem rather than on damaged ones. A wrecked run is a run whose state has left the basin the intact run sits in.

The four values are not arbitrary #

They are 65.25, about 137.3, 182.84 and 209.56 degrees. Two of those are recognisable: 137.3 is near the golden angle the intact stem settles to, and 209.56 folds to 150.4, which is a destination the settling table reaches at several rises.

So a wrecked run is not going anywhere; it is going to one of a handful of places. That is consistent with the picture the settling work gives and it is not a measurement of anything on this rung, because which of the four it picks is not stable.

What that rules out #

The reading the transition changes where the stem finishes is unavailable, and so is the reading the transition does not change where the stem finishes. Both would be statements about a quantity that changes at rises where nothing else does.

That is a stronger negative than it was looked for and did not change, and it is worth distinguishing. A quantity with no reproducibility supports no claim in either direction.

What it does to the slot table #

Nothing, because the slot table never used the endpoint. Its four cells are displacements of first organs and its interaction is arithmetic on those.

The reason to report this is that the endpoint is in the data — every cell carries it — and a reader with the table in front of them would reach for it. The honest label on that column is that it is not a measurement on this rung.

And to the rest of the thread #

Also nothing, but the check is worth stating. The ablation census reads a wrecked stem’s displacement profile, folded onto the lag it kept, over a window at the top of the run. That is a different quantity from a single settled divergence.

A profile is a difference between two runs organ by organ, and it can fall into a stable pattern even where neither run settles to a fixed angle. The census’s own periodicity classification is a reading of that pattern, and its five non-periodic rows are exactly the ones where it does not.

Which is a distinction worth having #

A run that never settles can still damage its control in a repeatable way. That sounds paradoxical and it is not: the difference between two runs of the same rule can be steady while both of them wander together.

Whether that is what happens here is not established. It is the reading that reconciles a stable displacement profile with an unstable endpoint, and testing it would mean looking at the control’s endpoint as well.

The control’s endpoint #

That check is cheap and it has not been run. The control shares the history below the hole and is continued without a removal, so it is an intact run and should settle to the rung’s own divergence — 137.438 degrees on this stretch.

If it does at every position, then the instability belongs entirely to the cut run and the comparison is between a settled control and an unsettled cut. If it does not, the reading is about the rule at these rises rather than about the damage.

Two rises apart is enough to change it #

The strongest form of the instability is worth stating with the numbers. At rises of 0.00997 and 0.00996 the endpoint is 65.25 degrees; at 0.00995 it is 137.25; at 0.00994 it is 65.25 again; at 0.00993 it is 137.26.

Those four rises span four parts in a hundred thousand. Everything else about them is identical to the digits this collection quotes.

What a stable endpoint would have bought #

It is worth saying what was hoped for. If the endpoint had been one value above the transition and another below it, the transition would have been a change in the arrangement the stem ends up in — a much larger claim than a change in one placement.

If it had been one value throughout, the transition would have been a transient: a placement that differs and washes out. Either would have been informative and the measurement supports neither.

The honest form of the result #

On the golden 5/8 rung, the doubled cut’s run does not have a reproducible endpoint, so whether the transition persists to the end of the run cannot be asked here.

That sentence is longer than either of the two it replaces and it is the one the data supports. Reporting a quantity’s instability is a result; reporting a number from it would not have been.

Where the question could be asked #

On a rung where the doubled cut does not wreck. Two of the twenty-nine positions recover — at 88 and 91 per cent of the rung — and on those the run returns to what the control does, so the endpoint is the rung’s own divergence by construction.

That is not helpful, because a recovering cut has no transition to be on either side of. A rung where the pair removal wrecks below a transition and recovers above it would be, and whether one exists is not known.

What this is an instance of #

A quantity that is recorded, looks like a measurement, and is not. The collection has found several: a block that was the grid rounding a constant, an onset that was a run length, a residual that was a window.

This one is the least subtle of them, because the instability is visible the moment the quantity is plotted against anything. It went unreported for as long as nobody plotted it.

How many placements stand between the two #

That is the quantity worth naming, because it is what separates the stable half of the run from the unstable half. The first organ after the cut is one placement downstream of a neighbourhood that differs from its neighbour rise’s by a fraction of a degree.

The endpoint is three hundred placements downstream, each against a neighbourhood the previous ones built. A rule of that shape amplifies a grid step into a different destination, and the number of steps it takes to do so is what nobody has measured.

Measuring it would mean asking, at each rise, how far along the run the two neighbouring rises’ placements first diverge. That is arithmetic on runs already grown and it would turn this instability from an obstacle into a reading.

Which would be the interesting measurement #

If two rises a thousandth apart stay together for two hundred organs and then separate, that is a horizon and it has a length. If they separate at the tenth organ, the endpoint is essentially unrelated to the rise and the four values are a lottery over four basins.

Both are testable and neither has been tested. It is the best question this reading leaves and it costs nothing beyond what is already grown.

What this says about reading a table #

The practical lesson is about columns rather than about rungs. A design records what it happens to record, and a column that the design does not use has never been checked for being a measurement at all.

The slot table has such a column and now it has a label. Any table in this collection may have others, and the cheapest way to find out is to plot a column against the thing the table is swept over and see whether it does anything.

Two rungs, and only one has been asked #

Everything here is about the golden 5/8 rung, because that is the rung the transition is on. Whether a wrecked run’s endpoint is unstable on every rung or on this one is not known, and it is the sort of thing that could easily differ.

The slot table holds twenty-four rows across eight rungs and every one of them carries an endpoint. Plotting all twenty-four against the rise they sit at would take a minute and would say whether this is a property of wrecked runs or of a stretch of one rung.

That is the cheapest of the three open questions this reading leaves and it is the one most likely to change how the result is stated.

What a reader should carry #

That the slot design measures one organ, and that the other end of the same run is not a measurement on this rung — four values, changing between rises a thousandth apart, three times inside a nine-rise sweep.

And that this is why the transition is described as a change in a placement rather than as a change in the stem. The narrower description is not modesty; it is the only one the data carries.

What the picture at the top shows #

The divergence the doubled cut’s run finishes at, rise by rise, coarse on the left, with a horizontal gridline at each of the values it takes.

The line jumps between two of those values several times in the left half of the plot, and the vertical rule — the transition, where the first organ’s displacement falls by a hundred and fifty-four degrees — passes through a stretch where the endpoint is doing nothing in particular.

The one line #

The doubled cut’s run ends at 65.25, about 137.3, 182.84 or 209.56 degrees across twenty-nine rises of the golden 5/8 rung, changing between values at rises one part in a thousand apart and three times inside the nine-rise fine sweep.

So does the stem finish somewhere different? has no answer here in either direction, and the transition stays what it is: a change in where one organ goes.