The column nobody read

The slot design removes the smaller of a lattice’s two contact neighbours, the larger, or both, and measures how far the first organ placed afterwards moves. It runs on thirty lattices across all eight rungs of the ladder, and every one of its cells also carries where the run finished — the divergence the stem was still running at three hundred organs later.

That column has been read once, on one rung, and reported as unusable. Reading it everywhere costs nothing: the numbers are already computed and sitting in the cache.

What was said about it before #

That across twenty-seven positions of the golden 5/8 rung the doubled cut’s endpoint takes four values, changes between rises one part in a thousand apart, and changes three times inside a nine-rise fine sweep — where the walls, the block, the intact stem’s divergence and the rigid lags are all held.

The conclusion was that a wrecked run’s final divergence is not a measurement on that rung, in either direction, and the file that said it was careful to say on that rung because it had nothing to compare against.

This is the comparison. Thirty lattices by three cut cells is ninety readings and every one is a cache hit.

The control #

Twenty-seven of the ninety cells recover — the pattern above the hole returns to what it was — and every one of those finishes at the divergence its own intact stem settles to, to within 0.03 degrees.

That is not quite trivial. The azimuth grid is 1,536 samples of the circle, a quarter of a degree a step, so 0.03 degrees is an eighth of one step: the recovered runs land exactly where the control lands.

So the endpoint column reads the run rather than the reading. Whatever the wrecked runs are doing, the instrument is not adding it.

The wrecked cells #

Sixty-three of them, at 43 distinct endpoints, which gather into seventeen groups no more than two degrees wide.

So the runs do not scatter. Sixty-three readings landing on seventeen places is a discrete answer, not a continuum, and it is the first thing the one-rung reading could not see: on that rung the four values looked like four arbitrary numbers.

The largest group holds twelve runs, all at a half turn. The next holds nine, at about 209 degrees.

Two rungs where it is exact #

On the Lucas 1/3 rung every wrecked run finishes at exactly 180.000 degrees — six cells, three lattices, one value. On the golden 2/3 rung the five wrecked cells finish at 180.00 or 179.97, a spread of three hundredths of a degree.

That is the answer to the question the round asked. A property of wrecked runs would put every rung near the golden 5/8’s spread; a stretch of the golden 5/8 would put every other rung near nothing. Two rungs are near nothing and six are not.

So the endpoint is neither always unstable nor unstable in one place. It is a quantity that is sharp at the coarse end of the ladder and wanders at the fine end.

And six where it is not #

The spreads, in degrees: 0.0 on the Lucas 1/3, 0.0 on the golden 2/3, then 45 on the golden 8/13, 58 on the Lucas 3/4, 73 on the golden 3/5, 103 on the Lucas 7/11, 129 on the Lucas 4/7 and 215 on the golden 5/8.

Six rungs spread by 45 degrees or more, which is far too much for the endpoint to be a property of the lattice. And the largest is 1.7 times the next largest, so the golden 5/8 is not merely at the top of a smooth range.

The rung the endpoint was read on, and called unusable on, is the worst of the eight by a margin.

Which is worth saying plainly #

The round that produced the caveat chose the golden 5/8 rung for reasons that had nothing to do with endpoints: it was one of two rungs where a slot loses its second wall, and the only one where that happens somewhere inside rather than everywhere.

It then read the endpoint column there, found it took four values across a fine sweep, and concluded that a wrecked run’s endpoint is not a stable quantity.

That conclusion is right about that rung and overstated as a general claim. It also sat beside a much sharper positive result from the same sweep — the fourth cell’s displacement changing by a hundred and seventy degrees in one step of the grid — so the caveat was doing real work there, and the work it was doing was local. The rung it was drawn on is the one where the quantity is least stable, and nothing in that round could have known it.

The pattern in the spreads #

Roughly, the coarser the rung the sharper the endpoint. The two exact rungs are the two coarsest on their branches — the Lucas 1/3 at rises around 0.067 and the golden 2/3 around 0.057 — and the widest spreads are on rungs at rises three to ten times finer.

That is a correlation over eight points and it is not perfect: the golden 8/13, which is the finest golden rung, spreads by 45 degrees where the golden 5/8 above it spreads by 215.

What it is consistent with is the general behaviour of these stems. At coarse rises the neighbourhood is short and a disturbed pattern has few ways to go; at fine rises the front is deeper and there are more arrangements available.

What a half turn is #

A divergence of exactly 180 degrees is two files of organs, alternating, with no spiral at all. It is a degenerate arrangement rather than a lattice with a large counted pair.

Twelve of the sixty-three wrecked runs finish there, and every one is on the two coarsest rungs plus one cell of the Lucas 3/4. That is the largest single destination in the table by a factor of four over the next.

Which makes the two exact rungs exact for a specific reason: they do not have one stable endpoint, they collapse to the same degenerate one.

The value is exact #

Eight of the twelve read 180.000 and the rest sit within 0.35 degrees. On a grid of 1,536 samples, 180 degrees is sample 768 exactly, so the arrangement is one the grid can represent without rounding.

That matters because it rules out an accident of the grid. A run drifting towards 180 and being rounded there would produce values scattered within a grid step; these are the same value.

So a wrecked run at the coarse end of the ladder does not wander to a new arrangement. It falls into the one arrangement that has no spiral in it.

What the column is not #

A measurement of the cut. The whole slot table is built on how far the first organ placed after the cut moves, and that quantity is reproducible, sits exactly on the azimuth grid, and changes once across twenty-seven positions of a rung where the endpoint changes nineteen times.

The two halves of the same run behave differently and the difference is the point, and it is the same separation the ablation thread makes between a displacement and a destination. One organ’s placement is decided by the cut; three hundred organs later the run has been decided by everything since.

So the endpoint is a property of a wrecked run’s whole history, and the displacement is a property of the cut. Reading the first as though it were the second is the mistake this essay is about.

The one rung that has been swept finely #

The golden 5/8, at twenty-seven positions, is the only rung where the endpoint has been read at more than three rises. It gives eighteen distinct endpoints and nineteen changes between them.

The slot table reads three positions a rung, so it sees at most three values a cell and between three and fourteen a rung. That is a much coarser view and it is the one available everywhere.

Nothing here says what the other seven rungs would do under a fine sweep. The two exact rungs might stay exact or might turn out to be exact only at three positions, and half an hour a rung would say.

What the three cells say #

The smaller wall removed, the larger removed, and both. Twenty-four smaller-wall cells wreck and land on 22 distinct endpoints; ten larger-wall cells wreck and land on 8; twenty-nine doubled cells wreck and land on 26.

So no cell is systematically sharper than another. The doubled cut, which is the most violent of the three, does not produce a wider spread of endpoints than the single removals — it produces more wrecked runs.

That is worth knowing because the doubled cut is the one the original caveat was read on, and it is not the reason the endpoints wandered.

What can now be said about a wrecked run #

That it finishes somewhere from a small list rather than anywhere, on every rung. That the list is one item long on two rungs and up to seven items long on others. That the items are sharp: seventeen groups no wider than two degrees hold all sixty-three readings.

And that at the coarse end of the ladder the item is a half turn — the arrangement with no spiral — while at the fine end the items are places the settling table already reaches, which is a separate finding and the more surprising one.

None of that was available from one rung, and none of it needed a new stem.

What the caveat should say now #

Not that a wrecked run’s endpoint is unstable, but that its spread is a property of the rung — nothing on two of them, tens of degrees on six, and 215 degrees on the one the caveat was written from.

And that on any rung the endpoint takes a small number of sharp values rather than varying continuously, so unstable is the wrong word for it in a second way: it is multi-valued over a known list.

The original file’s conclusion about its own rung stands unchanged. What changes is the generality it was stated with.

Why this cost nothing #

The slot design’s results are banked against the code that produced them, so asking for a column nobody had read is a file read rather than a computation. Thirty lattices at three cut cells and one control each is 120 grown stems, and every one of them was grown when the slot table was built.

That is the argument for a sweep returning everything it measured rather than the summary the question needed. The table was built to compare three removals; the endpoint came along because the run had to be grown to the end anyway, and it sat in the cache for two rounds.

The same is true of most of what this round reports. Four of the six leavings the previous round named were answered by re-reading tables already on disk, and this is the cheapest of the four.

What the thirty lattices are #

Three rises on each of the ladder’s eight rungs, at 15, 50 and 85 per cent of the rung’s span in the logarithm of the rise — plus the six lattices an earlier round ran the design on, kept so that the twenty-four are a superset of the six rather than a replacement.

The Lucas 1/3 rung is short enough that its three positions round to nearly the same rise, so it contributes three lattices that sit within four per cent of each other. That is a fact about the rung and is left in the table rather than padded out.

So the eight rungs are not equally sampled in the rise: a rung spanning a factor of 1.7 gets three positions and so does one spanning a factor of 1.08. Any statement about a rung’s spread is a statement over its own three positions.

What a spread of zero is worth over three positions #

Less than it sounds, and the arithmetic is worth doing. Two rungs give one endpoint value each, over six and five wrecked cells respectively.

If a rung’s endpoints were drawn at random from that rung’s own list of available places, six draws landing on one place would be strong evidence the list has one item. If they are drawn from the same place for a structural reason — because a half turn is where a disturbed coarse stem goes — then the six draws are one fact repeated.

The second reading is the right one here, because all twelve half-turn endpoints are on the coarse rungs and none is anywhere else. So the two exact rungs are exact because their runs collapse to a degenerate arrangement, not because their endpoints are well determined.

Three positions against twenty-seven #

The golden 5/8 rung appears twice in this table: as three positions in the slot design, and as twenty-seven in the fine sweep of the same rung.

Its three positions give six wrecked cells at six values spanning 215 degrees. Its twenty-seven positions give eighteen values and nineteen changes. So the coarse sampling already sees most of the spread, and what the fine sweep adds is the changes rather than the range.

That is a useful calibration for the other seven rungs. Three positions is enough to see whether a rung’s endpoint moves at all; it is not enough to see how often.

The one thing three positions cannot show #

Whether the exact rungs stay exact. Six cells on one value is consistent with a rung whose endpoint never moves and with a rung whose endpoint moves at rises nobody sampled.

Half an hour a rung would settle it, and the prediction has a direction: on the evidence of the ladder’s other rungs, and of every instrument setting this collection has varied, the expected answer is that a finer sweep finds more values.

That is the shape this site’s predictions take when they can be stated at all — a direction with a cost attached — and it is worth writing down before anyone runs it.

What a column read late is worth #

Everything it says was computed two rounds ago and sat unread, which is the ordinary return on a sweep that returns what it measured rather than what its question needed.

The slot design was built to compare three removals against a control. Its measurement is the displacement of the first organ placed after the cut; the endpoint came along because the run had to be grown to the end anyway. Nothing was added to the design to make this reading possible and nothing had to be recomputed to take it.

That is worth stating as a design principle rather than as luck. A sweep that returns only the summary its question needs is a sweep whose other readings are gone, and the cost of keeping them is a few numbers per row in a cache that is already on disk.

What the column still cannot say #

Whether any of these endpoints is where the run would stay. Three hundred organs is short by the settling table’s standards, and a run’s last divergence is not a destination.

The strongest indirect evidence is the half turns: eight of twelve read the grid’s exact 180 degrees, and a run passing through a value reads it exactly only by coincidence. The weakest is the three that finish where their own intact stem settles, where a run crossing its own divergence at organ three hundred would read identically.

Sixty-three long runs would settle it. That is a few minutes and it is the obvious next reading, and it is not taken here because this round’s question was whether the spread is a property of wrecking or of one rung.

What is claimed #

That the slot table’s endpoint column, read across all thirty lattices and three cut cells, gives sixty-three wrecked readings at 43 distinct values in seventeen groups no wider than two degrees.

That the twenty-seven recovering cells finish at their own settled divergence to within an eighth of a grid step, so the column reads the run rather than the reading.

That the spread over a rung is zero on two rungs and 45 to 215 degrees on six, with the widest on the rung the quantity was previously read on — which was chosen for a reason unrelated to endpoints.