A list that was a rounding

Three destinations are reached at falloff exponents four and five and at neither of the shallower ones: 42.3°, 47.9° and 148.1°. That list has been quoted through two rounds.

It depends entirely on how the destinations are grouped, and there are three defensible groupings. They give three, thirteen and four.

None of the three is a mistake and none is obviously the right one. What is a mistake is quoting a list without saying which grouping produced it, and that is what the thread has been doing — the number three has appeared in two rounds of essays with no tolerance attached to it, and the tolerance is doing all the work.

Thirteen, at a tenth of a degree #

That is the resolution the settling table prints. Read that way, the destinations only a steep exponent reaches are 42.3°, 47.9°, 78.6°, 78.9°, 79.2°, 99.0°, 99.6°, 99.7°, 136.6°, 138.8°, 139.0°, 148.1° and 151.5°.

Ten of those thirteen sit within a fraction of a degree of a destination a shallow exponent also reaches: 99.0° against 99.1°, 139.0° against 139.1°, 136.6° against 136.8°. They are the same arrangement measured on runs grown at slightly different rises.

Why they differ at all #

Because a settled divergence is not a constant of an arrangement. It slides across a rung: a stem on the golden ladder settles at 136.8° at one rise and 139.1° at another, and both count as 3/5.

So two runs reaching “the same arrangement” at different rises report different angles, and a grouping at a tenth of a degree separates them. The list of thirteen is mostly a list of rises.

The slide is not incidental to this subject; it is most of what a rung is. A stem holds one counted pair across a stretch of rise and its divergence moves by two or three degrees inside that stretch, so any grouping of divergences finer than the slide will split a single rung into several entries.

Three, at two grid steps #

Half a degree is two steps of the grid the azimuths are chosen from, which is 1,536 samples of the circle and a quarter of a degree a step. It is the smallest gap at which two settled values can be told apart at all.

Grouped that way the fifteen destinations of the table are fifteen, and the steep-only ones are three. It is a stated tolerance rather than a rounding, and it is the one the counting essays use.

Which of the two angle groupings is right #

Neither, and that is the point of the essay rather than a dodge. Half a degree is defensible because it is the instrument’s own floor; a tenth is defensible because the settled value is a mean over sixty divergences and a mean can resolve finer than one step of the grid it is built from.

Both are answers to “how far apart must two settled values be before they are two values”, and that question has no answer independent of what the values are for. For “has the rule found somewhere new”, two angles a tenth of a degree apart on the same arrangement are not somewhere new. For “does the settled divergence slide across a rung”, they are the whole measurement.

Which is still indexed by an angle #

And that is the deeper problem. A destination is a divergence; an arrangement is a counted pair; and the two do not correspond. Three of the fifteen destinations return more than one pair, and several pairs occur at more than one destination.

So a list of destinations only a steep rule reaches is not a list of arrangements only a steep rule reaches, however carefully the destinations are grouped.

Four, indexed by arrangement #

Group the 117 settled runs by the pair the counter returns and the table has twenty arrangements. Four of them are reached at exponents four and five and at neither shallow one: 8/9, 8/15, 9/16 and 2/5.

Three of those are the three destinations. The fourth, 9/16, does not appear on the angle list at all.

Why the angle reading hides it #

9/16 is reached at 157.7°, and 157.7° is also reached at exponent two — with 7/9. Two runs, one angle to within a tenth of a degree, two different arrangements: one found by a shallow rule and one only by a steep one.

So the angle says “a shallow exponent gets here too” and the count says “not to this”. Both are true and they are answers to different questions, and only one of them is the question the thread was asking.

The pair of runs is worth looking at directly. At exponent two a stem at one rise settles on 157.6° and counts 7/9; at exponents four and five stems at other rises settle on 157.7° and count 9/16. Seven and nine and sixteen are consecutive terms of one sequence, so this is not two arrangements that happen to share an angle — it is two adjacent rungs of one ladder, whose divergences differ by a tenth of a degree because the ladder’s angle barely moves between them.

A fourth grouping, which needs no tolerance at all #

Three groupings and no way to choose between them is an uncomfortable place to leave a list, and there is a fourth that the thread has since acquired and that has a property none of the three has: it needs no tolerance.

Every pair the counter returns is two terms of an additive sequence, and the table’s pairs sort into ten such sequences, each with a limit divergence of its own. A sequence is a ladder and a pair is a rung of it. So two runs on one sequence are the same attractor read at different rises, and grouping by sequence asks the question the thread was actually asking — does a steep rule reach a ladder a shallow one does not? — rather than the question about rungs, which is a question about which rises were swept.

That grouping is an integer fact. Two pairs are on one sequence or they are not; nothing has to be within anything of anything, and no census resolution decides it.

And it gives two #

Apply it to the four arrangements the pair grouping calls steep-only and two of them fall away.

2/5 is on 2, 5, 7, 12, 19, which is one of the two largest sequences in the table and is reached at all four exponents — at the 5/7 and 7/12 rungs by the shallow ones. A shallow rule finds that ladder; it simply does not find that rung of it.

9/16 is on 2, 7, 9, 16, 25, and the same table has a shallow-rule run at 7/9, which is the rung below. Again the ladder is reached and the rung is not — and this is the same pair of runs the section above uses to argue that the angle grouping hides an arrangement. Under the sequence grouping it hides nothing, because the two runs are two rungs of one ladder and that is what their nearly-equal divergences were saying all along.

8/9 is on 1, 8, 9, 17, 26 and 8/15 is on 1, 7, 8, 15, 23, and neither sequence is reached at any rung by a shallow exponent.

So the sequence grouping gives two: two ladders that only a steep falloff finds. That is a smaller claim than three or four or thirteen and a much more robust one, and it is the claim to quote — with the note that a rung a shallow rule misses is a statement about the rises swept and not about the rule.

Which is a small and clean correction #

The list of steep-only findings is four rather than three, and the fourth is on the same sequence as one of the others — 2, 7, 9, 16 supplies both 7/9 and 9/16, so the steep rule reaches a finer rung of a sequence the shallow rule already reaches a coarser rung of.

That is a shape worth naming, because it is the same shape 148.1° turned out to have: a coarser rung of a sequence the table already had. Two of the four steep-only arrangements are new rungs on old sequences and two are new sequences.

And the question was never asked in the other direction #

If a steeper falloff reaches arrangements a shallow one does not, the mirror question is whether a shallow falloff reaches arrangements a steep one does not. Nobody asked it, and the answer is not zero: 9/14 and 4/9 are reached at exponents two or three and never at four or five.

Two each way. The asymmetry the thread has been describing — that a steeper rule finds new destinations — is, at the level of arrangements, no asymmetry at all.

And the two shallow-only arrangements are as well-evidenced as the weaker two of the steep-only ones: 9/14 at two runs and 4/9 at one, against 9/16 at two and 2/5 at one. So the four-against-two split is really one well-evidenced steep-only arrangement and five findings of one or two runs each, three of which point one way and two the other.

How that happened #

Because the sweep was run to test whether a steeper falloff moves the wall, and the destinations were a by-product noticed while reading the table. A by-product gets described in whatever direction it was noticed in.

The mirror question needed no extra runs at all — it is a filter over a table already computed — and it took a round to be asked because nobody was looking that way.

Both directions are weakly evidenced #

Four steep-only arrangements and two shallow-only ones, and the run counts are 4, 2, 2, 1, 2 and 1. Every one of them is a handful of runs from nine starting angles a cell.

So the honest statement is that the exponent moves which arrangements appear in a sparse sample, in both directions, by about the amount a sparse sample would move anyway. Whether the effect is real needs twenty starting angles rather than nine, which is 640 runs against 288.

The same lesson as the shares #

The settling shares were four columns of one column: a spread of 0.056 against a binomial error of 0.058, reported as a difference between exponents. The response was to stop reading the share and read the clock and the map instead.

This is the same lesson one level down. A destination is a measured angle with a resolution; an arrangement is a count with a run behind it; and a list of either is a list at a stated tolerance from a stated sample. Neither is a fact about the rule until the tolerance and the sample are quoted with it.

What a run count adds #

Every list in this essay is more honest with a run count beside it, and none of them had one. Three destinations reads as three findings; three destinations at four, two and one runs reads as one finding and two hints.

That is a change to how the table prints rather than to what it computes, and it is the same correction the ablation thread made about run lengths — report the instrument beside the number, because a constant that appears in every row stops being visible.

What the three groupings agree about #

More than they disagree about. All three say the steep exponents reach 42.3° and 47.9° and the shallow ones do not, and all three say those two are countable and off both ladders.

The disagreements are entirely at the margins: whether near-duplicate angles are one destination or several, and whether an arrangement hidden behind a shared angle counts. That is the usual shape of a resolution problem and it is worth saying that the core is not in dispute.

What to quote #

The arrangement list, with its run counts. Four arrangements reached only at the steep exponents, at 4, 2, 2 and 1 runs; two reached only at the shallow ones, at 2 and 1. Twenty arrangements in all, from 117 settled runs.

That is longer than “three destinations only a steep rule reaches” and it is the sentence that does not change when somebody re-groups the table — which is the test a quotable sentence has to pass, and the one a summary statistic that outlives its scope always fails.

What it does to the wall #

Nothing, which is worth saying because the wall is the thread’s substantial result and the destinations are the ornament on it. Below a certain rise no starting angle reaches any lattice at all, at any of the four exponents, and that is a statement about runs that fail rather than about which arrangement the successes land on.

Every grouping in this essay leaves it untouched. A regrouping that changed a negative result about failures would be a much more serious matter than one that changes a list of successes, and this is not one.

The general form #

A list of things one condition produces and another does not is a list of set differences, and a set difference is exquisitely sensitive to how the members are identified. Group them finely and the difference grows; group them coarsely and it shrinks; change what the members are and it changes shape.

There is no neutral grouping. What there is instead is the obligation to say which one is in use, and to check that the finding survives the others — which here it mostly does, and where it does not is stated.

It is worth noting how little this cost. All three groupings run over the same 117 counted runs, which were computed once; re-grouping them is a second and produces a different answer. A finding that changes when the grouping does, at that price, has no excuse for being quoted at one grouping only.

What is not claimed #

That a tenth of a degree is a wrong resolution to print. It is the right resolution to print a measured angle at; it is the wrong one to group by, and printing and grouping had not been separated.

Nor that twenty arrangements is the table’s true number. A denser sweep of starting angles would very likely find more, and a different counting window might merge or split some of these.

The arrangement grouping has its own tolerance #

It is worth being even-handed. Grouping by counted pair looks tolerance-free — a pair is two whole numbers — and it is not: the counter reads a window of two hundred organs and a count depends on where it is taken. Two runs on the same arrangement can return different pairs if the window catches different parts of the pattern.

There is direct evidence of that in the table. At 79.2° the counter returns 4/5, 4/9 and 5/9 across nineteen runs, and 4/9 skips a term of the sequence the other two sit on — which is what a count taken at the wrong place produces. So the arrangement grouping has a soft edge too, in a different place from the angle grouping’s.

Neither grouping is clean. What can be said is that they fail differently, and a finding surviving both is a finding that has been checked.

The one line #

The list of destinations only a steep falloff reaches is three at half a degree, thirteen at a tenth of a degree, and — indexed by the arrangement a counter returns rather than by the angle — four, one of which the angle grouping hides behind a divergence a shallow exponent also reaches.

And the mirror question, never asked, has an answer: two arrangements are reached only by the shallow exponents, so at the level of arrangements the exponent’s effect is symmetric.